{"id":2961,"date":"2018-12-11T13:46:46","date_gmt":"2018-12-11T18:46:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/dividing-polynomials\/"},"modified":"2018-12-11T13:46:46","modified_gmt":"2018-12-11T18:46:46","slug":"dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/dividing-polynomials\/","title":{"raw":"Dividing Polynomials","rendered":"Dividing Polynomials"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Dividing monomials<\/li><li>Dividing a polynomial by a monomial<\/li><li>Dividing polynomials using long division<\/li><li>Dividing polynomials using synthetic division<\/li><li>Dividing polynomial functions<\/li><li>Use the remainder and factor theorems<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835337220\" class=\"be-prepared\"><p id=\"fs-id1167834539128\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167835173019\" type=\"1\"><li>Add: \\(\\frac{3}{d}+\\frac{x}{d}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(\\frac{30x{y}^{3}}{5xy}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Combine like terms: \\(8{a}^{2}+12a+1+3{a}^{2}-5a+4.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836652573\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831919748\"><h3 data-type=\"title\">Dividing Monomials<\/h3><p id=\"fs-id1167831890636\">We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we\u2019ll use these properties to divide monomials and polynomials.<\/p><div data-type=\"example\" id=\"fs-id1167835381198\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835357231\"><p id=\"fs-id1167835354274\">Find the quotient: \\(54{a}^{2}{b}^{3}\u00f7\\left(-6a{b}^{5}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826783568\"><p id=\"fs-id1167831835715\">When we divide monomials with more than one variable, we write one fraction for each variable.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\hfill 54{a}^{2}{b}^{3}\u00f7\\left(-6a{b}^{5}\\right)\\hfill \\\\ \\text{Rewrite as a fraction.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{54{a}^{2}{b}^{3}}{-6a{b}^{5}}\\hfill \\\\ \\text{Use fraction multiplication.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{54}{-6}\u00b7\\frac{{a}^{2}}{a}\u00b7\\frac{{b}^{3}}{{b}^{5}}\\hfill \\\\ \\text{Simplify and use the Quotient Property.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -9\u00b7a\u00b7\\frac{1}{{b}^{2}}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -\\frac{9a}{{b}^{2}}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835343714\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835349346\"><div data-type=\"problem\" id=\"fs-id1167834396049\"><p id=\"fs-id1167834429517\">Find the quotient: \\(-72{a}^{7}{b}^{3}\u00f7\\left(8{a}^{12}{b}^{4}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832151093\"><p id=\"fs-id1167835304216\">\\(-\\frac{9}{{a}^{5}b}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834505374\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826938133\"><div data-type=\"problem\" id=\"fs-id1167834556655\"><p id=\"fs-id1167835297451\">Find the quotient: \\(-63{c}^{8}{d}^{3}\u00f7\\left(7{c}^{12}{d}^{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834178752\"><p id=\"fs-id1167826778816\">\\(\\frac{-9d}{{c}^{4}}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167834527744\">Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.<\/p><div data-type=\"example\" id=\"fs-id1167832059550\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835418160\">Find the quotient: \\(\\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835515459\"><p>Be very careful to simplify \\(\\frac{14}{21}\\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\\hfill \\\\ \\text{Simplify and use the Quotient Property.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{2{y}^{6}}{3{x}^{4}}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831884435\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830698259\"><div data-type=\"problem\" id=\"fs-id1167834397893\"><p id=\"fs-id1167835229142\">Find the quotient: \\(\\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835356175\">\\(\\frac{4{y}^{2}}{7{x}^{4}}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835331022\"><div data-type=\"problem\"><p id=\"fs-id1167826782319\">Find the quotient: \\(\\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}.\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\frac{5}{8{m}^{5}{n}^{3}}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834473198\"><h3 data-type=\"title\">Divide a Polynomial by a Monomial<\/h3><p id=\"fs-id1167826803000\">Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.<\/p><p id=\"fs-id1167835514437\">The method we\u2019ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we\u2019ll start with an example to review fraction addition. The sum \\(\\frac{y}{5}+\\frac{2}{5}\\) simplifies to \\(\\frac{y+2}{5}.\\)<\/p><p id=\"fs-id1167827940168\">Now we will do this in reverse to split a single fraction into separate fractions. For example, \\(\\frac{y+2}{5}\\) can be written \\(\\frac{y}{5}+\\frac{2}{5}.\\)<\/p><p id=\"fs-id1167831892828\">This is the \u201creverse\u201d of fraction addition and it states that if <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where \\(c\\ne 0,\\) then \\(\\frac{a+b}{c}=\\frac{a}{c}+\\frac{b}{c}.\\) We will use this to divide polynomials by monomials.<\/p><div data-type=\"note\" id=\"fs-id1167830914794\"><div data-type=\"title\">Division of a Polynomial by a Monomial<\/div><p id=\"fs-id1167832058574\">To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167834226284\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832059371\"><div data-type=\"problem\" id=\"fs-id1167827958319\"><p id=\"fs-id1167826987582\">Find the quotient: \\(\\left(18{x}^{3}y-36x{y}^{2}\\right)\u00f7\\left(-3xy\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834536604\"><p id=\"fs-id1167835513598\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\left(18{x}^{3}y-36x{y}^{2}\\right)\u00f7\\left(-3xy\\right)\\hfill \\\\ \\text{Rewrite as a fraction.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{18{x}^{3}y-36x{y}^{2}}{-3xy}\\hfill \\\\ \\text{Divide each term by the divisor. Be careful with the signs!}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{18{x}^{3}y}{-3xy}-\\frac{36x{y}^{2}}{-3xy}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -6{x}^{2}+12y\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834536188\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835511504\"><p id=\"fs-id1167834299802\">Find the quotient: \\(\\left(32{a}^{2}b-16a{b}^{2}\\right)\u00f7\\left(-8ab\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826788581\"><p id=\"fs-id1167835319208\">\\(-4a+2b\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832134190\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826937939\"><div data-type=\"problem\" id=\"fs-id1167827966997\"><p id=\"fs-id1167835368739\">Find the quotient: \\(\\left(-48{a}^{8}{b}^{4}-36{a}^{6}{b}^{5}\\right)\u00f7\\left(-6{a}^{3}{b}^{3}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835354089\"><p id=\"fs-id1167830964251\">\\(8{a}^{5}b+6{a}^{3}{b}^{2}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832043205\"><h3 data-type=\"title\">Divide Polynomials Using Long Division<\/h3><p id=\"fs-id1167834394791\">Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let\u2019s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.<\/p><span data-type=\"media\" data-alt=\"This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.\"><\/span><p id=\"fs-id1167835479146\">We check division by multiplying the quotient by the divisor.<\/p><p id=\"fs-id1167832127974\">If we did the division correctly, the product should equal the dividend.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}35\u00b725\\hfill \\\\ 875\u2713\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835327385\">Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.<\/p><div data-type=\"example\" id=\"fs-id1167832051608\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831908410\"><div data-type=\"problem\" id=\"fs-id1167834473947\"><p id=\"fs-id1167834517539\">Find the quotient: \\(\\left({x}^{2}+9x+20\\right)\u00f7\\left(x+5\\right).\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167827957025\" class=\"unnumbered unstyled can-break\" summary=\"This image shows the process for polynomial long division using the example x squared plus 9 x plus 20 divided by x plus 5. Writing the problem as a long division problem we make sure the dividend is in standard form. The dividend of x squared plus 9 x plus 20 is put in the long division symbol. The divisor of x plus 5 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the x squared by x to get x. Then multiplying the x from the quotient with the entire divisor we get x squared plus 5 x. This is written below the x squared plus 9 x in the dividend and subtracted to get 4 x. The 20 in the dividend is dropped down to make this 4 x plus 20. To find the next term in the quotient we divide the 4 x by the x in the divisor to get 4. The quotient is now x plus 4. The 4 is multiplied to the entire divisor to get 4 x plus 20. This is subtracted from the 4 x plus 20 to give a remainder of 0. The process can be checked by multiplying the quotient of x plus 4 with the divisor of x plus 5 to get the dividend of x squared plus 9 x plus 20.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356577\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write it as a long division problem.<div data-type=\"newline\"><br><\/div>Be sure the dividend is in standard form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835343759\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide \\({x}^{2}\\) by \\(x.\\) It may help to ask yourself, \u201cWhat do I need<div data-type=\"newline\"><br><\/div>to multiply \\(x\\) by to get \\({x}^{2}\\)?\u201d<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830701188\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Put the answer, \\(x,\\) in the quotient over the \\(x\\) term.<div data-type=\"newline\"><br><\/div>Multiply \\(x\\) times \\(x+5.\\) Line up the like terms under the dividend.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\({x}^{2}+5x\\) from \\({x}^{2}+9x.\\)<div data-type=\"newline\"><br><\/div>You may find it easier to change the signs and then add.<div data-type=\"newline\"><br><\/div>Then bring down the last term, 20.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327295\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><div data-type=\"newline\"><br><\/div>Divide \\(4x\\) by \\(x.\\) It may help to ask yourself, \u201cWhat do I<div data-type=\"newline\"><br><\/div>need to multiply \\(x\\) by to get \\(4x\\)?\u201d<div data-type=\"newline\"><br><\/div>Put the answer, \\(4\\), in the quotient over the constant term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421672\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply 4 times \\(x+5.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254539\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\(4x+20\\) from \\(4x+20.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830757353\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div>Multiply the quotient by the divisor. \\(\\phantom{\\rule{1em}{0ex}}\\left(x+4\\right)\\left(x+5\\right)\\)<div data-type=\"newline\"><br><\/div>You should get the dividend. \\(\\phantom{\\rule{4em}{0ex}}{x}^{2}+9x+20\u2713\\)<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831887197\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831882387\"><div data-type=\"problem\" id=\"fs-id1167831872394\"><p id=\"fs-id1167834346516\">Find the quotient: \\(\\left({y}^{2}+10y+21\\right)\u00f7\\left(y+3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831891731\"><p id=\"fs-id1167834300936\">\\(y+7\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830924020\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834345678\"><div data-type=\"problem\" id=\"fs-id1167834053551\"><p id=\"fs-id1167834048906\">Find the quotient: \\(\\left({m}^{2}+9m+20\\right)\u00f7\\left(m+4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835321097\"><p>\\(m+5\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167828395729\">When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we\u2019ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.<\/p><p id=\"fs-id1167834423838\">Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be \\({x}^{4}-{x}^{2}+5x-6.\\) It is missing an \\({x}^{3}\\) term. We will add in \\(0{x}^{3}\\) as a placeholder.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835358491\"><p id=\"fs-id1167834182485\">Find the quotient: \\(\\left({x}^{4}-{x}^{2}+5x-6\\right)\u00f7\\left(x+2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831955744\"><p id=\"fs-id1167834395936\">Notice that there is no \\({x}^{3}\\) term in the dividend. We will add \\(0{x}^{3}\\) as a placeholder.<\/p><table id=\"fs-id1167835216851\" class=\"unnumbered unstyled can-break\" summary=\"This image shows the process for polynomial long division using the example x to the fourth power minus x squared plus 5 x minus 6 divided by x plus 2. Writing the problem as a long division problem we include placeholders for any missing terms in the dividend. The dividend of x to the fourth power plus 0 x cubed minus x squared plus 5 x minus 6 is put inside the long division symbol. The divisor of x plus 2 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the x to the fourth power by x to get x cubed. Then multiplying the x cubed from the quotient with the entire divisor we get x to the fourth power plus 2 x cubed. This is written below the x to the fourth power plus 0 x cubed in the dividend and subtracted to get negative 2 x cubed. Like terms are lined up in vertical columns. A note is made that it may be helpful to change the signs of the terms you are subtracting and add them. The negative x squared in the dividend is dropped down to make this 2 x cubed minus x squared. To find the next term in the quotient we divide the negative 2 x cubed by the x in the divisor to get negative 2 x squared. The quotient is now x cubed minus 2 x squared. The negative 2 x squared is multiplied to the entire divisor to get negative 2 x cubed minus 4 x squared. This is subtracted from the 2 x cubed minus x squared to give a remainder of 3 x squared. The 5 x in the dividend is dropped down to make this 3 x squared plus 5 x. Like terms are lined up in vertical columns. To find the next term in the quotient we divide the negative 3 x squared by the x in the divisor to get 3 x. The quotient is now x cubed minus 2 x squared plus 3 x. The 3 x is multiplied to the entire divisor to get 3 x squared plus 6 x. This is subtracted from the 3 x squared plus 5 x to give a remainder of negative x. The negative 6 in the dividend is brought down to make this negative x minus 6. The next term in the quotient is found by dividing the negative x by the x in the divisor to get negative 1. The quotient is now x cubed minus 2 x squared plus 3 x minus 1. The negative 1 is multiplied to the entire divisor to get negative x minus 2. This is subtracted from the negative x minus 6 to get the remainder of negative 4. The remainder is then written over the divisor as negative 4 divided by the quantity x plus 2. The process can be checked by multiplying the quotient of x cubed minus 2 x squared plus 3 x minus 1 minus 4 divided by the quantity x plus 2 with the divisor of x plus 2 to get the dividend of x to the fourth power minus x squared plus 5 x minus 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832116304\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835622876\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide \\({x}^{4}\\) by \\(x.\\)<div data-type=\"newline\"><br><\/div>Put the answer, \\({x}^{3},\\) in the quotient over the \\({x}^{3}\\) term.<div data-type=\"newline\"><br><\/div>Multiply \\({x}^{3}\\) times \\(x+2.\\) Line up the like terms.<div data-type=\"newline\"><br><\/div>Subtract and then bring down the next term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835303915\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide \\(-2{x}^{3}\\) by \\(x.\\)<div data-type=\"newline\"><br><\/div>Put the answer, \\(-2{x}^{2},\\) in the quotient over the \\({x}^{2}\\) term.<div data-type=\"newline\"><br><\/div>Multiply \\(-2{x}^{2}\\) times \\(x+1.\\) Line up the like terms<div data-type=\"newline\"><br><\/div>Subtract and bring down the next term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide \\(3{x}^{2}\\) by \\(x.\\)<div data-type=\"newline\"><br><\/div>Put the answer, \\(3x,\\) in the quotient over the \\(x\\) term.<div data-type=\"newline\"><br><\/div>Multiply \\(3x\\) times \\(x+1.\\) Line up the like terms.<div data-type=\"newline\"><br><\/div>Subtract and bring down the next term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835277231\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide \\(\\text{\u2212}x\\) by \\(x.\\)<div data-type=\"newline\"><br><\/div>Put the answer, \\(-1,\\) in the quotient over the constant term.<div data-type=\"newline\"><br><\/div>Multiply \\(-1\\) times \\(x+1.\\) Line up the like terms.<div data-type=\"newline\"><br><\/div>Change the signs, add.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Write the remainder as a fraction with the divisor as the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831836336\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To check, multiply \\(\\left(x+2\\right)\\left({x}^{3}-2{x}^{2}+3x-1-\\frac{4}{x+2}\\right)\\).<div data-type=\"newline\"><br><\/div>The result should be \\({x}^{4}-{x}^{2}+5x-6.\\)<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830962268\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835325550\"><div data-type=\"problem\" id=\"fs-id1167830702262\"><p id=\"fs-id1167835526976\">Find the quotient: \\(\\left({x}^{4}-7{x}^{2}+7x+6\\right)\u00f7\\left(x+3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835328234\"><p id=\"fs-id1167835187803\">\\({x}^{3}-3{x}^{2}+2x+1+\\frac{3}{x+3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834516130\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835377337\"><p id=\"fs-id1167835515166\">Find the quotient: \\(\\left({x}^{4}-11{x}^{2}-7x-6\\right)\u00f7\\left(x+3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835337664\"><p id=\"fs-id1167835365759\">\\({x}^{3}-3{x}^{2}-2x-1-\\frac{3}{x+3}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835218208\">In the next example, we will divide by \\(2a-3.\\) As we divide, we will have to consider the constants as well as the variables.<\/p><div data-type=\"example\" id=\"fs-id1167827957929\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835327751\"><div data-type=\"problem\" id=\"fs-id1167834394687\"><p id=\"fs-id1167835639946\">Find the quotient: \\(\\left(8{a}^{3}+27\\right)\u00f7\\left(2a+3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827958423\"><p>This time we will show the division all in one step. We need to add two placeholders in order to divide.<\/p><table id=\"fs-id1167834394541\" class=\"unnumbered unstyled\" summary=\"This image shows the process for polynomial long division using the example 8 a to the third power plus 27 divided by 2 a plus 3. Writing the problem as a long division problem we include placeholders for any missing terms in the dividend. The dividend of 8 a to the third power plus 0 a squared plus 0 a plus 27 is put inside the long division symbol. The divisor of 2 a plus 3 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the 8 a to the third power by 2 a to get 4 a squared. Then multiplying the 4 a squared from the quotient with the entire divisor we get 8 a to the third power plus 12 a squared. This is written below the 8 a to the third power plus 0 a squared in the dividend and subtracted to get negative 12 a squared. The 0 a in the dividend is dropped down to make this negative 12 a squared plus 0 a. To find the next term in the quotient we divide the negative 12 a squared by the 2 a in the divisor to get negative 6 a. The quotient is now 4 a squared minus 6 a. The negative 6 a is multiplied to the entire divisor to get negative 12 a squared minus 18 a. This is subtracted from the negative 12 a squared plus 0 a to give a remainder of 18 a. The 27 in the dividend is dropped down to make this 18 a plus 27. To find the next term in the quotient we divide the 18 a by the 2 a in the divisor to get 9. The quotient is now 4 a squared minus 6 a plus 9. The 9 is multiplied to the entire divisor to get 18 a plus 27. This is subtracted from the 18 a plus 27 to give a remainder of 0. The process can be checked by multiplying the quotient of 4 a squared minus 6 a plus 9 with the divisor of 2 a plus 3 to get the dividend of 8 a cubed plus 27.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835306127\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835218078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835433732\">To check, multiply \\(\\left(2a+3\\right)\\left(4{a}^{2}-6a+9\\right).\\)<\/p><p id=\"fs-id1167835346793\">The result should be \\(8{a}^{3}+27.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835331749\"><div data-type=\"problem\" id=\"fs-id1167835213271\"><p>Find the quotient: \\(\\left({x}^{3}-64\\right)\u00f7\\left(x-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834228807\"><p id=\"fs-id1167828434942\">\\({x}^{2}+4x+16\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834238976\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835187268\"><p id=\"fs-id1167835596618\">Find the quotient: \\(\\left(125{x}^{3}-8\\right)\u00f7\\left(5x-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830873696\"><p id=\"fs-id1167835268280\">\\(25{x}^{2}+10x+4\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Divide Polynomials using Synthetic Division<\/h3><p>As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let\u2019s look back at the long division we did in <a href=\"#fs-id1167832051608\" class=\"autogenerated-content\">(Figure)<\/a> and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.<\/p><span data-type=\"media\" id=\"fs-id1167826799263\" data-alt=\"The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.\"><\/span><p>Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the \\(x\\) and \\({x}^{2}\\) are removed. as well as the \\(\\text{\u2212}{x}^{2}\\) and \\(-4x\\) as they are opposite the term above.<\/p><p id=\"fs-id1167827958506\">The first row of the synthetic division is the coefficients of the dividend. The \\(-5\\) is the opposite of the 5 in the divisor.<\/p><p id=\"fs-id1167826782884\">The second row of the synthetic division are the numbers shown in red in the division problem.<\/p><p>The third row of the synthetic division are the numbers shown in blue in the division problem.<\/p><p id=\"fs-id1167834234162\">Notice the quotient and remainder are shown in the third row.<\/p><div data-type=\"equation\" id=\"fs-id1167832055859\" class=\"unnumbered\" data-label=\"\">\\({\\text{Synthetic division only works when the divisor is of the form}}\\phantom{\\rule{0.2em}{0ex}}x-c.\\)<\/div><p id=\"fs-id1167834189229\">The following example will explain the process.<\/p><div data-type=\"example\" id=\"fs-id1167831883197\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835338684\"><div data-type=\"problem\"><p id=\"fs-id1167834234098\">Use synthetic division to find the quotient and remainder when \\(2{x}^{3}+3{x}^{2}+x+8\\) is divided by \\(x+2.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167831923551\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows how to perform polynomial long division with the synthetic division method using the example 2 x cubed plus 3 x squared plus x plus 8 divided by x plus 2. The first step is to write the dividend with decreasing powers of x, which is already done. Next the coefficients of the terms are written as the first row 2 3 1 8. Writing the divisor as x minus c we would have x minus negative 2. This negative 2 is written in front of the first row negative 2 vertical line 2 3 1 8. The next step is to bring down the first coefficient to the third row. The third row is now 2 blank blank blank. Multiplying that coefficient with the divisor number of negative 2 we get the number negative 4. The negative 4 is written in the second column of the second row, directly below the 3. Adding the second column results in 3 plus negative 4 equals negative 1. The negative 1 goes in the third row. The third row now consists of 2 negative 1 blank blank. Multiplying the negative 1 with the divisor number of negative 2 results in 2. The 2 goes in the second row of the third column. The second row is now blank negative 4 2 blank. Adding the third column 1 plus 2 equals 3. The third row is now 2 negative 1 3 blank. Multiplying the 3 with the negative 2 in the divisor yields negative 6. The negative 6 is written in the fourth column of the second row. The second row is now blank negative 4 2 negative 6. Adding the fourth column 8 plus negative 6 equals 2. The 2 is goes in the fourth column of the third row. The third row is now 2 negative 1 3 2. The third row gives the coefficients for the quotient and remainder. The quotient is then 2 x squared minus 1 x plus 3 with remainder 2. The result can be checked by multiplying the quotient of 2 x squared minus 1 x plus 3 with the divisor of x plus 2 and adding the remainder of 2 to give the dividend of 2 x cubed plus 3 x squared plus x plus 8.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the dividend with decreasing powers of \\(x.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517517\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the coefficients of the terms as the first<div data-type=\"newline\"><br><\/div>row of the synthetic division.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835513928\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the divisor as \\(x-c\\) and place <em data-effect=\"italics\">c<\/em><div data-type=\"newline\"><br><\/div>in the synthetic division in the divisor box.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233324\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Bring down the first coefficient to the third row.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511938\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply that coefficient by the divisor and place the<div data-type=\"newline\"><br><\/div>result in the second row under the second coefficient.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835420451\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the second column, putting the result in the third row.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826870078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply that result by the divisor and place the<div data-type=\"newline\"><br><\/div>result in the second row under the third coefficient.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891887\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the third column, putting the result in the third row.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509767\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply that result by the divisor and place the<div data-type=\"newline\"><br><\/div>result in the third row under the third coefficient.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831881160\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the final column, putting the result in the third row.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831957032\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The quotient is \\(2{x}^{2}-1x+3\\) and the remainder is 2.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167826783725\">The division is complete. The numbers in the third row give us the result. The \\(2\\phantom{\\rule{1em}{0ex}}-1\\phantom{\\rule{1em}{0ex}}3\\) are the coefficients of the quotient. The quotient is \\(2{x}^{2}-1x+3.\\) The 2 in the box in the third row is the remainder.<\/p><p id=\"fs-id1167832066951\">Check:<\/p><p id=\"fs-id1167835351453\">\\(\\phantom{\\rule{2em}{0ex}}\\begin{array}{}\\\\ \\\\ \\hfill \\text{(quotient)(divisor)}+\\text{remainder}&amp; =\\hfill &amp; \\text{dividend}\\hfill \\\\ \\hfill \\left(2{x}^{2}-1x+3\\right)\\left(x+2\\right)+2&amp; \\stackrel{?}{=}\\hfill &amp; 2{x}^{3}+3{x}^{2}+x+8\\hfill \\\\ \\hfill 2{x}^{3}-{x}^{2}+3x+4{x}^{2}-2x+6+2&amp; \\stackrel{?}{=}\\hfill &amp; 2{x}^{3}+3{x}^{2}+x+8\\hfill \\\\ \\hfill 2{x}^{3}+3{x}^{2}+x+8&amp; =\\hfill &amp; 2{x}^{3}+3{x}^{2}+x+8\u2713\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835176301\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832059968\"><div data-type=\"problem\" id=\"fs-id1167828432131\"><p id=\"fs-id1167835350239\">Use synthetic division to find the quotient and remainder when \\(3{x}^{3}+10{x}^{2}+6x-2\\) is divided by \\(x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831833340\"><p id=\"fs-id1167835420346\">\\(3{x}^{2}+4x-2\\text{;}\\phantom{\\rule{0.2em}{0ex}}2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835355648\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835304949\"><div data-type=\"problem\" id=\"fs-id1167835231056\"><p id=\"fs-id1167830914780\">Use synthetic division to find the quotient and remainder when \\(4{x}^{3}+5{x}^{2}-5x+3\\) is divided by \\(x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832054107\"><p id=\"fs-id1167830960800\">\\(4{x}^{2}-3x+1\\text{;}\\phantom{\\rule{0.2em}{0ex}}1\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835240016\">In the next example, we will do all the steps together.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835333308\"><p id=\"fs-id1167831923651\">Use synthetic division to find the quotient and remainder when \\({x}^{4}-16{x}^{2}+3x+12\\) is divided by \\(x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835302292\"><p id=\"fs-id1167834512817\">The polynomial \\({x}^{4}-16{x}^{2}+3x+12\\) has its term in order with descending degree but we notice there is no \\({x}^{3}\\) term. We will add a 0 as a placeholder for the \\({x}^{3}\\) term. In \\(x-c\\) form, the divisor is \\(x-\\left(-4\\right).\\)<\/p><span data-type=\"media\" id=\"fs-id1167827967155\" data-alt=\"The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.\"><\/span><p>We divided a 4<sup>th<\/sup> degree polynomial by a 1<sup>st<\/sup> degree polynomial so the quotient will be a 3<sup>rd<\/sup> degree polynomial.<\/p><p id=\"fs-id1167828095848\">Reading from the third row, the quotient has the coefficients \\(1\\phantom{\\rule{1em}{0ex}}-4\\phantom{\\rule{1em}{0ex}}0\\phantom{\\rule{1em}{0ex}}3,\\) which is \\({x}^{3}-4{x}^{2}+3.\\) The remainder<\/p><div data-type=\"newline\"><br><\/div>is 0.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834244175\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834244178\"><div data-type=\"problem\" id=\"fs-id1167834244181\"><p id=\"fs-id1167834308454\">Use synthetic division to find the quotient and remainder when \\({x}^{4}-16{x}^{2}+5x+20\\) is divided by \\(x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831890958\"><p id=\"fs-id1167831890960\">\\({x}^{3}-4{x}^{2}+5\\text{;}\\phantom{\\rule{0.2em}{0ex}}0\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826967440\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826967443\"><div data-type=\"problem\" id=\"fs-id1167826967445\"><p id=\"fs-id1167826967447\">Use synthetic division to find the quotient and remainder when \\({x}^{4}-9{x}^{2}+2x+6\\) is divided by \\(x+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319100\"><p id=\"fs-id1167835319102\">\\({x}^{3}-3{x}^{2}+2\\text{;}\\phantom{\\rule{0.2em}{0ex}}0\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831923266\"><h3 data-type=\"title\">Divide Polynomial Functions<\/h3><p id=\"fs-id1167831923272\">Just as polynomials can be divided, polynomial functions can also be divided.<\/p><div data-type=\"note\" id=\"fs-id1167830961359\"><div data-type=\"title\">Division of Polynomial Functions<\/div><p>For functions \\(f\\left(x\\right)\\) and \\(g\\left(x\\right),\\) where \\(g\\left(x\\right)\\ne 0,\\)<\/p><div data-type=\"equation\" id=\"fs-id1167835301998\" class=\"unnumbered\" data-label=\"\">\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1167835366411\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835366413\"><div data-type=\"problem\" id=\"fs-id1167835366415\"><p>For functions \\(f\\left(x\\right)={x}^{2}-5x-14\\) and \\(g\\left(x\\right)=x+2,\\) find: <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-4\\right).\\)<\/p><\/div><div data-type=\"solution\"><p><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1171792359023\" data-alt=\"Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_023_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}\\text{Substitute for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.6em}{0ex}}\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{{x}^{2}-5x-14}{x+2}\\hfill \\\\ \\text{Divide the polynomials.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.6em}{0ex}}\\left(\\frac{f}{g}\\right)\\left(x\\right)=x-7\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> In part <span class=\"token\">\u24d0<\/span> we found \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) and now are asked to find \\(\\left(\\frac{f}{g}\\right)\\left(-4\\right).\\)<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\left(\\frac{f}{g}\\right)\\left(x\\right)&amp; =\\hfill &amp; x-7\\hfill \\\\ \\text{To find}\\phantom{\\rule{0.2em}{0ex}}\\left(\\frac{f}{g}\\right)\\left(-4\\right),\\phantom{\\rule{0.2em}{0ex}}\\text{substitute}\\phantom{\\rule{0.2em}{0ex}}x=-4.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\left(\\frac{f}{g}\\right)\\left(-4\\right)&amp; =\\hfill &amp; -4-7\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\left(\\frac{f}{g}\\right)\\left(-4\\right)&amp; =\\hfill &amp; -11\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830898676\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830898679\"><div data-type=\"problem\" id=\"fs-id1167830898681\"><p id=\"fs-id1167830898683\">For functions \\(f\\left(x\\right)={x}^{2}-5x-24\\) and \\(g\\left(x\\right)=x+3,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834438973\"><p id=\"fs-id1167834438975\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=x-8\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(-3\\right)=-11\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834161591\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834161594\"><div data-type=\"problem\" id=\"fs-id1167835322059\"><p id=\"fs-id1167835322061\">For functions \\(f\\left(x\\right)={x}^{2}-5x-36\\) and \\(g\\left(x\\right)=x+4,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834372261\"><p><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=x-9\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(-5\\right)=-14\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834131282\"><h3 data-type=\"title\">Use the Remainder and Factor Theorem<\/h3><p id=\"fs-id1167834131287\">Let\u2019s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as \\(x-c,\\) the value of the function at \\(c,f\\left(c\\right),\\) is the same as the remainder from the division problem.<\/p><table id=\"fs-id1167832053898\" summary=\"The table has four rows and five columns. The first row is a header row with the headers \u201cdividend\u201d, \u201cDivisor\u201d, \u201cRemainder\u201d, and f of c. The second row contains the expressions x to the fourth power minus x squared plus 5 x minus 6, x minus negative 2, negative 4, f of x equals x to the fourth power minus x squared plus 5 x minus 6, negative 4. The third row contains the expressions 3 x cubed minus 2 x squared minus 10 x plus 8, x minus 2, 4, f of x equals 3 x cubed minus 2 x squared minus 10 x plus 8, 4. The fourth row contains the expressions x to the fourth power minus 16 x squared plus 3 x plus 15, x minus negative 4, 3, f of x equals x to the fourth minus 16 x squared plus 3 x plus 15, 3.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Dividend<\/th><th data-valign=\"top\" data-align=\"left\">Divisor \\(x-c\\)<\/th><th data-valign=\"top\" data-align=\"center\">Remainder<\/th><th data-valign=\"top\" data-align=\"left\">Function<\/th><th data-valign=\"top\" data-align=\"center\">\\(f\\left(c\\right)\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\({x}^{4}-{x}^{2}+5x-6\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(x-\\left(-2\\right)\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(-4\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)={x}^{4}-{x}^{2}+5x-6\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(-4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(3{x}^{3}-2{x}^{2}-10x+8\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(x-2\\)<\/td><td data-valign=\"top\" data-align=\"center\">4<\/td><td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)=3{x}^{3}-2{x}^{2}-10x+8\\)<\/td><td data-valign=\"top\" data-align=\"center\">4<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\({x}^{4}-16{x}^{2}+3x+15\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(x-\\left(-4\\right)\\)<\/td><td data-valign=\"top\" data-align=\"center\">3<\/td><td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)={x}^{4}-16{x}^{2}+3x+15\\)<\/td><td data-valign=\"top\" data-align=\"center\">3<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167831884184\">To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend \\(f\\left(x\\right),\\) we multiply the quotient, \\(q\\left(x\\right)\\) times the divisor, \\(x-c,\\) and add the remainder, <em data-effect=\"italics\">r<\/em>.<\/p><table id=\"fs-id1167828446791\" class=\"unnumbered unstyled\" summary=\"The figure shows four equations in function notation. The first equation is f of x equals q of x times the quantity x minus c in parentheses plus r. The second equation replaces x with c. The second equation is f of c equals q of c times the quantity c minus c in parentheses plus r. The third equation simplifies the difference c minus c. The third equation is f of c equals q of c times 0 plus r. The fourth equation simplifies the product q of c times 0. The fourth equation is f of c equals r.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831823924\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">If we evaluate this at \\(c,\\) we get:<\/td><td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831117285\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831871512\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831894452\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835595747\">This leads us to the Remainder Theorem.<\/p><div data-type=\"note\" id=\"fs-id1167834345955\"><div data-type=\"title\">Remainder Theorem<\/div><p id=\"fs-id1167834345960\">If the polynomial function \\(f\\left(x\\right)\\) is divided by \\(x-c,\\) then the remainder is \\(f\\left(c\\right).\\)<\/p><\/div><div data-type=\"example\" id=\"fs-id1167831893284\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831893286\"><div data-type=\"problem\"><p>Use the Remainder Theorem to find the remainder when \\(f\\left(x\\right)={x}^{3}+3x+19\\) is divided by \\(x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835339264\"><p id=\"fs-id1167835339266\">To use the Remainder Theorem, we must use the divisor in the \\(x-c\\) form. We can write the divisor \\(x+2\\) as \\(x-\\left(-2\\right).\\) So, our \\(c\\) is \\(-2.\\)<\/p><p id=\"fs-id1167834534788\">To find the remainder, we evaluate \\(f\\left(c\\right)\\) which is \\(f\\left(-2\\right).\\)<\/p><table id=\"fs-id1167830884469\" class=\"unnumbered unstyled\" summary=\"The figure shows a series of equations. The first equation is the function f of x equals x cubed plus 3 x plus 19. The second equation evaluates this function when x equals negative 2. The second equation is f of negative 2 equals negative 2 cubed plus 3 times negative 2 plus 19. The equation is simplified in the next two lines. The third equation is f of negative 2 equals negative 8 minus 6 plus 19. The fourth equation is f of negative 2 equals 5. The remainder is 5 when f of x equals x cubed plus 3 x plus 19 is divided by x plus 2. Synthetic division is used to check. The divisor number is negative 2. The first row is 1 0 3 19. The first column is 1 blank 1. The second column is 0 negative 2 negative 2. The third column is 3 4 7. The fourth column is 19 negative 14 5. The remainder is 5.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831909359\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To evaluate \\(f\\left(-2\\right),\\) substitute \\(x=-2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835352701\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834130194\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832042288\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The remainder is 5 when \\(f\\left(x\\right)={x}^{3}+3x+19\\) is divided by \\(x+2.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div>Use synthetic division to check.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061609\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The remainder is 5.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834227330\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834227333\"><div data-type=\"problem\" id=\"fs-id1167834227336\"><p id=\"fs-id1167834227338\">Use the Remainder Theorem to find the remainder when \\(f\\left(x\\right)={x}^{3}+4x+15\\) is divided by \\(x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826995236\"><p id=\"fs-id1167826995238\">\\(-1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835529941\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835529944\"><div data-type=\"problem\" id=\"fs-id1167834062981\"><p id=\"fs-id1167834062983\">Use the Remainder Theorem to find the remainder when \\(f\\left(x\\right)={x}^{3}-7x+12\\) is divided by \\(x+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830873681\"><p id=\"fs-id1167830873684\">\\(6\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835231506\">When we divided \\(8{a}^{3}+27\\) by \\(2a+3\\) in <a href=\"#fs-id1167827957929\" class=\"autogenerated-content\">(Figure)<\/a> the result was \\(4{a}^{2}-6a+9.\\) To check our work, we multiply \\(4{a}^{2}-6a+9\\) by \\(2a+3\\) to get \\(8{a}^{3}+27\\).<\/p><div data-type=\"equation\" id=\"fs-id1167826981579\" class=\"unnumbered\" data-label=\"\">\\(\\left(4{a}^{2}-6a+9\\right)\\left(2a+3\\right)=8{a}^{3}+27\\)<\/div><p id=\"fs-id1167835420333\">Written this way, we can see that \\(4{a}^{2}-6a+9\\) and \\(2a+3\\) are factors of \\(8{a}^{3}+27.\\) When we did the division, the remainder was zero.<\/p><p id=\"fs-id1167835414666\">Whenever a divisor, \\(x-c,\\) divides a polynomial function, \\(f\\left(x\\right),\\) and resulting in a remainder of zero, we say \\(x-c\\) is a factor of \\(f\\left(x\\right).\\)<\/p><p id=\"fs-id1167832151280\">The reverse is also true. If \\(x-c\\) is a factor of \\(f\\left(x\\right)\\) then \\(x-c\\) will divide the polynomial function resulting in a remainder of zero.<\/p><p id=\"fs-id1167835416931\">We will state this in the Factor Theorem.<\/p><div data-type=\"note\" id=\"fs-id1167835416934\"><div data-type=\"title\">Factor Theorem<\/div><p id=\"fs-id1167835416939\">For any polynomial function \\(f\\left(x\\right),\\)<\/p><ul id=\"fs-id1167832081798\" data-bullet-style=\"bullet\"><li>if \\(x-c\\) is a factor of \\(f\\left(x\\right),\\) then \\(f\\left(c\\right)=0\\)<\/li><li>if \\(f\\left(c\\right)=0,\\) then \\(x-c\\) is a factor of \\(f\\left(x\\right)\\)<\/li><\/ul><\/div><div data-type=\"example\" id=\"fs-id1167834134228\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834134232\"><p id=\"fs-id1167834134235\">Use the Remainder Theorem to determine if \\(x-4\\) is a factor of \\(f\\left(x\\right)={x}^{3}-64.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826940836\"><p id=\"fs-id1167826940838\">The Factor Theorem tells us that \\(x-4\\) is a factor of \\(f\\left(x\\right)={x}^{3}-64\\) if \\(f\\left(4\\right)=0.\\)<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; f\\left(x\\right)={x}^{3}-64\\hfill \\\\ \\text{To evaluate}\\phantom{\\rule{0.2em}{0ex}}f\\left(4\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{substitute}\\phantom{\\rule{0.2em}{0ex}}x=4.\\hfill &amp; &amp; &amp; &amp; &amp; f\\left(4\\right)={4}^{3}-64\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; f\\left(4\\right)=64-64\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; &amp; &amp; f\\left(4\\right)=0\\hfill \\end{array}\\)<p id=\"fs-id1167827987846\">Since \\(f\\left(4\\right)=0,\\) \\(x-4\\) is a factor of \\(f\\left(x\\right)={x}^{3}-64.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834309001\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834309004\"><div data-type=\"problem\" id=\"fs-id1167834309007\"><p id=\"fs-id1167834309009\">Use the Factor Theorem to determine if \\(x-5\\) is a factor of \\(f\\left(x\\right)={x}^{3}-125.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834138014\"><p id=\"fs-id1167834403332\">yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834403338\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834403341\"><div data-type=\"problem\" id=\"fs-id1167834403343\"><p id=\"fs-id1167831913523\">Use the Factor Theorem to determine if \\(x-6\\) is a factor of \\(f\\left(x\\right)={x}^{3}-216.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834130270\"><p id=\"fs-id1167834130272\">yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834130279\" class=\"media-2\"><p id=\"fs-id1167832053520\">Access these online resources for additional instruction and practice with dividing polynomials.<\/p><ul id=\"fs-id1167832053523\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37Polybybinom\">Dividing a Polynomial by a Binomial<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37SynDivision\">Synthetic Division &amp; Remainder Theorem<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835331576\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835360912\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Division of a Polynomial by a Monomial<\/strong><ul id=\"fs-id1167835360921\" data-bullet-style=\"bullet\"><li>To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Division of Polynomial Functions<\/strong><ul id=\"fs-id1167834394759\" data-bullet-style=\"bullet\"><li>For functions \\(f\\left(x\\right)\\) and \\(g\\left(x\\right),\\) where \\(g\\left(x\\right)\\ne 0,\\)<div data-type=\"newline\"><br><\/div> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Remainder Theorem<\/strong><ul id=\"fs-id1167835596561\" data-bullet-style=\"bullet\"><li>If the polynomial function \\(f\\left(x\\right)\\) is divided by \\(x-c,\\) then the remainder is \\(f\\left(c\\right).\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Factor Theorem:<\/strong> For any polynomial function \\(f\\left(x\\right),\\) <ul id=\"fs-id1167830702182\" data-bullet-style=\"bullet\"><li>if \\(x-c\\) is a factor of \\(f\\left(x\\right),\\) then \\(f\\left(c\\right)=0\\)<\/li><li>if \\(f\\left(c\\right)=0,\\) then \\(x-c\\) is a factor of \\(f\\left(x\\right)\\)<\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831881547\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831881550\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167834190287\"><strong data-effect=\"bold\">Divide Monomials<\/strong><\/p><p id=\"fs-id1167834190293\">In the following exercises, divide the monomials.<\/p><div data-type=\"exercise\" id=\"fs-id1167834190296\"><div data-type=\"problem\" id=\"fs-id1167834190298\"><p id=\"fs-id1167835381049\">\\(15{r}^{4}{s}^{9}\u00f7\\left(15{r}^{4}{s}^{9}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826998342\"><div data-type=\"problem\"><p id=\"fs-id1167834583341\">\\(20{m}^{8}{n}^{4}\u00f7\\left(30{m}^{5}{n}^{9}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831910154\"><p id=\"fs-id1167831910156\">\\(\\frac{2{m}^{3}}{3{n}^{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835345812\"><div data-type=\"problem\" id=\"fs-id1167835345814\"><p id=\"fs-id1167835345816\">\\(\\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834431626\"><div data-type=\"problem\" id=\"fs-id1167834431628\"><p id=\"fs-id1167834431630\">\\(\\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835262298\"><p id=\"fs-id1167835262300\">\\(\\frac{-3{y}^{3}}{4{x}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834129910\"><div data-type=\"problem\" id=\"fs-id1167834129912\"><p id=\"fs-id1167834129914\">\\(\\frac{\\left(10{m}^{5}{n}^{4}\\right)\\left(5{m}^{3}{n}^{6}\\right)}{25{m}^{7}{n}^{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834306723\"><div data-type=\"problem\" id=\"fs-id1167834306725\"><p id=\"fs-id1167835303706\">\\(\\frac{\\left(-18{p}^{4}{q}^{7}\\right)\\left(-6{p}^{3}{q}^{8}\\right)}{-36{p}^{12}{q}^{10}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834131380\"><p id=\"fs-id1167835305571\">\\(\\frac{-3{q}^{5}}{{p}^{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835595411\"><div data-type=\"problem\" id=\"fs-id1167835287286\"><p id=\"fs-id1167835287288\">\\(\\frac{\\left(6{a}^{4}{b}^{3}\\right)\\left(4a{b}^{5}\\right)}{\\left(12{a}^{2}b\\right)\\left({a}^{3}b\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834527594\"><div data-type=\"problem\" id=\"fs-id1167834527596\"><p id=\"fs-id1167834527598\">\\(\\frac{\\left(4{u}^{2}{v}^{5}\\right)\\left(15{u}^{3}v\\right)}{\\left(12{u}^{3}v\\right)\\left({u}^{4}v\\right)}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835348433\"><p id=\"fs-id1167835348435\">\\(\\frac{5{v}^{4}}{{u}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1167832081934\"><strong data-effect=\"bold\">Divide a Polynomial by a Monomial<\/strong><\/p><p id=\"fs-id1167832081939\">In the following exercises, divide each polynomial by the monomial.<\/p><div data-type=\"exercise\" id=\"fs-id1167832081943\"><div data-type=\"problem\" id=\"fs-id1167835621211\"><p id=\"fs-id1167835621213\">\\(\\left(9{n}^{4}+6{n}^{3}\\right)\u00f73n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064316\"><div data-type=\"problem\" id=\"fs-id1167831955569\"><p id=\"fs-id1167831955571\">\\(\\left(8{x}^{3}+6{x}^{2}\\right)\u00f72x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835410265\"><p id=\"fs-id1167835410267\">\\(4{x}^{2}+3x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831908934\"><div data-type=\"problem\" id=\"fs-id1167831908937\"><p id=\"fs-id1167831872409\">\\(\\left(63{m}^{4}-42{m}^{3}\\right)\u00f7\\left(-7{m}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831148777\"><div data-type=\"problem\" id=\"fs-id1167831148779\"><p id=\"fs-id1167831148781\">\\(\\left(48{y}^{4}-24{y}^{3}\\right)\u00f7\\left(-8{y}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835378740\"><p id=\"fs-id1167835378742\">\\(-6{y}^{2}+3y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831180598\"><div data-type=\"problem\" id=\"fs-id1167832015955\"><p id=\"fs-id1167832015957\">\\(\\frac{66{x}^{3}{y}^{2}-110{x}^{2}{y}^{3}-44{x}^{4}{y}^{3}}{11{x}^{2}{y}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831228859\"><div data-type=\"problem\" id=\"fs-id1167831228861\"><p id=\"fs-id1167835312251\">\\(\\frac{72{r}^{5}{s}^{2}+132{r}^{4}{s}^{3}-96{r}^{3}{s}^{5}}{12{r}^{2}{s}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834184213\"><p id=\"fs-id1167834184216\">\\(6{r}^{3}+11{r}^{2}s-8r{s}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834505956\"><div data-type=\"problem\" id=\"fs-id1167834505958\"><p id=\"fs-id1167834505961\">\\(\\frac{10{x}^{2}+5x-4}{-5x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831921573\"><div data-type=\"problem\" id=\"fs-id1167831921575\"><p id=\"fs-id1167831921578\">\\(\\frac{20{y}^{2}+12y-1}{-4y}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834246538\"><p id=\"fs-id1167834246540\">\\(-5y-3+\\frac{1}{4y}\\)<\/p><\/div><\/div><p id=\"fs-id1167835352938\"><strong data-effect=\"bold\">Divide Polynomials using Long Division<\/strong><\/p><p id=\"fs-id1167831223959\">In the following exercises, divide each polynomial by the binomial.<\/p><div data-type=\"exercise\" id=\"fs-id1167831223962\"><div data-type=\"problem\" id=\"fs-id1167831223964\"><p id=\"fs-id1167831223966\">\\(\\left({y}^{2}+7y+12\\right)\u00f7\\left(y+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831832490\"><div data-type=\"problem\" id=\"fs-id1167831832493\"><p id=\"fs-id1167831832495\">\\(\\left({a}^{2}-2a-35\\right)\u00f7\\left(a+5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828420175\"><p id=\"fs-id1167828420177\">\\(a-7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835376266\"><div data-type=\"problem\" id=\"fs-id1167835376268\"><p id=\"fs-id1167835376270\">\\(\\left(6{m}^{2}-19m-20\\right)\u00f7\\left(m-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835389933\"><div data-type=\"problem\" id=\"fs-id1167835389935\"><p id=\"fs-id1167835389938\">\\(\\left(4{x}^{2}-17x-15\\right)\u00f7\\left(x-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831909823\"><p id=\"fs-id1167831909825\">\\(4x+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922637\"><div data-type=\"problem\" id=\"fs-id1167831922639\"><p id=\"fs-id1167831922641\">\\(\\left({q}^{2}+2q+20\\right)\u00f7\\left(q+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920042\"><div data-type=\"problem\" id=\"fs-id1167831920044\"><p id=\"fs-id1167831920046\">\\(\\left({p}^{2}+11p+16\\right)\u00f7\\left(p+8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834182566\"><p id=\"fs-id1167834182568\">\\(p+3-\\frac{8}{p+8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834473248\"><div data-type=\"problem\" id=\"fs-id1167834473250\"><p id=\"fs-id1167834473252\">\\(\\left(3{b}^{3}+{b}^{2}+4\\right)\u00f7\\left(b+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835343861\"><p id=\"fs-id1167835343864\">\\(\\left(2{n}^{3}-10n+28\\right)\u00f7\\left(n+3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834534989\"><p id=\"fs-id1167834534991\">\\(\\frac{2{n}^{2}-6n+8+4}{n+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830836783\"><div data-type=\"problem\" id=\"fs-id1167830836785\"><p id=\"fs-id1167830836787\">\\(\\left({z}^{3}+1\\right)\u00f7\\left(z+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835517770\"><div data-type=\"problem\" id=\"fs-id1167835517772\"><p id=\"fs-id1167835517774\">\\(\\left({m}^{3}+1000\\right)\u00f7\\left(m+10\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830836658\"><p id=\"fs-id1167826828355\">\\({m}^{2}-10m+100\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834462189\"><div data-type=\"problem\" id=\"fs-id1167834462191\"><p id=\"fs-id1167826828716\">\\(\\left(64{x}^{3}-27\\right)\u00f7\\left(4x-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834377048\"><div data-type=\"problem\" id=\"fs-id1167835332391\"><p id=\"fs-id1167835332393\">\\(\\left(125{y}^{3}-64\\right)\u00f7\\left(5y-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831066045\"><p id=\"fs-id1167831066047\">\\(25{y}^{2}+20x+16\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167830954378\"><p id=\"fs-id1167830954380\"><strong data-effect=\"bold\">Divide Polynomials using Synthetic Division<\/strong><\/p><p id=\"fs-id1167830954386\">In the following exercises, use synthetic Division to find the quotient and remainder.<\/p><div data-type=\"exercise\" id=\"fs-id1167830954389\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167834403353\">\\({x}^{3}-6{x}^{2}+5x+14\\) is divided by \\(x+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834473913\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834473915\"><p id=\"fs-id1167834473917\">\\({x}^{3}-3{x}^{2}-4x+12\\) is divided by \\(x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831023411\"><p id=\"fs-id1167831023413\">\\({x}^{2}-5x+6\\text{;}\\phantom{\\rule{0.2em}{0ex}}0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832068295\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832068298\"><p id=\"fs-id1167834196119\">\\(2{x}^{3}-11{x}^{2}+11x+12\\) is divided by \\(x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831224757\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831224759\"><p id=\"fs-id1167831224761\">\\(2{x}^{3}-11{x}^{2}+16x-12\\) is divided by \\(x-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831115415\"><p id=\"fs-id1167831115417\">\\(2{x}^{2}-3x+4\\text{;}\\phantom{\\rule{0.2em}{0ex}}4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834189660\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834189662\"><p id=\"fs-id1167834461629\">\\({x}^{4}-5{x}^{2}+2+13x+3\\) is divided by \\(x+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834111683\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834111685\"><p id=\"fs-id1167834111687\">\\({x}^{4}+{x}^{2}+6x-10\\) is divided by \\(x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835629122\"><p id=\"fs-id1167835629124\">\\({x}^{3}-2{x}^{2}+5x-4\\text{;}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835550509\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835550511\"><p id=\"fs-id1167835550514\">\\(2{x}^{4}-9{x}^{3}+5{x}^{2}-3x-6\\) is divided by \\(x-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835515273\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835515276\"><p id=\"fs-id1167835515278\">\\(3{x}^{4}-11{x}^{3}+2{x}^{2}+10x+6\\) is divided by \\(x-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830696545\"><p id=\"fs-id1167830696547\">\\(3{x}^{3}-2{x}^{2}-4x-2\\text{;}\\phantom{\\rule{0.2em}{0ex}}0\\)<\/p><\/div><\/div><p id=\"fs-id1167831922265\"><strong data-effect=\"bold\">Divide Polynomial Functions<\/strong><\/p><p id=\"fs-id1167831922271\">In the following exercises, divide.<\/p><div data-type=\"exercise\" id=\"fs-id1167831922274\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834193793\"><p id=\"fs-id1167834193795\">For functions \\(f\\left(x\\right)={x}^{2}-13x+36\\) and \\(g\\left(x\\right)=x-4,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835367404\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835367406\"><p id=\"fs-id1167835367408\">For functions \\(f\\left(x\\right)={x}^{2}-15x+54\\) and \\(g\\left(x\\right)=x-9,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826978521\"><p id=\"fs-id1167826978523\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=x-6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(-5\\right)=-11\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834061693\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834061695\"><p id=\"fs-id1167834061697\">For functions \\(f\\left(x\\right)={x}^{3}+{x}^{2}-7x+2\\) and \\(g\\left(x\\right)=x-2,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835229376\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835229378\"><p id=\"fs-id1167835229380\">For functions \\(f\\left(x\\right)={x}^{3}+2{x}^{2}-19x+12\\) and \\(g\\left(x\\right)=x-3,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(0\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828436158\"><p id=\"fs-id1167828436160\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)={x}^{2}+5x-4\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(0\\right)=-4\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834279190\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834279192\"><p id=\"fs-id1167834279194\">For functions \\(f\\left(x\\right)={x}^{2}-5x+2\\) and \\(g\\left(x\\right)={x}^{2}-3x-1,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834229293\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834229295\"><p id=\"fs-id1167834229297\">For functions \\(f\\left(x\\right)={x}^{2}+4x-3\\) and \\(g\\left(x\\right)={x}^{2}+2x+4,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834201920\"><p id=\"fs-id1167834201922\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>&nbsp;\\(\\left(f\u00b7g\\right)\\left(x\\right)={x}^{4}+6{x}^{3}+9{x}^{2}+10x-12;\\)<span class=\"token\">\u24d1<\/span>\\(\\left(f\u00b7g\\right)\\left(1\\right)=14\\)<\/div><\/div><p id=\"fs-id1167835384523\"><strong data-effect=\"bold\">Use the Remainder and Factor Theorem<\/strong><\/p><p id=\"fs-id1167835384529\">In the following exercises, use the Remainder Theorem to find the remainder.<\/p><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167834527642\">\\(f\\left(x\\right)={x}^{3}-8x+7\\) is divided by \\(x+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826995963\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826995965\"><p id=\"fs-id1167826995967\">\\(f\\left(x\\right)={x}^{3}-4x-9\\) is divided by \\(x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831833226\"><p id=\"fs-id1167831833228\">\\(-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834300708\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834300710\"><p id=\"fs-id1167834300712\">\\(f\\left(x\\right)=2{x}^{3}-6x-24\\) divided by \\(x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832015452\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167832015457\">\\(f\\left(x\\right)=7{x}^{2}-5x-8\\) divided by \\(x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834084792\"><p id=\"fs-id1167834084794\">\u22126<\/p><\/div><\/div><p id=\"fs-id1167834084800\">In the following exercises, use the Factor Theorem to determine if \\(x-c\\) is a factor of the polynomial function.<\/p><div data-type=\"exercise\" id=\"fs-id1167834063320\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834063322\"><p id=\"fs-id1167834063324\">Determine whether \\(x+3\\) a factor of \\({x}^{3}+8{x}^{2}+21x+18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834593508\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834593510\"><p id=\"fs-id1167834593512\">Determine whether \\(x+4\\) a factor of \\({x}^{3}+{x}^{2}-14x+8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834137634\"><p id=\"fs-id1167834137636\">no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834137641\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834137643\"><p id=\"fs-id1167834137645\">Determine whether \\(x-2\\) a factor of \\({x}^{3}-7{x}^{2}+7x-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881950\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831881952\"><p id=\"fs-id1167831881954\">Determine whether \\(x-3\\) a factor of \\({x}^{3}-7{x}^{2}+11x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834464355\"><p id=\"fs-id1167834464357\">yes<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834464364\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167835326159\"><div data-type=\"problem\" id=\"fs-id1167835326161\"><p id=\"fs-id1167835326163\">James divides \\(48y+6\\) by 6 this way: \\(\\frac{48y+\\overline{)6}}{\\overline{)6}}=48y.\\) What is wrong with his reasoning?<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831239169\"><p id=\"fs-id1167831239172\">Divide \\(\\frac{10{x}^{2}+x-12}{2x}\\) and explain with words how you get each term of the quotient.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835596587\"><p id=\"fs-id1167835596589\">answer will vary<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831923173\"><p id=\"fs-id1167831923176\">Explain when you can use synthetic division.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831923187\"><div data-type=\"problem\" id=\"fs-id1167835367980\"><p id=\"fs-id1167835367982\">In your own words, write the steps for synthetic division for \\({x}^{2}+5x+6\\) divided by \\(x-2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826863906\"><p id=\"fs-id1167826863909\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826863915\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167831923022\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section<\/p><span data-type=\"media\" id=\"fs-id1167831923030\" data-alt=\"The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is &quot;confidently&quot;, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cdivide monomials\u201d, \u201cdivide a polynomial by using a monomial\u201d, \u201cdivide polynomials using long division\u201d, \u201cdivide polynomials using synthetic division\u201d, \u201cdivide polynomial functions\u201d, and \u201cuse the Remainder and Factor Theorem\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is &quot;confidently&quot;, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cdivide monomials\u201d, \u201cdivide a polynomial by using a monomial\u201d, \u201cdivide polynomials using long division\u201d, \u201cdivide polynomials using synthetic division\u201d, \u201cdivide polynomial functions\u201d, and \u201cuse the Remainder and Factor Theorem\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><\/span><p id=\"fs-id1167835333892\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167834063173\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834063180\"><h4 data-type=\"title\"><a href=\"\/contents\/9e9d60d5-b373-4675-a019-2b4220bd93f4\" class=\"target-chapter\">Add and Subtract Polynomials<\/a><\/h4><p id=\"fs-id1167835503854\"><strong data-effect=\"bold\">Determine the Degree of Polynomials<\/strong><\/p><p id=\"fs-id1167835503860\">In the following exercises, determine the type of polynomial.<\/p><div data-type=\"exercise\" id=\"fs-id1167835503863\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835503865\"><p id=\"fs-id1167834387376\">\\(16{x}^{2}-40x-25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835333090\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835333092\"><p id=\"fs-id1167835333095\">\\(5m+9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835361921\"><p id=\"fs-id1167835361924\">binomial<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835361929\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835361931\"><p id=\"fs-id1167835361933\">\\(-15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834185093\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834185095\"><p id=\"fs-id1167828409519\">\\({y}^{2}+6{y}^{3}+9{y}^{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834438834\"><p id=\"fs-id1167832074368\">other polynomial<\/p><\/div><\/div><p id=\"fs-id1167832074373\"><strong data-effect=\"bold\">Add and Subtract Polynomials<\/strong><\/p><p id=\"fs-id1167832074378\">In the following exercises, add or subtract the polynomials.<\/p><div data-type=\"exercise\" id=\"fs-id1167832074382\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826996499\"><p id=\"fs-id1167826996501\">\\(4p+11p\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831949126\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831949128\"><p id=\"fs-id1167831949130\">\\(-8{y}^{3}-5{y}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831023954\"><p id=\"fs-id1167831023956\">\\(-13{y}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835348487\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835348489\"><p>\\(\\left(4{a}^{2}+9a-11\\right)+\\left(6{a}^{2}-5a+10\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835283104\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835283106\"><p id=\"fs-id1167835283109\">\\(\\left(8{m}^{2}+12m-5\\right)-\\left(2{m}^{2}-7m-1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826986067\"><p id=\"fs-id1167830700148\">\\(6{m}^{2}+19m-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167834141339\">\\(\\left({y}^{2}-3y+12\\right)+\\left(5{y}^{2}-9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832220498\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832220501\"><p id=\"fs-id1167832220503\">\\(\\left(5{u}^{2}+8u\\right)-\\left(4u-7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835335643\"><p id=\"fs-id1167835335645\">\\(5{u}^{2}+4u+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834376141\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834376143\"><p id=\"fs-id1167834376146\">Find the sum of \\(8{q}^{3}-27\\) and \\({q}^{2}+6q-2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831086752\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835281919\"><p id=\"fs-id1167835281921\">Find the difference of \\({x}^{2}+6x+8\\) and \\({x}^{2}-8x+15.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828094974\"><p id=\"fs-id1167831880085\">\\(2{x}^{2}-2x+23\\)<\/p><\/div><\/div><p id=\"fs-id1167831239103\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167835498297\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835498299\"><p id=\"fs-id1167835498301\">\\(17m{n}^{2}-\\left(-9m{n}^{2}\\right)+3m{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835513167\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835513169\"><p id=\"fs-id1167835513171\">\\(18a-7b-21a\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834448873\"><p id=\"fs-id1167834448875\">\\(-7b-3a\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834111736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831944093\"><p id=\"fs-id1167831944095\">\\(2p{q}^{2}-5p-3{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830865736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827943112\"><p id=\"fs-id1167827943114\">\\(\\left(6{a}^{2}+7\\right)+\\left(2{a}^{2}-5a-9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835345856\"><p id=\"fs-id1167835345858\">\\(8{a}^{2}-5a-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826977613\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826977615\"><p id=\"fs-id1167826977617\">\\(\\left(3{p}^{2}-4p-9\\right)+\\left(5{p}^{2}+14\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834079481\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834079483\"><p id=\"fs-id1167834079485\">\\(\\left(7{m}^{2}-2m-5\\right)-\\left(4{m}^{2}+m-8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834376894\"><p id=\"fs-id1167834376896\">\\(-3m+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835361109\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835361111\"><p id=\"fs-id1167826799039\">\\(\\left(7{b}^{2}-4b+3\\right)-\\left(8{b}^{2}-5b-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834189748\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834189750\"><p id=\"fs-id1167831883136\">Subtract \\(\\left(8{y}^{2}-y+9\\right)\\) from \\(\\left(11{y}^{2}-9y-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831944069\"><p id=\"fs-id1167831944071\">\\(3{y}^{2}-8y-14\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319081\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831887260\"><p id=\"fs-id1167831887263\">Find the difference of \\(\\left({z}^{2}-4z-12\\right)\\) and \\(\\left(3{z}^{2}+2z-11\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536274\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834536276\"><p id=\"fs-id1167834536278\">\\(\\left({x}^{3}-{x}^{2}y\\right)-\\left(4x{y}^{2}-{y}^{3}\\right)+\\left(3{x}^{2}y-x{y}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834539157\"><p id=\"fs-id1167835334046\">\\({x}^{3}+2{x}^{2}y-4x{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831837524\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831837526\"><p id=\"fs-id1167831837528\">\\(\\left({x}^{3}-2{x}^{2}y\\right)-\\left(x{y}^{2}-3{y}^{3}\\right)-\\left({x}^{2}y-4x{y}^{2}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167835517802\"><strong data-effect=\"bold\">Evaluate a Polynomial Function for a Given Value of the Variable<\/strong><\/p><p id=\"fs-id1167835517808\">In the following exercises, find the function values for each polynomial function.<\/p><div data-type=\"exercise\" id=\"fs-id1167835517811\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835517813\"><p id=\"fs-id1167834461779\">For the function \\(f\\left(x\\right)=7{x}^{2}-3x+5\\) find:<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(f\\left(5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(f\\left(-2\\right)\\) <span class=\"token\">\u24d2<\/span> \\(f\\left(0\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167830769619\"><p id=\"fs-id1167832054054\"><span class=\"token\">\u24d0<\/span> 165 <span class=\"token\">\u24d1<\/span> 39 <span class=\"token\">\u24d2<\/span> 5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832054071\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167834448839\">For the function \\(g\\left(x\\right)=15-16{x}^{2},\\) find:<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(g\\left(-1\\right)\\) <span class=\"token\">\u24d1<\/span> \\(g\\left(0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(g\\left(2\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831832538\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831832540\"><p id=\"fs-id1167831832542\">A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function \\(h\\left(t\\right)=-16{t}^{2}+640\\) gives the height of the glasses <em data-effect=\"italics\">t<\/em> seconds after they were dropped. Find the height of the glasses when \\(t=6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830703200\"><p id=\"fs-id1167834062151\">The height is \\(64\\phantom{\\rule{0.2em}{0ex}}\\text{feet}\\).<\/p><\/div><\/div><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834062168\"><p id=\"fs-id1167835621478\">A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of \\(p\\) dollars each is given by the polynomial \\(R\\left(p\\right)=-5{p}^{2}+360p.\\) Find the revenue received when \\(p=110\\) dollars.<\/p><\/div><\/div><p id=\"fs-id1167835327964\"><strong data-effect=\"bold\">Add and Subtract Polynomial Functions<\/strong><\/p><p id=\"fs-id1167835327969\">In the following exercises, find <span class=\"token\">\u24d0<\/span> (<em data-effect=\"italics\">f<\/em> + <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>)\u2003<span class=\"token\">\u24d1<\/span> (<em data-effect=\"italics\">f<\/em> + <em data-effect=\"italics\">g<\/em>)(3)\u2003<span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">f<\/em> \u2212 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>)\u2003<span class=\"token\">\u24d3<\/span> (<em data-effect=\"italics\">f<\/em> \u2212 <em data-effect=\"italics\">g<\/em>)(\u22122)<\/p><div data-type=\"exercise\" id=\"fs-id1167835361598\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835361600\"><p id=\"fs-id1167835361602\">\\(f\\left(x\\right)=2{x}^{2}-4x-7\\) and \\(g\\left(x\\right)=2{x}^{2}-x+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834395618\"><p id=\"fs-id1167835318878\"><span class=\"token\">\u24d0<\/span>\\(\\left(f+g\\right)\\left(x\\right)=4{x}^{2}-5x-2\\)<span class=\"token\">\u24d1<\/span>\\(\\left(f+g\\right)\\left(3\\right)=19\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left(f-g\\right)\\left(x\\right)=-3x-12\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\left(f-g\\right)\\left(-2\\right)=-6\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826828950\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826828953\"><p id=\"fs-id1167826828955\">\\(f\\left(x\\right)=4{x}^{3}-3{x}^{2}+x-1\\) and \\(g\\left(x\\right)=8{x}^{3}-1\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832042597\"><h4 data-type=\"title\"><a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb\" class=\"target-chapter\">Properties of Exponents and Scientific Notation<\/a><\/h4><p id=\"fs-id1167827964080\"><strong data-effect=\"bold\">Simplify Expressions Using the Properties for Exponents<\/strong><\/p><p id=\"fs-id1167827964086\">In the following exercises, simplify each expression using the properties for exponents.<\/p><div data-type=\"exercise\" id=\"fs-id1167826967130\"><div data-type=\"problem\" id=\"fs-id1167826967132\"><p id=\"fs-id1167826967134\">\\({p}^{3}\u00b7{p}^{10}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834134987\"><p id=\"fs-id1167834134989\">\\({p}^{13}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834135002\"><div data-type=\"problem\" id=\"fs-id1167835361742\"><p id=\"fs-id1167835361744\">\\(2\u00b7{2}^{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828095825\"><div data-type=\"problem\" id=\"fs-id1167828095827\"><p id=\"fs-id1167828095829\">\\(a\u00b7{a}^{2}\u00b7{a}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828349228\"><p id=\"fs-id1167834104663\">\\({a}^{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834104675\"><div data-type=\"problem\" id=\"fs-id1167834104677\"><p id=\"fs-id1167834104679\">\\(x\u00b7{x}^{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834191995\"><div data-type=\"problem\" id=\"fs-id1167834191998\"><p id=\"fs-id1167834192000\">\\({y}^{a}\u00b7{y}^{b}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826778556\"><p id=\"fs-id1167826778558\">\\({y}^{a+b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828327170\"><div data-type=\"problem\" id=\"fs-id1167828327172\"><p id=\"fs-id1167828327174\">\\(\\frac{{2}^{8}}{{2}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194457\"><div data-type=\"problem\" id=\"fs-id1167834194459\"><p id=\"fs-id1167834194461\">\\(\\frac{{a}^{6}}{a}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832226727\"><p id=\"fs-id1167832226729\">\\({a}^{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834537677\"><div data-type=\"problem\" id=\"fs-id1167834537679\"><p id=\"fs-id1167834537681\">\\(\\frac{{n}^{3}}{{n}^{12}}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831223730\"><p id=\"fs-id1167831223732\">\\(\\frac{1}{{x}^{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834346682\"><p id=\"fs-id1167834346684\">\\(\\frac{1}{{x}^{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834346700\"><div data-type=\"problem\" id=\"fs-id1167830924075\"><p id=\"fs-id1167830924077\">\\({3}^{0}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830924095\"><div data-type=\"problem\" id=\"fs-id1167828095986\"><p id=\"fs-id1167828095988\">\\({y}^{0}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828095999\"><p id=\"fs-id1167828096001\">1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834213951\"><div data-type=\"problem\" id=\"fs-id1167834213953\"><p id=\"fs-id1167834213955\">\\({\\left(14t\\right)}^{0}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830961462\"><div data-type=\"problem\" id=\"fs-id1167830961464\"><p id=\"fs-id1167830961466\">\\(12{a}^{0}-15{b}^{0}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831894232\"><p id=\"fs-id1167831894234\">\\(-3\\)<\/p><\/div><\/div><p id=\"fs-id1167830964426\"><strong data-effect=\"bold\">Use the Definition of a Negative Exponent<\/strong><\/p><p id=\"fs-id1167830964431\">In the following exercises, simplify each expression.<\/p><div data-type=\"exercise\" id=\"fs-id1167830964434\"><div data-type=\"problem\" id=\"fs-id1167830964436\"><p id=\"fs-id1167830964439\">\\({6}^{-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834225106\"><div data-type=\"problem\" id=\"fs-id1167834225108\"><p id=\"fs-id1167834225110\">\\({\\left(-10\\right)}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834554992\"><p id=\"fs-id1167834554994\">\\(-\\frac{1}{1000}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834431376\"><div data-type=\"problem\" id=\"fs-id1167834431378\"><p id=\"fs-id1167834431380\">\\(5\u00b7{2}^{-4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831847169\"><div data-type=\"problem\"><p id=\"fs-id1167831847173\">\\({\\left(8n\\right)}^{-1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835382032\"><p id=\"fs-id1167835382035\">\\(\\frac{1}{8n}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834282473\"><div data-type=\"problem\" id=\"fs-id1167834282475\"><p>\\({y}^{-5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831115481\"><div data-type=\"problem\" id=\"fs-id1167831115483\"><p id=\"fs-id1167831115486\">\\({10}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831896558\"><p id=\"fs-id1167831896560\">\\(\\frac{1}{1000}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831896573\"><div data-type=\"problem\" id=\"fs-id1167831896575\"><p id=\"fs-id1167831896577\">\\(\\frac{1}{{a}^{-4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826995212\"><div data-type=\"problem\" id=\"fs-id1167826995214\"><p id=\"fs-id1167826995216\">\\(\\frac{1}{{6}^{-2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831832832\"><p>36<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831832840\"><div data-type=\"problem\" id=\"fs-id1167831832842\"><p id=\"fs-id1167831832844\">\\(\\text{\u2212}{5}^{-3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831223770\"><div data-type=\"problem\" id=\"fs-id1167831223772\"><p id=\"fs-id1167831223774\">\\({\\left(-\\frac{1}{5}\\right)}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826987682\"><p id=\"fs-id1167826987684\">\\(-125\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826807787\"><div data-type=\"problem\" id=\"fs-id1167826807789\"><p id=\"fs-id1167826807791\">\\(\\text{\u2212}{\\left(\\frac{1}{2}\\right)}^{-3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835338425\"><div data-type=\"problem\" id=\"fs-id1167832149820\"><p id=\"fs-id1167832149822\">\\({\\left(-5\\right)}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832149841\"><p id=\"fs-id1167828401314\">\\(-\\frac{1}{125}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828401329\"><div data-type=\"problem\" id=\"fs-id1167828401331\"><p id=\"fs-id1167828401333\">\\({\\left(\\frac{5}{9}\\right)}^{-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835364690\"><div data-type=\"problem\" id=\"fs-id1167835364693\"><p id=\"fs-id1167835364695\">\\({\\left(-\\frac{3}{x}\\right)}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832153903\"><p id=\"fs-id1167835317208\">\\(\\frac{{x}^{3}}{27}\\)<\/p><\/div><\/div><p id=\"fs-id1167835317225\">In the following exercises, simplify each expression using the Product Property.<\/p><div data-type=\"exercise\" id=\"fs-id1167835317228\"><div data-type=\"problem\" id=\"fs-id1167834093160\"><p id=\"fs-id1167834093162\">\\({\\left({y}^{4}\\right)}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832066220\"><div data-type=\"problem\" id=\"fs-id1167832066222\"><p id=\"fs-id1167832066224\">\\({\\left({3}^{2}\\right)}^{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831892980\"><p id=\"fs-id1167831892982\">\\({3}^{10}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826801765\"><div data-type=\"problem\" id=\"fs-id1167826801767\"><p id=\"fs-id1167826801770\">\\({\\left({a}^{10}\\right)}^{y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831969943\"><div data-type=\"problem\"><p id=\"fs-id1167831969947\">\\({x}^{-3}\u00b7{x}^{9}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831228516\"><p id=\"fs-id1167831228519\">\\({x}^{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835347942\"><div data-type=\"problem\" id=\"fs-id1167835347945\"><p id=\"fs-id1167835347947\">\\({r}^{-5}\u00b7{r}^{-4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832151156\"><div data-type=\"problem\" id=\"fs-id1167832151158\"><p id=\"fs-id1167832151160\">\\(\\left(u{v}^{-3}\\right)\\left({u}^{-4}{v}^{-2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834227739\"><p id=\"fs-id1167828327038\">\\(\\frac{1}{{u}^{3}{v}^{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828327059\"><div data-type=\"problem\" id=\"fs-id1167830704814\"><p id=\"fs-id1167830704816\">\\({\\left({m}^{5}\\right)}^{-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987508\"><div data-type=\"problem\" id=\"fs-id1167826987510\"><p id=\"fs-id1167826987512\">\\({p}^{5}\u00b7{p}^{-2}\u00b7{p}^{-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831824568\"><p id=\"fs-id1167831824570\">\\(\\frac{1}{{m}^{5}}\\)<\/p><\/div><\/div><p id=\"fs-id1167831824586\">In the following exercises, simplify each expression using the Power Property.<\/p><div data-type=\"exercise\" id=\"fs-id1167831824589\"><div data-type=\"problem\" id=\"fs-id1167834429238\"><p id=\"fs-id1167834429240\">\\({\\left({k}^{-2}\\right)}^{-3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575829\"><div data-type=\"problem\" id=\"fs-id1167835575831\"><p id=\"fs-id1167835575833\">\\(\\frac{{q}^{4}}{{q}^{20}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834523693\"><p id=\"fs-id1167834523695\">\\(\\frac{1}{{q}^{16}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827966574\"><div data-type=\"problem\" id=\"fs-id1167827966577\"><p id=\"fs-id1167827966579\">\\(\\frac{{b}^{8}}{{b}^{-2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834099008\"><div data-type=\"problem\" id=\"fs-id1167834099010\"><p id=\"fs-id1167834099012\">\\(\\frac{{n}^{-3}}{{n}^{-5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832185134\"><p id=\"fs-id1167832185137\">\\({n}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167832185148\">In the following exercises, simplify each expression using the Product to a Power Property.<\/p><div data-type=\"exercise\" id=\"fs-id1167834472872\"><div data-type=\"problem\" id=\"fs-id1167834472874\"><p id=\"fs-id1167834472876\">\\({\\left(-5ab\\right)}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834537063\"><div data-type=\"problem\" id=\"fs-id1167834537065\"><p id=\"fs-id1167834537067\">\\({\\left(-4pq\\right)}^{0}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831833314\"><p id=\"fs-id1167831833316\">1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831833321\"><div data-type=\"problem\" id=\"fs-id1167831833323\"><p id=\"fs-id1167831833325\">\\({\\left(-6{x}^{3}\\right)}^{-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834402901\"><div data-type=\"problem\" id=\"fs-id1167834402903\"><p id=\"fs-id1167834402905\">\\({\\left(3{y}^{-4}\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834423591\"><p id=\"fs-id1167834423593\">\\(\\frac{9}{{y}^{8}}\\)<\/p><\/div><\/div><p id=\"fs-id1167834526657\">In the following exercises, simplify each expression using the Quotient to a Power Property.<\/p><div data-type=\"exercise\" id=\"fs-id1167834526660\"><div data-type=\"problem\" id=\"fs-id1167834526662\"><p id=\"fs-id1167834526664\">\\({\\left(\\frac{3}{5x}\\right)}^{-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533423\"><div data-type=\"problem\" id=\"fs-id1167834533425\"><p id=\"fs-id1167835311756\">\\({\\left(\\frac{3x{y}^{2}}{z}\\right)}^{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834527540\"><p id=\"fs-id1167834527543\">\\(\\frac{81{x}^{4}{y}^{8}}{{z}^{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835281638\"><div data-type=\"problem\" id=\"fs-id1167835281640\"><p id=\"fs-id1167835281642\">\\({\\left(\\frac{4{p}^{-3}}{{q}^{2}}\\right)}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167832092251\">In the following exercises, simplify each expression by applying several properties.<\/p><div data-type=\"exercise\" id=\"fs-id1167832092254\"><div data-type=\"problem\" id=\"fs-id1167832092256\"><p id=\"fs-id1167832092258\">\\({\\left({x}^{2}y\\right)}^{2}{\\left(3x{y}^{5}\\right)}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834525167\"><p id=\"fs-id1167834525170\">\\(27{x}^{7}{y}^{17}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826828297\"><div data-type=\"problem\" id=\"fs-id1167826828299\"><p id=\"fs-id1167826828301\">\\(\\frac{{\\left(-3{a}^{-2}\\right)}^{4}{\\left(2{a}^{4}\\right)}^{2}}{{\\left(-6{a}^{2}\\right)}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830698714\"><div data-type=\"problem\" id=\"fs-id1167830698716\"><p id=\"fs-id1167830698718\">\\({\\left(\\frac{3x{y}^{3}}{4{x}^{4}{y}^{-2}}\\right)}^{2}{\\left(\\frac{6x{y}^{4}}{8{x}^{3}{y}^{-2}}\\right)}^{-1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834226021\"><p id=\"fs-id1167834226023\">\\(\\frac{3{y}^{4}}{4{x}^{4}}\\)<\/p><\/div><\/div><p id=\"fs-id1167831921311\">In the following exercises, write each number in scientific notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831921315\"><div data-type=\"problem\" id=\"fs-id1167831921317\"><p id=\"fs-id1167831921319\">\\(2.568\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831910363\"><div data-type=\"problem\" id=\"fs-id1167831910365\"><p id=\"fs-id1167831910367\">5,300,000<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832150134\"><p id=\"fs-id1167832150136\">\\(5.3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832150158\"><div data-type=\"problem\"><p id=\"fs-id1167832153411\">\\(0.00814\\)<\/p><\/div><\/div><p id=\"fs-id1167826926708\">In the following exercises, convert each number to decimal form.<\/p><div data-type=\"exercise\" id=\"fs-id1167826926712\"><div data-type=\"problem\" id=\"fs-id1167826926714\"><p id=\"fs-id1167826926716\">\\(2.9\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834422979\"><p id=\"fs-id1167834422981\">\\(29,000\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834462078\"><div data-type=\"problem\" id=\"fs-id1167834462080\"><p id=\"fs-id1167834462082\">\\(3.75\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830963329\"><div data-type=\"problem\" id=\"fs-id1167830963331\"><p id=\"fs-id1167830963333\">\\(9.413\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834060992\"><p id=\"fs-id1167834060994\">\\(0.00009413\\)<\/p><\/div><\/div><p id=\"fs-id1167832057327\">In the following exercises, multiply or divide as indicated. Write your answer in decimal form.<\/p><div data-type=\"exercise\" id=\"fs-id1167832057332\"><div data-type=\"problem\" id=\"fs-id1167832057334\"><p id=\"fs-id1167832057336\">\\(\\left(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{7}\\right)\\left(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-4}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830703370\"><div data-type=\"problem\" id=\"fs-id1167828094930\"><p id=\"fs-id1167828094932\">\\(\\left(1.5\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-3}\\right)\\left(4.8\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-1}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826932451\"><p id=\"fs-id1167826932453\">\\(0.00072\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826932461\"><div data-type=\"problem\" id=\"fs-id1167826932464\"><p id=\"fs-id1167826932466\">\\(\\frac{6\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{9}}{2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-1}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826804661\"><div data-type=\"problem\" id=\"fs-id1167826804663\"><p id=\"fs-id1167826804665\">\\(\\frac{9\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-3}}{1\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-6}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831909571\"><p id=\"fs-id1167831909573\">\\(9,000\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831909582\"><h4 data-type=\"title\"><a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c\" class=\"target-chapter\">Multiply Polynomials<\/a><\/h4><p><strong data-effect=\"bold\">Multiply Monomials<\/strong><\/p><p id=\"fs-id1167835281070\">In the following exercises, multiply the monomials.<\/p><div data-type=\"exercise\" id=\"fs-id1167835281074\"><div data-type=\"problem\" id=\"fs-id1167835281076\"><p id=\"fs-id1167835281078\">\\(\\left(-6{p}^{4}\\right)\\left(9p\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834309057\"><div data-type=\"problem\" id=\"fs-id1167834309059\"><p id=\"fs-id1167834309061\">\\(\\left(\\frac{1}{3}{c}^{2}\\right)\\left(30{c}^{8}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834372718\"><p id=\"fs-id1167820628265\">\\(10{c}^{10}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167820628280\"><div data-type=\"problem\" id=\"fs-id1167820628282\"><p id=\"fs-id1167820628284\">\\(\\left(8{x}^{2}{y}^{5}\\right)\\left(7x{y}^{6}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835281800\"><div data-type=\"problem\" id=\"fs-id1167835281802\"><p id=\"fs-id1167835281804\">\\(\\left(\\frac{2}{3}{m}^{3}{n}^{6}\\right)\\left(\\frac{1}{6}{m}^{4}{n}^{4}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831882523\"><p id=\"fs-id1167834440810\">\\(\\frac{{m}^{7}{n}^{10}}{9}\\)<\/p><\/div><\/div><p id=\"fs-id1167834440833\"><strong data-effect=\"bold\">Multiply a Polynomial by a Monomial<\/strong><\/p><p id=\"fs-id1167835510599\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1167835510602\"><div data-type=\"problem\" id=\"fs-id1167835510604\"><p id=\"fs-id1167835510606\">\\(7\\left(10-x\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826997544\"><div data-type=\"problem\" id=\"fs-id1167826997546\"><p id=\"fs-id1167826997548\">\\({a}^{2}\\left({a}^{2}-9a-36\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834228231\"><p id=\"fs-id1167834228233\">\\({a}^{4}-9{a}^{3}-36{a}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832128806\"><div data-type=\"problem\" id=\"fs-id1167832128808\"><p id=\"fs-id1167832128810\">\\(-5y\\left(125{y}^{3}-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835352743\"><div data-type=\"problem\"><p>\\(\\left(4n-5\\right)\\left(2{n}^{3}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826870791\"><p id=\"fs-id1167826870793\">\\(8{n}^{4}-10{n}^{3}\\)<\/p><\/div><\/div><p><strong data-effect=\"bold\">Multiply a Binomial by a Binomial<\/strong><\/p><p id=\"fs-id1167826967168\">In the following exercises, multiply the binomials using:<\/p><p id=\"fs-id1167826967171\"><span class=\"token\">\u24d0<\/span> the Distributive Property <span class=\"token\">\u24d1<\/span> the FOIL method <span class=\"token\">\u24d2<\/span> the Vertical Method.<\/p><div data-type=\"exercise\" id=\"fs-id1167826779585\"><div data-type=\"problem\" id=\"fs-id1167826779587\"><p id=\"fs-id1167826779589\">\\(\\left(a+5\\right)\\left(a+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922866\"><div data-type=\"problem\" id=\"fs-id1167831922868\"><p id=\"fs-id1167831922870\">\\(\\left(y-4\\right)\\left(y+12\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834422466\"><p id=\"fs-id1167834422468\">\\({y}^{2}+8y-48\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831071328\"><div data-type=\"problem\" id=\"fs-id1167831071330\"><p id=\"fs-id1167831071332\">\\(\\left(3x+1\\right)\\left(2x-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834132827\"><div data-type=\"problem\" id=\"fs-id1167834132829\"><p id=\"fs-id1167834132831\">\\(\\left(6p-11\\right)\\left(3p-10\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831928882\"><p id=\"fs-id1167831040265\">\\(18{p}^{2}-93p+110\\)<\/p><\/div><\/div><p id=\"fs-id1167831040290\">In the following exercises, multiply the binomials. Use any method.<\/p><div data-type=\"exercise\" id=\"fs-id1167835281740\"><div data-type=\"problem\" id=\"fs-id1167835281742\"><p id=\"fs-id1167835281745\">\\(\\left(n+8\\right)\\left(n+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834593082\"><div data-type=\"problem\" id=\"fs-id1167834593084\"><p id=\"fs-id1167834593086\">\\(\\left(k+6\\right)\\left(k-9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826804090\"><p id=\"fs-id1167826804092\">\\({k}^{2}-3k-54\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831040418\"><div data-type=\"problem\" id=\"fs-id1167831040421\"><p id=\"fs-id1167831040423\">\\(\\left(5u-3\\right)\\left(u+8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064164\"><div data-type=\"problem\" id=\"fs-id1167834064166\"><p id=\"fs-id1167834064168\">\\(\\left(2y-9\\right)\\left(5y-7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834439047\"><p id=\"fs-id1167834439049\">\\(10{y}^{2}-59y+63\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826780810\"><div data-type=\"problem\" id=\"fs-id1167826780812\"><p id=\"fs-id1167826780814\">\\(\\left(p+4\\right)\\left(p+7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830838103\"><div data-type=\"problem\" id=\"fs-id1167830838105\"><p id=\"fs-id1167830838107\">\\(\\left(x-8\\right)\\left(x+9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834229179\"><p id=\"fs-id1167834229181\">\\({x}^{2}+x-72\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832150075\"><div data-type=\"problem\" id=\"fs-id1167832150077\"><p id=\"fs-id1167832150079\">\\(\\left(3c+1\\right)\\left(9c-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834094072\"><div data-type=\"problem\" id=\"fs-id1167834094074\"><p id=\"fs-id1167834094077\">\\(\\left(10a-1\\right)\\left(3a-3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832076252\"><p id=\"fs-id1167832076254\">\\(30{a}^{2}-33a+3\\)<\/p><\/div><\/div><p id=\"fs-id1167835517887\"><strong data-effect=\"bold\">Multiply a Polynomial by a Polynomial<\/strong><\/p><p id=\"fs-id1167835517893\">In the following exercises, multiply using <span class=\"token\">\u24d0<\/span> the Distributive Property <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p><div data-type=\"exercise\" id=\"fs-id1167835517904\"><div data-type=\"problem\" id=\"fs-id1167835517906\"><p id=\"fs-id1167835517909\">\\(\\left(x+1\\right)\\left({x}^{2}-3x-21\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831115525\"><div data-type=\"problem\" id=\"fs-id1167831115527\"><p id=\"fs-id1167831115529\">\\(\\left(5b-2\\right)\\left(3{b}^{2}+b-9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834463251\"><p id=\"fs-id1167834463253\">\\(15{b}^{3}-{b}^{2}-47b+18\\)<\/p><\/div><\/div><p id=\"fs-id1167831835374\">In the following exercises, multiply. Use either method.<\/p><div data-type=\"exercise\" id=\"fs-id1167835413972\"><div data-type=\"problem\" id=\"fs-id1167835413974\"><p id=\"fs-id1167835413976\">\\(\\left(m+6\\right)\\left({m}^{2}-7m-30\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834527258\"><div data-type=\"problem\" id=\"fs-id1167834527260\"><p id=\"fs-id1167834527263\">\\(\\left(4y-1\\right)\\left(6{y}^{2}-12y+5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834058955\"><p id=\"fs-id1167834058957\">\\(24{y}^{2}-54{y}^{2}+32y-5\\)<\/p><\/div><\/div><p id=\"fs-id1167832185177\"><strong data-effect=\"bold\">Multiply Special Products<\/strong><\/p><p id=\"fs-id1167832185182\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1167832185185\"><div data-type=\"problem\" id=\"fs-id1167832185187\"><p id=\"fs-id1167832185190\">\\({\\left(2x-y\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834137607\"><div data-type=\"problem\" id=\"fs-id1167834137609\"><p id=\"fs-id1167834137611\">\\({\\left(x+\\frac{3}{4}\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830769593\"><p id=\"fs-id1167830769596\">\\({x}^{2}+\\frac{3}{2}x+\\frac{9}{16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828096031\"><div data-type=\"problem\" id=\"fs-id1167828096033\"><p id=\"fs-id1167828096035\">\\({\\left(8{p}^{3}-3\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167834373426\">\\({\\left(5p+7q\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835410079\"><p id=\"fs-id1167835410082\">\\(25{p}^{2}+70pq+49{q}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167834120554\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates.<\/p><div data-type=\"exercise\" id=\"fs-id1167834535761\"><div data-type=\"problem\" id=\"fs-id1167834535764\"><p id=\"fs-id1167834535766\">\\(\\left(3y+5\\right)\\left(3y-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834141253\"><div data-type=\"problem\" id=\"fs-id1167834141255\"><p id=\"fs-id1167834141257\">\\(\\left(6x+y\\right)\\left(6x-y\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835311479\"><p id=\"fs-id1167835311481\">\\(36{x}^{2}-{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834515452\"><div data-type=\"problem\" id=\"fs-id1167834515454\"><p id=\"fs-id1167834515456\">\\(\\left(a+\\frac{2}{3}b\\right)\\left(a-\\frac{2}{3}b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301973\"><div data-type=\"problem\" id=\"fs-id1167835301976\"><p id=\"fs-id1167835301978\">\\(\\left(12{x}^{3}-7{y}^{2}\\right)\\left(12{x}^{3}+7{y}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828095962\"><p id=\"fs-id1167828095964\">\\(144{x}^{6}-49{y}^{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834461813\"><div data-type=\"problem\" id=\"fs-id1167834461815\"><p id=\"fs-id1167834461817\">\\(\\left(13{a}^{2}-8{b}^{4}\\right)\\left(13{a}^{2}+8{b}^{4}\\right)\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834462484\"><h4 data-type=\"title\"><a href=\"\/contents\/4102cdd7-8da4-4802-83c7-342d2f3cd3f5\" class=\"target-chapter\">Divide Monomials<\/a><\/h4><p id=\"fs-id1167831919324\"><strong data-effect=\"bold\">Divide Monomials<\/strong><\/p><p id=\"fs-id1167831919330\">In the following exercises, divide the monomials.<\/p><div data-type=\"exercise\" id=\"fs-id1167831919334\"><div data-type=\"problem\" id=\"fs-id1167831919336\"><p id=\"fs-id1167831919338\">\\(72{p}^{12}\u00f78{p}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835416860\"><p id=\"fs-id1167835416862\">\\(9{p}^{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835416876\"><div data-type=\"problem\" id=\"fs-id1167835416878\"><p id=\"fs-id1167835416880\">\\(-26{a}^{8}\u00f7\\left(2{a}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826799010\"><div data-type=\"problem\" id=\"fs-id1167826799012\"><p id=\"fs-id1167826799014\">\\(\\frac{45{y}^{6}}{-15{y}^{10}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835389999\"><p id=\"fs-id1167835390001\">\\(-\\frac{3}{{y}^{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835390019\"><div data-type=\"problem\" id=\"fs-id1167835390021\"><p id=\"fs-id1167835390023\">\\(\\frac{-30{x}^{8}}{-36{x}^{9}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832076183\"><div data-type=\"problem\" id=\"fs-id1167832076185\"><p id=\"fs-id1167832076187\">\\(\\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831824823\"><p id=\"fs-id1167831824826\">\\(\\frac{4{a}^{5}}{{b}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831824848\"><div data-type=\"problem\" id=\"fs-id1167831824850\"><p id=\"fs-id1167831824852\">\\(\\frac{11{u}^{6}{v}^{3}}{55{u}^{2}{v}^{8}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828096067\"><div data-type=\"problem\" id=\"fs-id1167828096069\"><p id=\"fs-id1167828096071\">\\(\\frac{\\left(5{m}^{9}{n}^{3}\\right)\\left(8{m}^{3}{n}^{2}\\right)}{\\left(10m{n}^{4}\\right)\\left({m}^{2}{n}^{5}\\right)}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834469637\"><p id=\"fs-id1167834469639\">\\(\\frac{4{m}^{9}}{{n}^{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834469661\"><div data-type=\"problem\" id=\"fs-id1167834469663\"><p id=\"fs-id1167835517726\">\\(\\frac{\\left(42{r}^{2}{s}^{4}\\right)\\left(54r{s}^{2}\\right)}{\\left(6r{s}^{3}\\right)\\left(9s\\right)}\\)<\/p><\/div><\/div><p id=\"fs-id1167831969743\"><strong data-effect=\"bold\">Divide a Polynomial by a Monomial<\/strong><\/p><p id=\"fs-id1167831890813\">In the following exercises, divide each polynomial by the monomial<\/p><div data-type=\"exercise\" id=\"fs-id1167831890816\"><div data-type=\"problem\" id=\"fs-id1167831890819\"><p id=\"fs-id1167831890821\">\\(\\left(54{y}^{4}-24{y}^{3}\\right)\u00f7\\left(-6{y}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828095649\"><p id=\"fs-id1167828095651\">\\(-9{y}^{2}+4y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832052514\"><div data-type=\"problem\" id=\"fs-id1167832052516\"><p id=\"fs-id1167832052518\">\\(\\frac{63{x}^{3}{y}^{2}-99{x}^{2}{y}^{3}-45{x}^{4}{y}^{3}}{9{x}^{2}{y}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832042166\"><div data-type=\"problem\" id=\"fs-id1167832042168\"><p id=\"fs-id1167832042170\">\\(\\frac{12{x}^{2}+4x-3}{-4x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832042416\"><p id=\"fs-id1167832042419\">\\(-3x-1+\\frac{3}{4x}\\)<\/p><\/div><\/div><p id=\"fs-id1167832042444\"><strong data-effect=\"bold\">Divide Polynomials using Long Division<\/strong><\/p><p id=\"fs-id1167832042450\">In the following exercises, divide each polynomial by the binomial.<\/p><div data-type=\"exercise\" id=\"fs-id1167828095056\"><div data-type=\"problem\" id=\"fs-id1167828095058\"><p id=\"fs-id1167828095060\">\\(\\left(4{x}^{2}-21x-18\\right)\u00f7\\left(x-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828095890\"><div data-type=\"problem\" id=\"fs-id1167828095892\"><p id=\"fs-id1167828095894\">\\(\\left({y}^{2}+2y+18\\right)\u00f7\\left(y+5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832144287\"><p id=\"fs-id1167832144289\">\\(y-3+\\frac{33}{q+6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832144316\"><div data-type=\"problem\" id=\"fs-id1167831023258\"><p id=\"fs-id1167831023260\">\\(\\left({n}^{3}-2{n}^{2}-6n+27\\right)\u00f7\\left(n+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832067631\"><div data-type=\"problem\" id=\"fs-id1167832067633\"><p id=\"fs-id1167832067635\">\\(\\left({a}^{3}-1\\right)\u00f7\\left(a+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834505002\"><p id=\"fs-id1167834505005\">\\({a}^{2}+a+1\\)<\/p><\/div><\/div><p id=\"fs-id1167834505025\"><strong data-effect=\"bold\">Divide Polynomials using Synthetic Division<\/strong><\/p><p id=\"fs-id1167834505030\"><em data-effect=\"italics\">In the following exercises<\/em>, use synthetic Division to find the quotient and remainder.<\/p><div data-type=\"exercise\" id=\"fs-id1167834535981\"><div data-type=\"problem\" id=\"fs-id1167834535983\"><p id=\"fs-id1167834535985\">\\({x}^{3}-3{x}^{2}-4x+12\\) is divided by \\(x+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826828663\"><div data-type=\"problem\" id=\"fs-id1167826828665\"><p id=\"fs-id1167826828667\">\\(2{x}^{3}-11{x}^{2}+11x+12\\) is divided by \\(x-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828409726\"><p id=\"fs-id1167828409728\">\\(2{x}^{2}-5x-4\\text{;}\\phantom{\\rule{0.2em}{0ex}}0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830697224\"><div data-type=\"problem\" id=\"fs-id1167830697226\"><p id=\"fs-id1167830697228\">\\({x}^{4}+{x}^{2}+6x-10\\) is divided by \\(x+2\\)<\/p><\/div><\/div><p id=\"fs-id1167827966958\"><strong data-effect=\"bold\">Divide Polynomial Functions<\/strong><\/p><p id=\"fs-id1167827966964\">In the following exercises, divide.<\/p><div data-type=\"exercise\" id=\"fs-id1167827966968\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827966970\"><p id=\"fs-id1167827966972\">For functions \\(f\\left(x\\right)={x}^{2}-15x+45\\) and \\(g\\left(x\\right)=x-9,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(-2\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167826829004\"><p id=\"fs-id1167826829006\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=x-6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(-2\\right)=-8\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835498376\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835498378\"><p id=\"fs-id1167835498380\">For functions \\(f\\left(x\\right)={x}^{3}+{x}^{2}-7x+2\\) and \\(g\\left(x\\right)=x-2,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(3\\right)\\)<\/div><\/div><p id=\"fs-id1167835526589\"><strong data-effect=\"bold\">Use the Remainder and Factor Theorem<\/strong><\/p><p id=\"fs-id1167835526596\">In the following exercises, use the Remainder Theorem to find the remainder.<\/p><div data-type=\"exercise\" id=\"fs-id1167835526599\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835526601\"><p id=\"fs-id1167835526603\">\\(f\\left(x\\right)={x}^{3}-4x-9\\) is divided by \\(x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834555205\"><p id=\"fs-id1167834555207\">\\(-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835468472\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835468474\"><p id=\"fs-id1167835468476\">\\(f\\left(x\\right)=2{x}^{3}-6x-24\\) divided by \\(x-3\\)<\/p><\/div><\/div><p id=\"fs-id1167828240690\">In the following exercises, use the Factor Theorem to determine if \\(x-c\\) is a factor of the polynomial function.<\/p><div data-type=\"exercise\" id=\"fs-id1167828240703\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167828240705\"><p id=\"fs-id1167828240707\">Determine whether \\(x-2\\) is a factor of \\({x}^{3}-7{x}^{2}+7x-6\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835530790\"><p id=\"fs-id1167835530792\">no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835530798\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835530800\"><p id=\"fs-id1167835530802\">Determine whether \\(x-3\\) is a factor of \\({x}^{3}-7{x}^{2}+11x+3\\).<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167828325874\"><h3 data-type=\"title\">Chapter Practice Test<\/h3><div data-type=\"exercise\" id=\"fs-id1167828325880\"><div data-type=\"problem\" id=\"fs-id1167828325882\"><p id=\"fs-id1167828325884\">For the polynomial \\(8{y}^{4}-3{y}^{2}+1\\)<\/p><p id=\"fs-id1167828325911\"><span class=\"token\">\u24d0<\/span> Is it a monomial, binomial, or trinomial? <span class=\"token\">\u24d1<\/span> What is its degree?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834461865\"><p id=\"fs-id1167834461868\"><span class=\"token\">\u24d0<\/span> trinomial <span class=\"token\">\u24d1<\/span> 4<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834461881\"><div data-type=\"problem\" id=\"fs-id1167834461883\"><p id=\"fs-id1167834461886\">\\(\\left(5{a}^{2}+2a-12\\right)\\left(9{a}^{2}+8a-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827957306\"><div data-type=\"problem\" id=\"fs-id1167827957308\"><p id=\"fs-id1167827957310\">\\(\\left(10{x}^{2}-3x+5\\right)-\\left(4{x}^{2}-6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834152035\"><p id=\"fs-id1167834152037\">\\(6{x}^{2}-3x+11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830959530\"><div data-type=\"problem\" id=\"fs-id1167830959533\"><p id=\"fs-id1167830959535\">\\({\\left(-\\frac{3}{4}\\right)}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830757432\"><div data-type=\"problem\" id=\"fs-id1167830757435\"><p id=\"fs-id1167830757437\">\\({x}^{-3}{x}^{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830757454\"><p id=\"fs-id1167830757456\">\\(x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834494725\"><div data-type=\"problem\" id=\"fs-id1167834494727\"><p id=\"fs-id1167834494729\">\\(\\frac{{5}^{6}}{{5}^{8}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834494763\"><div data-type=\"problem\" id=\"fs-id1167834494766\"><p id=\"fs-id1167834494768\">\\({\\left(47{a}^{18}{b}^{23}{c}^{5}\\right)}^{0}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834527804\"><p id=\"fs-id1167834527806\">\\(1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834527813\"><div data-type=\"problem\" id=\"fs-id1167835609183\"><p id=\"fs-id1167835609185\">\\({4}^{-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835609210\"><div data-type=\"problem\" id=\"fs-id1167835609212\"><p id=\"fs-id1167835609215\">\\({\\left(2y\\right)}^{-3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827897706\"><p id=\"fs-id1167827897708\">\\(\\frac{1}{8{y}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827897726\"><div data-type=\"problem\" id=\"fs-id1167827897728\"><p id=\"fs-id1167827897730\">\\({p}^{-3}\u00b7{p}^{-8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831891050\"><div data-type=\"problem\" id=\"fs-id1167831891052\"><p id=\"fs-id1167831891055\">\\(\\frac{{x}^{4}}{{x}^{-5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831224091\"><p id=\"fs-id1167831224093\">\\({x}^{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831224105\"><div data-type=\"problem\" id=\"fs-id1167831224107\"><p id=\"fs-id1167831224109\">\\({\\left(3{x}^{-3}\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834517697\"><div data-type=\"problem\"><p id=\"fs-id1167834517702\">\\(\\frac{24{r}^{3}s}{6{r}^{2}{s}^{7}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834097856\"><p id=\"fs-id1167834097859\">\\(\\frac{4r}{{s}^{6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834097878\"><div data-type=\"problem\" id=\"fs-id1167834097880\"><p id=\"fs-id1167834097882\">\\({\\left(\\frac{{x}^{4}{y}^{9}}{{x}^{-3}}\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834461621\"><div data-type=\"problem\" id=\"fs-id1167834461623\"><p id=\"fs-id1167835416122\">\\(\\left(8x{y}^{3}\\right)\\left(-6{x}^{4}{y}^{6}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835416162\"><p id=\"fs-id1167835416165\">\\(-48{x}^{5}{y}^{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831023446\"><div data-type=\"problem\" id=\"fs-id1167831023448\"><p id=\"fs-id1167831023450\">\\(4u\\left({u}^{2}-9u+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828409666\"><div data-type=\"problem\" id=\"fs-id1167828409668\"><p id=\"fs-id1167828409670\">\\(\\left(m+3\\right)\\left(7m-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826813806\"><p id=\"fs-id1167826813809\">\\(21{m}^{2}-19m-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835498878\"><div data-type=\"problem\" id=\"fs-id1167835498880\"><p id=\"fs-id1167835498882\">\\(\\left(n-8\\right)\\left({n}^{2}-4n+11\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828095700\"><div data-type=\"problem\" id=\"fs-id1167828095703\"><p id=\"fs-id1167828095705\">\\({\\left(4x-3\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830960556\"><p id=\"fs-id1167830960558\">\\(16{x}^{2}-24x+9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830960583\"><div data-type=\"problem\" id=\"fs-id1167830960585\"><p id=\"fs-id1167830960587\">\\(\\left(5x+2y\\right)\\left(5x-2y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831909422\"><div data-type=\"problem\" id=\"fs-id1167831909424\"><p id=\"fs-id1167831909426\">\\(\\left(15x{y}^{3}-35{x}^{2}y\\right)\u00f75xy\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832133966\"><p id=\"fs-id1167832133968\">\\(3{y}^{2}-7x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832133988\"><div data-type=\"problem\" id=\"fs-id1167832133990\"><p id=\"fs-id1167832133993\">\\(\\left(3{x}^{3}-10{x}^{2}+7x+10\\right)\u00f7\\left(3x+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826879986\"><div data-type=\"problem\" id=\"fs-id1167826879989\"><p id=\"fs-id1167826879991\">Use the Factor Theorem to determine if \\(x+3\\) a factor of \\({x}^{3}+8{x}^{2}+21x+18.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831224786\"><p id=\"fs-id1167831224788\">yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831224794\"><div data-type=\"problem\" id=\"fs-id1167831224796\"><p id=\"fs-id1167831224798\"><span class=\"token\">\u24d0<\/span> Convert 112,000 to scientific notation. <span class=\"token\">\u24d1<\/span> Convert \\(5.25\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-4}\\) to decimal form.<\/p><\/div><\/div><p id=\"fs-id1167831882323\">In the following exercises, simplify and write your answer in decimal form.<\/p><div data-type=\"exercise\" id=\"fs-id1167831882326\"><div data-type=\"problem\" id=\"fs-id1167831882329\"><p id=\"fs-id1167831882331\">\\(\\left(2.4\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{8}\\right)\\left(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-5}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834228714\"><p id=\"fs-id1167834228716\">\\(4.4\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834228739\"><div data-type=\"problem\" id=\"fs-id1167834228741\"><p id=\"fs-id1167834228743\">\\(\\frac{9\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{4}}{3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}{10}^{-1}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835496126\"><div data-type=\"problem\" id=\"fs-id1167835496128\"><p id=\"fs-id1167835496130\">For the function \\(f\\left(x\\right)=6{x}^{2}-3x-9\\) find:<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(f\\left(3\\right)\\) <span class=\"token\">\u24d1<\/span> \\(f\\left(-2\\right)\\) <span class=\"token\">\u24d2<\/span> \\(f\\left(0\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834525293\"><p id=\"fs-id1167834525295\"><span class=\"token\">\u24d0<\/span>\\(36\\)<span class=\"token\">\u24d1<\/span>\\(21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834525318\"><div data-type=\"problem\" id=\"fs-id1167834525320\"><p id=\"fs-id1167834525322\">For \\(f\\left(x\\right)=2{x}^{2}-3x-5\\) and \\(g\\left(x\\right)=3{x}^{2}-4x+1,\\) find<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(\\left(f+g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f+g\\right)\\left(1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(\\left(f-g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d3<\/span> \\(\\left(f-g\\right)\\left(-2\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831893243\"><div data-type=\"problem\" id=\"fs-id1167831893245\"><p id=\"fs-id1167831893247\">For functions<\/p><div data-type=\"newline\"><br><\/div>\\(f\\left(x\\right)=3{x}^{2}-23x-36\\) and<div data-type=\"newline\"><br><\/div>\\(g\\left(x\\right)=x-9,\\) find<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{f}{g}\\right)\\left(3\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167828094875\"><p id=\"fs-id1167828094877\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(x\\right)=3x+4\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{f}{g}\\right)\\left(3\\right)=13\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834505309\"><div data-type=\"problem\" id=\"fs-id1167834505311\"><p id=\"fs-id1167834505313\">A hiker drops a pebble from a bridge 240 feet above a canyon. The function \\(h\\left(t\\right)=-16{t}^{2}+240\\) gives the height of the pebble \\(t\\) seconds after it was dropped. Find the height when \\(t=3.\\)<\/p><\/div><\/div><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Dividing monomials<\/li>\n<li>Dividing a polynomial by a monomial<\/li>\n<li>Dividing polynomials using long division<\/li>\n<li>Dividing polynomials using synthetic division<\/li>\n<li>Dividing polynomial functions<\/li>\n<li>Use the remainder and factor theorems<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835337220\" class=\"be-prepared\">\n<p id=\"fs-id1167834539128\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835173019\" type=\"1\">\n<li>Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c8a5788f4e6519ac8913b7ff2327d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#100;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f4eeed73475043d4daf66afc0b5fdb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#53;&#120;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"41\" style=\"vertical-align: -9px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Combine like terms: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2fa4d0e4a11381ba28ef6b06ae6f53d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#97;&#43;&#49;&#43;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"227\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836652573\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831919748\">\n<h3 data-type=\"title\">Dividing Monomials<\/h3>\n<p id=\"fs-id1167831890636\">We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we\u2019ll use these properties to divide monomials and polynomials.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835381198\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835357231\">\n<p id=\"fs-id1167835354274\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2da5d124b9b6612b0751ee449bae9fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#97;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"124\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826783568\">\n<p id=\"fs-id1167831835715\">When we divide monomials with more than one variable, we write one fraction for each variable.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c4137a317dd9eb2f1b8de03d2c24bd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#97;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#97;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#45;&#54;&#97;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#105;&#99;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#52;&#125;&#123;&#45;&#54;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#97;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#97;&#110;&#100;&#32;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#81;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#57;&middot;&#97;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#97;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"508\" style=\"vertical-align: -54px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835343714\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835349346\">\n<div data-type=\"problem\" id=\"fs-id1167834396049\">\n<p id=\"fs-id1167834429517\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-956f962389b8f3c2f028c980b314c642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#50;&#123;&#97;&#125;&#94;&#123;&#55;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#97;&#125;&#94;&#123;&#49;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"137\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832151093\">\n<p id=\"fs-id1167835304216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-114c4a365b5179db2ff600286083a9a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"35\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834505374\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826938133\">\n<div data-type=\"problem\" id=\"fs-id1167834556655\">\n<p id=\"fs-id1167835297451\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d27abd74853bccdb396db3a89d738bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;&#123;&#99;&#125;&#94;&#123;&#56;&#125;&#123;&#100;&#125;&#94;&#123;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#123;&#99;&#125;&#94;&#123;&#49;&#50;&#125;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"137\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834178752\">\n<p id=\"fs-id1167826778816\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53c5602da26ee8ba909515f4a3f86aa5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#57;&#100;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"26\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834527744\">Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832059550\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835418160\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1877d9d3803cc06aee3074b569a9bc3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#55;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#50;&#125;&#125;&#123;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#49;&#49;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"54\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835515459\">\n<p>Be very careful to simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7f0cb6c12ea0bdbde00f9bf784f0c74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> by dividing out a common factor, and to simplify the variables by subtracting their exponents.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d25eaec46261f6776f2a674117a34caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#55;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#50;&#125;&#125;&#123;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#49;&#49;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#97;&#110;&#100;&#32;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#81;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"53\" width=\"444\" style=\"vertical-align: -22px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831884435\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830698259\">\n<div data-type=\"problem\" id=\"fs-id1167834397893\">\n<p id=\"fs-id1167835229142\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d065774091616757c58a4f23725413ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#56;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#52;&#125;&#125;&#123;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#57;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"54\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835356175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1dd285e29c59837eb10a57064f35522_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#55;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"22\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835331022\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167826782319\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad772ad9a34123a9b81b64d8762b665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#49;&#49;&#125;&#125;&#123;&#52;&#56;&#123;&#109;&#125;&#94;&#123;&#49;&#48;&#125;&#123;&#110;&#125;&#94;&#123;&#49;&#52;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"66\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1a0928fd09175e47d87f4f7f665b604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"41\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834473198\">\n<h3 data-type=\"title\">Divide a Polynomial by a Monomial<\/h3>\n<p id=\"fs-id1167826803000\">Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.<\/p>\n<p id=\"fs-id1167835514437\">The method we\u2019ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we\u2019ll start with an example to review fraction addition. The sum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-197b2288b6e9ce3306d6f9f30707d570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/> simplifies to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-381be94e50ae30839eb3a7a61ddadeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#50;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167827940168\">Now we will do this in reverse to split a single fraction into separate fractions. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abf3ab79cfd66cecf912d0060839d282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -6px;\" \/> can be written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7dfe50507a0abb11f171727fb17f5a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167831892828\">This is the \u201creverse\u201d of fraction addition and it states that if <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6fff7b8ea2aa79d8d5b4c39843a64a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6fe5481c7287f1cced87c4b03bf7d42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -6px;\" \/> We will use this to divide polynomials by monomials.<\/p>\n<div data-type=\"note\" id=\"fs-id1167830914794\">\n<div data-type=\"title\">Division of a Polynomial by a Monomial<\/div>\n<p id=\"fs-id1167832058574\">To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834226284\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832059371\">\n<div data-type=\"problem\" id=\"fs-id1167827958319\">\n<p id=\"fs-id1167826987582\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-050b63463c77abd78439401b14ca7947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#45;&#51;&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#120;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"194\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536604\">\n<p id=\"fs-id1167835513598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79322e200c923106f10a7eb8de6aebe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#45;&#51;&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#120;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#97;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#45;&#51;&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#45;&#51;&#120;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#116;&#101;&#114;&#109;&#32;&#98;&#121;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#118;&#105;&#115;&#111;&#114;&#46;&#32;&#66;&#101;&#32;&#99;&#97;&#114;&#101;&#102;&#117;&#108;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#125;&#123;&#45;&#51;&#120;&#121;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#45;&#51;&#120;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"726\" style=\"vertical-align: -42px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834536188\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835511504\">\n<p id=\"fs-id1167834299802\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-357c68643e5725e605dc3a49af267e8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;&#45;&#49;&#54;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#97;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"187\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826788581\">\n<p id=\"fs-id1167835319208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ae53580a06eeabc8eb6709f6e70c3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#97;&#43;&#50;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"70\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832134190\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826937939\">\n<div data-type=\"problem\" id=\"fs-id1167827966997\">\n<p id=\"fs-id1167835368739\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01be169aec446312a460d1ac2b939ae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#56;&#123;&#97;&#125;&#94;&#123;&#56;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#45;&#51;&#54;&#123;&#97;&#125;&#94;&#123;&#54;&#125;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"234\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835354089\">\n<p id=\"fs-id1167830964251\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e696306ec587c9a7ccdd6bba7d85536_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#98;&#43;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832043205\">\n<h3 data-type=\"title\">Divide Polynomials Using Long Division<\/h3>\n<p id=\"fs-id1167834394791\">Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let\u2019s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the long division of 875 divided by 25. 875 is labeled dividend and 25 is labeled divisor. The result of 35 is labeled quotient. The 3 in 35 is determined from the number of times we can divide 25 into 87. Multiplying 25 and 3 results in 75. 75 is subtracted from 87 to get 12. The 5 from 875 is dropped down to make 12 into 125. The 5 in 35 is determined from the number of times was can divide 25 into 125. Since 25 goes into 125 evenly there is no remainder. The result of subtracting 125 from 125 is 0 which is labeled remainder.\" \/><\/span><\/p>\n<p id=\"fs-id1167835479146\">We check division by multiplying the quotient by the divisor.<\/p>\n<p id=\"fs-id1167832127974\">If we did the division correctly, the product should equal the dividend.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89484a9907632add077676a18f574882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#53;&middot;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#56;&#55;&#53;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"35\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"fs-id1167835327385\">Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832051608\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831908410\">\n<div data-type=\"problem\" id=\"fs-id1167834473947\">\n<p id=\"fs-id1167834517539\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b3ff9d40381e318c830f5fec7802327_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"179\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167827957025\" class=\"unnumbered unstyled can-break\" summary=\"This image shows the process for polynomial long division using the example x squared plus 9 x plus 20 divided by x plus 5. Writing the problem as a long division problem we make sure the dividend is in standard form. The dividend of x squared plus 9 x plus 20 is put in the long division symbol. The divisor of x plus 5 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the x squared by x to get x. Then multiplying the x from the quotient with the entire divisor we get x squared plus 5 x. This is written below the x squared plus 9 x in the dividend and subtracted to get 4 x. The 20 in the dividend is dropped down to make this 4 x plus 20. To find the next term in the quotient we divide the 4 x by the x in the divisor to get 4. The quotient is now x plus 4. The 4 is multiplied to the entire divisor to get 4 x plus 20. This is subtracted from the 4 x plus 20 to give a remainder of 0. The process can be checked by multiplying the quotient of x plus 4 with the divisor of x plus 5 to get the dividend of x squared plus 9 x plus 20.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356577\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write it as a long division problem.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Be sure the dividend is in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835343759\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/> It may help to ask yourself, \u201cWhat do I need<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> by to get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>?\u201d<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830701188\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> in the quotient over the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> term.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b15f93e4636326f69503613638cb6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/> Line up the like terms under the dividend.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adaefc20015aa58781891fe1763a03d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"58\" style=\"vertical-align: -2px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f2ab2cbb26fab27f6d3ee87a40f2cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"62\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>You may find it easier to change the signs and then add.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Then bring down the last term, 20.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327295\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\n<div data-type=\"newline\"><\/div>\n<p>Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fa4d35b5c4377934245a54e07e43e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: -1px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/> It may help to ask yourself, \u201cWhat do I<\/p>\n<div data-type=\"newline\"><\/div>\n<p>need to multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> by to get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fa4d35b5c4377934245a54e07e43e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: -1px;\" \/>?\u201d<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/>, in the quotient over the constant term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421672\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply 4 times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b15f93e4636326f69503613638cb6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254539\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a974c99b6e476658bde7b78ed63a575_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"59\" style=\"vertical-align: -2px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d67939747a6b3ce7b7564adfa4b672d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830757353\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply the quotient by the divisor. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc9f6bd0e372160547c044d3a24f60a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>You should get the dividend. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a64d095fa572e0ae4219ecb48ff0325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#50;&#48;&#10003;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831887197\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831882387\">\n<div data-type=\"problem\" id=\"fs-id1167831872394\">\n<p id=\"fs-id1167834346516\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a63292249c1b6a5f30a5422dca89e835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#121;&#43;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"186\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831891731\">\n<p id=\"fs-id1167834300936\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebed38f331e3e5f9c21b4f5ec6846336_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830924020\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834345678\">\n<div data-type=\"problem\" id=\"fs-id1167834053551\">\n<p id=\"fs-id1167834048906\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66d38bc483916d0affd704fdd79497fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#109;&#43;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"196\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835321097\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22ed76139ab25c60bcabdf5810fd52cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167828395729\">When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we\u2019ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.<\/p>\n<p id=\"fs-id1167834423838\">Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd1b01374677d5ea16594a5cf8ab597_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/> It is missing an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term. We will add in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04fc23735ccd8919bdf69e4d8410762f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> as a placeholder.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835358491\">\n<p id=\"fs-id1167834182485\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beed610b83c1f89f55d331bded61ef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"210\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831955744\">\n<p id=\"fs-id1167834395936\">Notice that there is no <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term in the dividend. We will add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04fc23735ccd8919bdf69e4d8410762f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> as a placeholder.<\/p>\n<table id=\"fs-id1167835216851\" class=\"unnumbered unstyled can-break\" summary=\"This image shows the process for polynomial long division using the example x to the fourth power minus x squared plus 5 x minus 6 divided by x plus 2. Writing the problem as a long division problem we include placeholders for any missing terms in the dividend. The dividend of x to the fourth power plus 0 x cubed minus x squared plus 5 x minus 6 is put inside the long division symbol. The divisor of x plus 2 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the x to the fourth power by x to get x cubed. Then multiplying the x cubed from the quotient with the entire divisor we get x to the fourth power plus 2 x cubed. This is written below the x to the fourth power plus 0 x cubed in the dividend and subtracted to get negative 2 x cubed. Like terms are lined up in vertical columns. A note is made that it may be helpful to change the signs of the terms you are subtracting and add them. The negative x squared in the dividend is dropped down to make this 2 x cubed minus x squared. To find the next term in the quotient we divide the negative 2 x cubed by the x in the divisor to get negative 2 x squared. The quotient is now x cubed minus 2 x squared. The negative 2 x squared is multiplied to the entire divisor to get negative 2 x cubed minus 4 x squared. This is subtracted from the 2 x cubed minus x squared to give a remainder of 3 x squared. The 5 x in the dividend is dropped down to make this 3 x squared plus 5 x. Like terms are lined up in vertical columns. To find the next term in the quotient we divide the negative 3 x squared by the x in the divisor to get 3 x. The quotient is now x cubed minus 2 x squared plus 3 x. The 3 x is multiplied to the entire divisor to get 3 x squared plus 6 x. This is subtracted from the 3 x squared plus 5 x to give a remainder of negative x. The negative 6 in the dividend is brought down to make this negative x minus 6. The next term in the quotient is found by dividing the negative x by the x in the divisor to get negative 1. The quotient is now x cubed minus 2 x squared plus 3 x minus 1. The negative 1 is multiplied to the entire divisor to get negative x minus 2. This is subtracted from the negative x minus 6 to get the remainder of negative 4. The remainder is then written over the divisor as negative 4 divided by the quantity x plus 2. The process can be checked by multiplying the quotient of x cubed minus 2 x squared plus 3 x minus 1 minus 4 divided by the quantity x plus 2 with the divisor of x plus 2 to get the dividend of x to the fourth power minus x squared plus 5 x minus 6.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832116304\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835622876\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-159899034a68d14dd24abc2121923568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8322c3c196fa4c7fc837e5e93222c1ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> in the quotient over the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/> Line up the like terms.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Subtract and then bring down the next term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835303915\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-074f765ddcd7bb9ff892b3924601c315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27d98b3359f7ff9a50f82b98db095177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -4px;\" \/> in the quotient over the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dee62ab3fe8deaa2480c81dc03e58907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: 0px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f514cc9361efcbf72d7e90ff5087feb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/> Line up the like terms<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Subtract and bring down the next term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9608403b176cb023606ca01493d9f883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acf48f02997503ddab9453c8694c7cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"23\" style=\"vertical-align: -4px;\" \/> in the quotient over the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> term.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f514cc9361efcbf72d7e90ff5087feb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/> Line up the like terms.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Subtract and bring down the next term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835277231\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1921bc6f8d905df045b27a1dadb4ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Put the answer, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df802eaa8e08eb73e8cd4d30d23d4b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> in the quotient over the constant term.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> times <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f514cc9361efcbf72d7e90ff5087feb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/> Line up the like terms.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Change the signs, add.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Write the remainder as a fraction with the divisor as the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831836336\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To check, multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4c68e778dfc09936a9d31ab63f1e9b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#43;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"263\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The result should be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd1b01374677d5ea16594a5cf8ab597_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830962268\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835325550\">\n<div data-type=\"problem\" id=\"fs-id1167830702262\">\n<p id=\"fs-id1167835526976\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3941ca8a22dbe2c82ce4a676bda22ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"219\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835328234\">\n<p id=\"fs-id1167835187803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c210edf68124f6467ddd1cecb341cc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"187\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834516130\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835377337\">\n<p id=\"fs-id1167835515166\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96c9d68f6158560e6f99c33a8d8d7a36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"228\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835337664\">\n<p id=\"fs-id1167835365759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c09c1c4fa47c0b770ab10d5752d817d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"187\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835218208\">In the next example, we will divide by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-367ba36b79d261ad8a66d3427ce287ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> As we divide, we will have to consider the constants as well as the variables.<\/p>\n<div data-type=\"example\" id=\"fs-id1167827957929\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835327751\">\n<div data-type=\"problem\" id=\"fs-id1167834394687\">\n<p id=\"fs-id1167835639946\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-faebbb2cf42fa896554bfac2e9781fb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827958423\">\n<p>This time we will show the division all in one step. We need to add two placeholders in order to divide.<\/p>\n<table id=\"fs-id1167834394541\" class=\"unnumbered unstyled\" summary=\"This image shows the process for polynomial long division using the example 8 a to the third power plus 27 divided by 2 a plus 3. Writing the problem as a long division problem we include placeholders for any missing terms in the dividend. The dividend of 8 a to the third power plus 0 a squared plus 0 a plus 27 is put inside the long division symbol. The divisor of 2 a plus 3 is written to the left of the long division symbol. The quotient is written above the long division symbol. To find the first term in the quotient we divide the 8 a to the third power by 2 a to get 4 a squared. Then multiplying the 4 a squared from the quotient with the entire divisor we get 8 a to the third power plus 12 a squared. This is written below the 8 a to the third power plus 0 a squared in the dividend and subtracted to get negative 12 a squared. The 0 a in the dividend is dropped down to make this negative 12 a squared plus 0 a. To find the next term in the quotient we divide the negative 12 a squared by the 2 a in the divisor to get negative 6 a. The quotient is now 4 a squared minus 6 a. The negative 6 a is multiplied to the entire divisor to get negative 12 a squared minus 18 a. This is subtracted from the negative 12 a squared plus 0 a to give a remainder of 18 a. The 27 in the dividend is dropped down to make this 18 a plus 27. To find the next term in the quotient we divide the 18 a by the 2 a in the divisor to get 9. The quotient is now 4 a squared minus 6 a plus 9. The 9 is multiplied to the entire divisor to get 18 a plus 27. This is subtracted from the 18 a plus 27 to give a remainder of 0. The process can be checked by multiplying the quotient of 4 a squared minus 6 a plus 9 with the divisor of 2 a plus 3 to get the dividend of 8 a cubed plus 27.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835306127\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835218078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835433732\">To check, multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-912063a318a3b0830f35875c5a6929db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"184\" style=\"vertical-align: -7px;\" \/><\/p>\n<p id=\"fs-id1167835346793\">The result should be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ca90198b9908403e2f7d8e37603809b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835331749\">\n<div data-type=\"problem\" id=\"fs-id1167835213271\">\n<p>Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5d3552d48b7a7fa2a896c20da3f416c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228807\">\n<p id=\"fs-id1167828434942\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-117511911c2ff89fa0b74fb980235d34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834238976\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835187268\">\n<p id=\"fs-id1167835596618\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5358ca9daa71db593602210d32c628e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830873696\">\n<p id=\"fs-id1167835268280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-235c4d51b7bea3c1ac6692fd53ab480e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Divide Polynomials using Synthetic Division<\/h3>\n<p>As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let\u2019s look back at the long division we did in <a href=\"#fs-id1167832051608\" class=\"autogenerated-content\">(Figure)<\/a> and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826799263\" data-alt=\"The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the long division of 1 x squared plus 9 x plus 20 divided by x plus 5 right next to the same problem done with synthetic division. In the long division problem, the coefficients of the dividend are 1 and 9 and 20 and the zero of the divisor is negative 5. In the synthetic division problem, we just write the numbers negative 5 1 9 20 with a line separating the negative 5. In the long division problem, the subtracted terms are 5 x and 20. In the synthetic division problem the second line is the numbers negative 5 and negative 20. The remainder of the problem is 0 and the quotient is x plus 4. The synthetic division puts these coefficients as the last line 1 4 0.\" \/><\/span><\/p>\n<p>Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> are removed. as well as the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37f27aba51f41963a10954c11a543eef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02432f7c0c440e71281741bcabb2d7a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> as they are opposite the term above.<\/p>\n<p id=\"fs-id1167827958506\">The first row of the synthetic division is the coefficients of the dividend. The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/> is the opposite of the 5 in the divisor.<\/p>\n<p id=\"fs-id1167826782884\">The second row of the synthetic division are the numbers shown in red in the division problem.<\/p>\n<p>The third row of the synthetic division are the numbers shown in blue in the division problem.<\/p>\n<p id=\"fs-id1167834234162\">Notice the quotient and remainder are shown in the third row.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832055859\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7479e10dd3e35e1f8929c052b4a05b71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#121;&#110;&#116;&#104;&#101;&#116;&#105;&#99;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#111;&#110;&#32;&#111;&#110;&#108;&#121;&#32;&#119;&#111;&#114;&#107;&#115;&#32;&#119;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#118;&#105;&#115;&#111;&#114;&#32;&#105;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#111;&#114;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#45;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"520\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1167834189229\">The following example will explain the process.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831883197\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835338684\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834234098\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83bd94acaa760831ef4cedee4028c6a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167831923551\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows how to perform polynomial long division with the synthetic division method using the example 2 x cubed plus 3 x squared plus x plus 8 divided by x plus 2. The first step is to write the dividend with decreasing powers of x, which is already done. Next the coefficients of the terms are written as the first row 2 3 1 8. Writing the divisor as x minus c we would have x minus negative 2. This negative 2 is written in front of the first row negative 2 vertical line 2 3 1 8. The next step is to bring down the first coefficient to the third row. The third row is now 2 blank blank blank. Multiplying that coefficient with the divisor number of negative 2 we get the number negative 4. The negative 4 is written in the second column of the second row, directly below the 3. Adding the second column results in 3 plus negative 4 equals negative 1. The negative 1 goes in the third row. The third row now consists of 2 negative 1 blank blank. Multiplying the negative 1 with the divisor number of negative 2 results in 2. The 2 goes in the second row of the third column. The second row is now blank negative 4 2 blank. Adding the third column 1 plus 2 equals 3. The third row is now 2 negative 1 3 blank. Multiplying the 3 with the negative 2 in the divisor yields negative 6. The negative 6 is written in the fourth column of the second row. The second row is now blank negative 4 2 negative 6. Adding the fourth column 8 plus negative 6 equals 2. The 2 is goes in the fourth column of the third row. The third row is now 2 negative 1 3 2. The third row gives the coefficients for the quotient and remainder. The quotient is then 2 x squared minus 1 x plus 3 with remainder 2. The result can be checked by multiplying the quotient of 2 x squared minus 1 x plus 3 with the divisor of x plus 2 and adding the remainder of 2 to give the dividend of 2 x cubed plus 3 x squared plus x plus 8.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the dividend with decreasing powers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517517\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the coefficients of the terms as the first<\/p>\n<div data-type=\"newline\"><\/div>\n<p>row of the synthetic division.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835513928\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the divisor as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> and place <em data-effect=\"italics\">c<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p>in the synthetic division in the divisor box.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233324\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Bring down the first coefficient to the third row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511938\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply that coefficient by the divisor and place the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>result in the second row under the second coefficient.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835420451\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the second column, putting the result in the third row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826870078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply that result by the divisor and place the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>result in the second row under the third coefficient.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891887\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the third column, putting the result in the third row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509767\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply that result by the divisor and place the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>result in the third row under the third coefficient.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831881160\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the final column, putting the result in the third row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831957032\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The quotient is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5530dcd701d16361b65a15efde4a9123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/> and the remainder is 2.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167826783725\">The division is complete. The numbers in the third row give us the result. The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-216ddf5f82e17ec82279db9df68d8838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"84\" style=\"vertical-align: -1px;\" \/> are the coefficients of the quotient. The quotient is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed27dd0b1fc6565422cb7ca70295c706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/> The 2 in the box in the third row is the remainder.<\/p>\n<p id=\"fs-id1167832066951\">Check:<\/p>\n<p id=\"fs-id1167835351453\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae96fdfafbd91e1a65caf70f7e62b49d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#41;&#40;&#100;&#105;&#118;&#105;&#115;&#111;&#114;&#41;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#109;&#97;&#105;&#110;&#100;&#101;&#114;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#118;&#105;&#100;&#101;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#54;&#43;&#50;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#56;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#56;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"269\" width=\"258\" style=\"vertical-align: -150px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835176301\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832059968\">\n<div data-type=\"problem\" id=\"fs-id1167828432131\">\n<p id=\"fs-id1167835350239\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f70b658b55cfb796bcca3e53820e8ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831833340\">\n<p id=\"fs-id1167835420346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a52796002f44fba8d623deac0f9d7af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835355648\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835304949\">\n<div data-type=\"problem\" id=\"fs-id1167835231056\">\n<p id=\"fs-id1167830914780\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c58e051d2d993f98d56d8c23658f11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832054107\">\n<p id=\"fs-id1167830960800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d4d97817b288afec89ccad84cfb6918_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835240016\">In the next example, we will do all the steps together.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835333308\">\n<p id=\"fs-id1167831923651\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eefed4b8f926cb083d0e0dbac06fae3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8344d700415f34f743486a051215f1ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302292\">\n<p id=\"fs-id1167834512817\">The polynomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eefed4b8f926cb083d0e0dbac06fae3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/> has its term in order with descending degree but we notice there is no <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term. We will add a 0 as a placeholder for the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29a18debcfeba0c22f81de96b6c65e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term. In <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> form, the divisor is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33241e8bda875ac18ba76ee758b59b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167827967155\" data-alt=\"The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0.\" \/><\/span><\/p>\n<p>We divided a 4<sup>th<\/sup> degree polynomial by a 1<sup>st<\/sup> degree polynomial so the quotient will be a 3<sup>rd<\/sup> degree polynomial.<\/p>\n<p id=\"fs-id1167828095848\">Reading from the third row, the quotient has the coefficients <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-036a56718d299efd74f7c2e5ef8a405b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\" \/> which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9e872f09098b0b5da8eb3c9d3bf650a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/> The remainder<\/p>\n<div data-type=\"newline\"><\/div>\n<p>is 0.<\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834244175\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834244178\">\n<div data-type=\"problem\" id=\"fs-id1167834244181\">\n<p id=\"fs-id1167834308454\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52a6e89fe648577d3cfe0535d86ad4cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8344d700415f34f743486a051215f1ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831890958\">\n<p id=\"fs-id1167831890960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01ac66ece92a29635856d00e39225279_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826967440\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826967443\">\n<div data-type=\"problem\" id=\"fs-id1167826967445\">\n<p id=\"fs-id1167826967447\">Use synthetic division to find the quotient and remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79f663e619216a84938c51b09631764e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fff851a0bf058fd49c29dd3e3040b5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319100\">\n<p id=\"fs-id1167835319102\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45397f8a9c4b53ec0a497f82a43c879a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831923266\">\n<h3 data-type=\"title\">Divide Polynomial Functions<\/h3>\n<p id=\"fs-id1167831923272\">Just as polynomials can be divided, polynomial functions can also be divided.<\/p>\n<div data-type=\"note\" id=\"fs-id1167830961359\">\n<div data-type=\"title\">Division of Polynomial Functions<\/div>\n<p>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-344231958daaa52c1755b0b6892d4016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11586e035d0c3f45a2c8589e19f9ba92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167835301998\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d90b951b5d30947ed8e41f2a98635d90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"110\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835366411\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835366413\">\n<div data-type=\"problem\" id=\"fs-id1167835366415\">\n<p>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a41098a64693eecd4c90d0e248fa0a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4037233681386a566898ec8cd25ee016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05e39106fb71eb61385a28ebea95236a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"76\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1171792359023\" data-alt=\"Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_023_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Equation shows f over g of x equals f of x divided by g of x. This is translated into a division problem showing x squared minus 5x minus 14 divided by x plus 2. The quotient is x minus 7.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49d6e85b9b6e94e4d0eb47d9286e5311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#49;&#52;&#125;&#123;&#120;&#43;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#108;&#121;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"547\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> In part <span class=\"token\">\u24d0<\/span> we found <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> and now are asked to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05e39106fb71eb61385a28ebea95236a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"76\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3be5500d201661241cebea80914ce8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#52;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"544\" style=\"vertical-align: -44px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830898676\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830898679\">\n<div data-type=\"problem\" id=\"fs-id1167830898681\">\n<p id=\"fs-id1167830898683\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a270d64726a89ecc688d7e0fb5b1191a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a43c8bcabe131623b6b42731236c7932_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5fc9157f3c6f0ed16a78416fd1d5d7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"76\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834438973\">\n<p id=\"fs-id1167834438975\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92fd325bf41a77b958aff93cf8f50c46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"121\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eecde55c9701cdde60fc7916adb5e1cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"124\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834161591\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834161594\">\n<div data-type=\"problem\" id=\"fs-id1167835322059\">\n<p id=\"fs-id1167835322061\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97054947fba81bdf602af66acbca1267_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ea2d50ecd38cf2c19778ce5da7cc884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc6838cbb8eb232c62b160757e47e368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"76\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834372261\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f8eb67c09818d3ecc622a1c034095de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"121\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88422d5e11e3d2c0a4af7fd27c7d5c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"125\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834131282\">\n<h3 data-type=\"title\">Use the Remainder and Factor Theorem<\/h3>\n<p id=\"fs-id1167834131287\">Let\u2019s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313ea17cde610801c80050dc03ef0cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: -4px;\" \/> the value of the function at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f24a7d0ae83a3167af284686a9b2ae7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#44;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> is the same as the remainder from the division problem.<\/p>\n<table id=\"fs-id1167832053898\" summary=\"The table has four rows and five columns. The first row is a header row with the headers \u201cdividend\u201d, \u201cDivisor\u201d, \u201cRemainder\u201d, and f of c. The second row contains the expressions x to the fourth power minus x squared plus 5 x minus 6, x minus negative 2, negative 4, f of x equals x to the fourth power minus x squared plus 5 x minus 6, negative 4. The third row contains the expressions 3 x cubed minus 2 x squared minus 10 x plus 8, x minus 2, 4, f of x equals 3 x cubed minus 2 x squared minus 10 x plus 8, 4. The fourth row contains the expressions x to the fourth power minus 16 x squared plus 3 x plus 15, x minus negative 4, 3, f of x equals x to the fourth minus 16 x squared plus 3 x plus 15, 3.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Dividend<\/th>\n<th data-valign=\"top\" data-align=\"left\">Divisor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"center\">Remainder<\/th>\n<th data-valign=\"top\" data-align=\"left\">Function<\/th>\n<th data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d0f55a755b6cc824702ec5a6907062a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67eeef9a1c0898ddf238ae98bd485a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1cfa00ca0b1bb8a6196eaa078ecdeb19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c23ef6ea471498d6d8e85956f8dfcd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-076a75aadcef22141e874b2bc228b780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79c51dce88ed8deaaab9733f311dd7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9be3a8c4a60b62ef14764d63bc9126f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"216\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c793dd82c86672a54fc14c49ad9f5ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae8a89710934d195e395473d820f0d2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e39ba5e26aff0e087d099d571029c49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"215\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167831884184\">To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> we multiply the quotient, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-428f18d16b4058d1236554965e5de78f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> times the divisor, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313ea17cde610801c80050dc03ef0cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: -4px;\" \/> and add the remainder, <em data-effect=\"italics\">r<\/em>.<\/p>\n<table id=\"fs-id1167828446791\" class=\"unnumbered unstyled\" summary=\"The figure shows four equations in function notation. The first equation is f of x equals q of x times the quantity x minus c in parentheses plus r. The second equation replaces x with c. The second equation is f of c equals q of c times the quantity c minus c in parentheses plus r. The third equation simplifies the difference c minus c. The third equation is f of c equals q of c times 0 plus r. The fourth equation simplifies the product q of c times 0. The fourth equation is f of c equals r.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831823924\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\">If we evaluate this at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-818255b758e790541f4c10eee83ef800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: -4px;\" \/> we get:<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831117285\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831871512\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831894452\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835595747\">This leads us to the Remainder Theorem.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834345955\">\n<div data-type=\"title\">Remainder Theorem<\/div>\n<p id=\"fs-id1167834345960\">If the polynomial function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313ea17cde610801c80050dc03ef0cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: -4px;\" \/> then the remainder is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77bb6761b8ea56a0f3f155df279ea752_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831893284\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831893286\">\n<div data-type=\"problem\">\n<p>Use the Remainder Theorem to find the remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87ecabee5d1c4767198e34a0b9ceb05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#120;&#43;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835339264\">\n<p id=\"fs-id1167835339266\">To use the Remainder Theorem, we must use the divisor in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> form. We can write the divisor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-845840ba157bce2df424620b48293a7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> So, our <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d36c434a1919e0aaa1a4125fdaa40853_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167834534788\">To find the remainder, we evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d0f55a755b6cc824702ec5a6907062a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ccfd7ec77d7d595dca58ab727de8ffd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"fs-id1167830884469\" class=\"unnumbered unstyled\" summary=\"The figure shows a series of equations. The first equation is the function f of x equals x cubed plus 3 x plus 19. The second equation evaluates this function when x equals negative 2. The second equation is f of negative 2 equals negative 2 cubed plus 3 times negative 2 plus 19. The equation is simplified in the next two lines. The third equation is f of negative 2 equals negative 8 minus 6 plus 19. The fourth equation is f of negative 2 equals 5. The remainder is 5 when f of x equals x cubed plus 3 x plus 19 is divided by x plus 2. Synthetic division is used to check. The divisor number is negative 2. The first row is 1 0 3 19. The first column is 1 blank 1. The second column is 0 negative 2 negative 2. The third column is 3 4 7. The fourth column is 19 negative 14 5. The remainder is 5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831909359\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05b7a6d51e75f2f334ccb86c6d2c8ec4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/> substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86f872935a384592f05d5fdc077a0a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835352701\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834130194\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832042288\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The remainder is 5 when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87ecabee5d1c4767198e34a0b9ceb05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#120;&#43;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Use synthetic division to check.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834061609\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The remainder is 5.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834227330\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834227333\">\n<div data-type=\"problem\" id=\"fs-id1167834227336\">\n<p id=\"fs-id1167834227338\">Use the Remainder Theorem to find the remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47944f577b5f5fcd58ea642037999aec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430e7aa075d9795e934ea8e1595b79f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826995236\">\n<p id=\"fs-id1167826995238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835529941\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835529944\">\n<div data-type=\"problem\" id=\"fs-id1167834062981\">\n<p id=\"fs-id1167834062983\">Use the Remainder Theorem to find the remainder when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df01495fdcc45e8f659787314e5eb99d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fff851a0bf058fd49c29dd3e3040b5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830873681\">\n<p id=\"fs-id1167830873684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835231506\">When we divided <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e257533d949d55e5dfc6578aab6e2fbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -2px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d5580553a8edb8b5abbda8e6b4321cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/> in <a href=\"#fs-id1167827957929\" class=\"autogenerated-content\">(Figure)<\/a> the result was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-278dc990b0e82aa73f342e28b688f1af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/> To check our work, we multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a4a989fa8328f738fbaf10d0723b6b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d5580553a8edb8b5abbda8e6b4321cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/> to get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e257533d949d55e5dfc6578aab6e2fbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167826981579\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9635bf44c1a18f21aa3964e872c1530a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"265\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1167835420333\">Written this way, we can see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a4a989fa8328f738fbaf10d0723b6b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d5580553a8edb8b5abbda8e6b4321cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/> are factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ca90198b9908403e2f7d8e37603809b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -2px;\" \/> When we did the division, the remainder was zero.<\/p>\n<p id=\"fs-id1167835414666\">Whenever a divisor, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313ea17cde610801c80050dc03ef0cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: -4px;\" \/> divides a polynomial function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> and resulting in a remainder of zero, we say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-066aa5868267297977626df4011032f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832151280\">The reverse is also true. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> will divide the polynomial function resulting in a remainder of zero.<\/p>\n<p id=\"fs-id1167835416931\">We will state this in the Factor Theorem.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835416934\">\n<div data-type=\"title\">Factor Theorem<\/div>\n<p id=\"fs-id1167835416939\">For any polynomial function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<ul id=\"fs-id1167832081798\" data-bullet-style=\"bullet\">\n<li>if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87cc8616a3f6b202ac163d4cea4080a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a341c13b617d5560f8c1deb52756913c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834134228\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834134232\">\n<p id=\"fs-id1167834134235\">Use the Remainder Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f5f3c92307a15605824e3f359350bb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826940836\">\n<p id=\"fs-id1167826940838\">The Factor Theorem tells us that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-088f37732f6e7b8ba3ab9bd38c45d371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -4px;\" \/> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70974b4e7e7224c09eddefd961927385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-396e4e3fee4fd1f8dbbf04c3c70ee9db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#101;&#118;&#97;&#108;&#117;&#97;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#52;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#52;&#45;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"464\" style=\"vertical-align: -37px;\" \/><\/p>\n<p id=\"fs-id1167827987846\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4c8336e8f66d588431d1848b7a5debf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f5f3c92307a15605824e3f359350bb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834309001\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834309004\">\n<div data-type=\"problem\" id=\"fs-id1167834309007\">\n<p id=\"fs-id1167834309009\">Use the Factor Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c307cdabd1259e3f52730a26a4e77eaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e5b33e84acb3e9efbdada0595b0ee65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834138014\">\n<p id=\"fs-id1167834403332\">yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834403338\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834403341\">\n<div data-type=\"problem\" id=\"fs-id1167834403343\">\n<p id=\"fs-id1167831913523\">Use the Factor Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a977b4ac50dc35ecc9b44e0b3ba4f7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e32e7f7f544872b7840244ca6fdba84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834130270\">\n<p id=\"fs-id1167834130272\">yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834130279\" class=\"media-2\">\n<p id=\"fs-id1167832053520\">Access these online resources for additional instruction and practice with dividing polynomials.<\/p>\n<ul id=\"fs-id1167832053523\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37Polybybinom\">Dividing a Polynomial by a Binomial<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37SynDivision\">Synthetic Division &amp; Remainder Theorem<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835331576\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835360912\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Division of a Polynomial by a Monomial<\/strong>\n<ul id=\"fs-id1167835360921\" data-bullet-style=\"bullet\">\n<li>To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Division of Polynomial Functions<\/strong>\n<ul id=\"fs-id1167834394759\" data-bullet-style=\"bullet\">\n<li>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-344231958daaa52c1755b0b6892d4016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11586e035d0c3f45a2c8589e19f9ba92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d90b951b5d30947ed8e41f2a98635d90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"110\" style=\"vertical-align: -12px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Remainder Theorem<\/strong>\n<ul id=\"fs-id1167835596561\" data-bullet-style=\"bullet\">\n<li>If the polynomial function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313ea17cde610801c80050dc03ef0cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: -4px;\" \/> then the remainder is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77bb6761b8ea56a0f3f155df279ea752_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Factor Theorem:<\/strong> For any polynomial function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/>\n<ul id=\"fs-id1167830702182\" data-bullet-style=\"bullet\">\n<li>if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87cc8616a3f6b202ac163d4cea4080a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a341c13b617d5560f8c1deb52756913c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831881547\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831881550\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167834190287\"><strong data-effect=\"bold\">Divide Monomials<\/strong><\/p>\n<p id=\"fs-id1167834190293\">In the following exercises, divide the monomials.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834190296\">\n<div data-type=\"problem\" id=\"fs-id1167834190298\">\n<p id=\"fs-id1167835381049\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1102541c13b70588ce8e91fdb64f6378_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#114;&#125;&#94;&#123;&#52;&#125;&#123;&#115;&#125;&#94;&#123;&#57;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#114;&#125;&#94;&#123;&#52;&#125;&#123;&#115;&#125;&#94;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"116\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826998342\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834583341\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ce76e8f45090169d404b6a4d63674d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#109;&#125;&#94;&#123;&#56;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#48;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"136\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831910154\">\n<p id=\"fs-id1167831910156\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da12054bc0363db8c65879a1c7eadc6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#51;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835345812\">\n<div data-type=\"problem\" id=\"fs-id1167835345814\">\n<p id=\"fs-id1167835345816\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca9873a929cc9db09b84be6deccbbc96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#45;&#50;&#55;&#123;&#97;&#125;&#94;&#123;&#57;&#125;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834431626\">\n<div data-type=\"problem\" id=\"fs-id1167834431628\">\n<p id=\"fs-id1167834431630\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cb820fe485b809ecfb668dac17a2120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#53;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#57;&#125;&#125;&#123;&#45;&#54;&#48;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"53\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835262298\">\n<p id=\"fs-id1167835262300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9bf2ae31c43a377048abf28d3c8b487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834129910\">\n<div data-type=\"problem\" id=\"fs-id1167834129912\">\n<p id=\"fs-id1167834129914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19356fdfe33ebf5af7dfdac3baabfdf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#50;&#53;&#123;&#109;&#125;&#94;&#123;&#55;&#125;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"117\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834306723\">\n<div data-type=\"problem\" id=\"fs-id1167834306725\">\n<p id=\"fs-id1167835303706\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ffbbd87d5f996fed174490c8edc2782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#123;&#112;&#125;&#94;&#123;&#52;&#125;&#123;&#113;&#125;&#94;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#123;&#113;&#125;&#94;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#51;&#54;&#123;&#112;&#125;&#94;&#123;&#49;&#50;&#125;&#123;&#113;&#125;&#94;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"124\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834131380\">\n<p id=\"fs-id1167835305571\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfc011a5350d8bb402a7d7cbebd7f48e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"31\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835595411\">\n<div data-type=\"problem\" id=\"fs-id1167835287286\">\n<p id=\"fs-id1167835287288\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e29c843a8ffac88951ea8ef6978a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"88\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834527594\">\n<div data-type=\"problem\" id=\"fs-id1167834527596\">\n<p id=\"fs-id1167834527598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1748952ead0200f746778afe3520b5a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"99\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835348433\">\n<p id=\"fs-id1167835348435\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec50c7ee0563a9bdbc0f75ab8ba0b31e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#118;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832081934\"><strong data-effect=\"bold\">Divide a Polynomial by a Monomial<\/strong><\/p>\n<p id=\"fs-id1167832081939\">In the following exercises, divide each polynomial by the monomial.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832081943\">\n<div data-type=\"problem\" id=\"fs-id1167835621211\">\n<p id=\"fs-id1167835621213\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46ef003aabefc7d79f9f2cad02f3111a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#43;&#54;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064316\">\n<div data-type=\"problem\" id=\"fs-id1167831955569\">\n<p id=\"fs-id1167831955571\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b3721d7adcdaf17a895aaeb79b35048_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835410265\">\n<p id=\"fs-id1167835410267\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecb23d23c41dc5a415b951a293beefb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"67\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831908934\">\n<div data-type=\"problem\" id=\"fs-id1167831908937\">\n<p id=\"fs-id1167831872409\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55ad7faa490512074314d455ad4102fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#51;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#52;&#50;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"184\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831148777\">\n<div data-type=\"problem\" id=\"fs-id1167831148779\">\n<p id=\"fs-id1167831148781\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f26f6cd1a9dd3a0c9e754fd7c989999b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#50;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378740\">\n<p id=\"fs-id1167835378742\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbbb9edf17e8927b341d9b5d040bd647_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831180598\">\n<div data-type=\"problem\" id=\"fs-id1167832015955\">\n<p id=\"fs-id1167832015957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c73a7afba1377038889cfd5df95c6c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"156\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831228859\">\n<div data-type=\"problem\" id=\"fs-id1167831228861\">\n<p id=\"fs-id1167835312251\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0d68921633eea3b0cd68c69a3d86d0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#123;&#114;&#125;&#94;&#123;&#53;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#50;&#123;&#114;&#125;&#94;&#123;&#52;&#125;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#45;&#57;&#54;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#123;&#115;&#125;&#94;&#123;&#53;&#125;&#125;&#123;&#49;&#50;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834184213\">\n<p id=\"fs-id1167834184216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57c997cfed5a0e1173e147cee379b4ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#49;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#115;&#45;&#56;&#114;&#123;&#115;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834505956\">\n<div data-type=\"problem\" id=\"fs-id1167834505958\">\n<p id=\"fs-id1167834505961\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a799b561b316bda62cca2558f4c7b56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#52;&#125;&#123;&#45;&#53;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831921573\">\n<div data-type=\"problem\" id=\"fs-id1167831921575\">\n<p id=\"fs-id1167831921578\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63dd79b45b13645b238cb44e332359f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#45;&#49;&#125;&#123;&#45;&#52;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"78\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246538\">\n<p id=\"fs-id1167834246540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3c9fe3fd2ecc04dbaaa71d024ff78cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#121;&#45;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835352938\"><strong data-effect=\"bold\">Divide Polynomials using Long Division<\/strong><\/p>\n<p id=\"fs-id1167831223959\">In the following exercises, divide each polynomial by the binomial.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831223962\">\n<div data-type=\"problem\" id=\"fs-id1167831223964\">\n<p id=\"fs-id1167831223966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20036db68d8c578050aab331215f128a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#121;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832490\">\n<div data-type=\"problem\" id=\"fs-id1167831832493\">\n<p id=\"fs-id1167831832495\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ef45140ce7576eb35ed6027d033eca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#45;&#51;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828420175\">\n<p id=\"fs-id1167828420177\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6f9776671486b21795f5115f09aa290_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835376266\">\n<div data-type=\"problem\" id=\"fs-id1167835376268\">\n<p id=\"fs-id1167835376270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0524a73d13af271724ddbdd192ff1b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#109;&#45;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"206\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835389933\">\n<div data-type=\"problem\" id=\"fs-id1167835389935\">\n<p id=\"fs-id1167835389938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a064902d1d55dfc356646a64aa4a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#55;&#120;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"189\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831909823\">\n<p id=\"fs-id1167831909825\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93621914630f7eefa8f40eb8b38b28ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922637\">\n<div data-type=\"problem\" id=\"fs-id1167831922639\">\n<p id=\"fs-id1167831922641\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bfd11ddf1ba614877a41f0d12c024a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#113;&#43;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"167\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920042\">\n<div data-type=\"problem\" id=\"fs-id1167831920044\">\n<p id=\"fs-id1167831920046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90b162dd7b922e5ae8b2a3c4104fb61b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#112;&#43;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"177\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834182566\">\n<p id=\"fs-id1167834182568\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6b8690862c7b45eb25d5d8499b57ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#43;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#112;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"89\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834473248\">\n<div data-type=\"problem\" id=\"fs-id1167834473250\">\n<p id=\"fs-id1167834473252\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-862df179f52cd190b67c49ceaa7dd2ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835343861\">\n<p id=\"fs-id1167835343864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a36756d5acd524784e692c5de65b2c63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#48;&#110;&#43;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"191\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534989\">\n<p id=\"fs-id1167834534991\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a73747075a0ebec9922de262014f5ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#110;&#43;&#56;&#43;&#52;&#125;&#123;&#110;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"84\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830836783\">\n<div data-type=\"problem\" id=\"fs-id1167830836785\">\n<p id=\"fs-id1167830836787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2900efdee9f4c2b6f26a2af385c0b228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#122;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835517770\">\n<div data-type=\"problem\" id=\"fs-id1167835517772\">\n<p id=\"fs-id1167835517774\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e18c61faf8ad5e9bebba58885fe60863_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830836658\">\n<p id=\"fs-id1167826828355\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c030b13cf715a04f8f7f411ee2b5aadc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#109;&#43;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834462189\">\n<div data-type=\"problem\" id=\"fs-id1167834462191\">\n<p id=\"fs-id1167826828716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a7760a1ebe5c2d3ade1f016ddd23ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"158\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834377048\">\n<div data-type=\"problem\" id=\"fs-id1167835332391\">\n<p id=\"fs-id1167835332393\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66e0a6942e6a03c35fa689e141301ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831066045\">\n<p id=\"fs-id1167831066047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e34e52dd9c0d57849aa204dc5f023d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#120;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167830954378\">\n<p id=\"fs-id1167830954380\"><strong data-effect=\"bold\">Divide Polynomials using Synthetic Division<\/strong><\/p>\n<p id=\"fs-id1167830954386\">In the following exercises, use synthetic Division to find the quotient and remainder.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830954389\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834403353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42de4554e7c12dabd78c2548f6f3e053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc79ac27a4802ea8e5f0a37a6889212b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834473913\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834473915\">\n<p id=\"fs-id1167834473917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b619c20476e1d34958da2cdebb360cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831023411\">\n<p id=\"fs-id1167831023413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d7983f7eaafdb0dacd7c0e805d41a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832068295\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832068298\">\n<p id=\"fs-id1167834196119\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c1e444fd47c0efa07e682eed398b9b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"172\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831224757\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831224759\">\n<p id=\"fs-id1167831224761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f1fd9d32fcfc028fe9cd9670321d471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"172\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831115415\">\n<p id=\"fs-id1167831115417\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50b7533a9b4a5bc65cb30f3b9d057c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834189660\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834189662\">\n<p id=\"fs-id1167834461629\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88bf18ecb5ac47281c9d104a4f9daf56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#43;&#49;&#51;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"177\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834111683\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834111685\">\n<p id=\"fs-id1167834111687\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f43ed3159f806c9458dc22259d054d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835629122\">\n<p id=\"fs-id1167835629124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e96b67e17da363f1b4aa60d0785e3812_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835550509\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835550511\">\n<p id=\"fs-id1167835550514\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdde39fd1ca5c80ce85a38aeda00c1dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"195\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835515273\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835515276\">\n<p id=\"fs-id1167835515278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8200a8c0dd6b066de8741ba457b68d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"212\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830696545\">\n<p id=\"fs-id1167830696547\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e920f22a33167753d7ee2f23f30e851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831922265\"><strong data-effect=\"bold\">Divide Polynomial Functions<\/strong><\/p>\n<p id=\"fs-id1167831922271\">In the following exercises, divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831922274\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834193793\">\n<p id=\"fs-id1167834193795\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e27af2a7914021968b4baaf0f832900_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#120;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6189ffd32c81a4cc577a996ea8901e12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44b02e7dab65152291b50f83127dc381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835367404\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835367406\">\n<p id=\"fs-id1167835367408\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce9bf71996af434b955e06f6b3e4edc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#120;&#43;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e76e2f980c4c84f953587ba94140a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98b1d153df9570914c2faf09e87876e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826978521\">\n<p id=\"fs-id1167826978523\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ec3166c7642e64f4521a246862addb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"121\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9948019e26a644b3dab906665b26a29f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"124\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834061693\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834061695\">\n<p id=\"fs-id1167834061697\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d4b3c18213548eb1a6f9fe942975e28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"189\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-def2c9f1007422c50cedbae34f85a27f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c06474b12f702f46857c4178b59cfa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835229376\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835229378\">\n<p id=\"fs-id1167835229380\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7d321cf4ff3045adc931ca477266fc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"215\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d94a2ed22025a2a17844da59f63d56c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f613b08183988d2ac9890d9f9b7209d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828436158\">\n<p id=\"fs-id1167828436160\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f08e862c459d4ebf10ca90d7d1a27c0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"170\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16cf1878b04c81acafa7af9c123c9d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"102\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834279190\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834279192\">\n<p id=\"fs-id1167834279194\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad144465c8d25106dc6667a1360c576f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f8742c9dc2df94a20460ce7242af2d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff77fc6d8ce39e0c137debd709c838d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4c984ba2bd226067e7eb82dee0648c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834229293\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834229295\">\n<p id=\"fs-id1167834229297\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-891fd734e25d265423cbe74cdfadfc6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86ff1189469b2b1b1dbca0e1d6e7c2b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff77fc6d8ce39e0c137debd709c838d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7eda147b8ca3b39575691fd76fe6d6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834201920\">\n<p id=\"fs-id1167834201922\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span>&nbsp;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a18b64b6af88fe5c4ebb8b74f533b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#49;&#50;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"291\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ae6dba550e8409c6a9eb3e9f92e554f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835384523\"><strong data-effect=\"bold\">Use the Remainder and Factor Theorem<\/strong><\/p>\n<p id=\"fs-id1167835384529\">In the following exercises, use the Remainder Theorem to find the remainder.<\/p>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834527642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47d29d407d3ba928b4429484a86ffc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826995963\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826995965\">\n<p id=\"fs-id1167826995967\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9b9f262a020c8f5b32ee5a347cb197c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831833226\">\n<p id=\"fs-id1167831833228\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300708\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834300710\">\n<p id=\"fs-id1167834300712\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7767e4ab46fad8049c168a00a7515729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#120;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/> divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832015452\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832015457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b141c852fc29ed826008b3cbe30768f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce48b2b21d45b1802bcce97645acc819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834084792\">\n<p id=\"fs-id1167834084794\">\u22126<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834084800\">In the following exercises, use the Factor Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of the polynomial function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834063320\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063322\">\n<p id=\"fs-id1167834063324\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4faca3263a080f9dcff513cb7d0be144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#120;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834593508\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834593510\">\n<p id=\"fs-id1167834593512\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cd392abb5a6fd3a660e3b5912589956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caa8e20b307b4cb41bfbff2c792901ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834137634\">\n<p id=\"fs-id1167834137636\">no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834137641\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834137643\">\n<p id=\"fs-id1167834137645\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79c51dce88ed8deaaab9733f311dd7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/> a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a8d1d3329a1f9a77998051fa7fb11ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881950\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881952\">\n<p id=\"fs-id1167831881954\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af091a1d73057f65807d15f36f171843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834464355\">\n<p id=\"fs-id1167834464357\">yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834464364\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835326159\">\n<div data-type=\"problem\" id=\"fs-id1167835326161\">\n<p id=\"fs-id1167835326163\">James divides <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4da16307aca990e293ab73641f2799ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#121;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -4px;\" \/> by 6 this way: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6aee5f0133ee2fa71e1dd31ec2528f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#56;&#121;&#43;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#54;&#125;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#54;&#125;&#125;&#61;&#52;&#56;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"101\" style=\"vertical-align: -12px;\" \/> What is wrong with his reasoning?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831239169\">\n<p id=\"fs-id1167831239172\">Divide <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8885b4e9f554a84512e832a3df58997_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#50;&#125;&#123;&#50;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"72\" style=\"vertical-align: -6px;\" \/> and explain with words how you get each term of the quotient.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835596587\">\n<p id=\"fs-id1167835596589\">answer will vary<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831923173\">\n<p id=\"fs-id1167831923176\">Explain when you can use synthetic division.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831923187\">\n<div data-type=\"problem\" id=\"fs-id1167835367980\">\n<p id=\"fs-id1167835367982\">In your own words, write the steps for synthetic division for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-796dbf4877c44d45b95e2f7845612068_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/> divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfdc25fb64ffc9ad0325b336b33b7ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826863906\">\n<p id=\"fs-id1167826863909\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826863915\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167831923022\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831923030\" data-alt=\"The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is &quot;confidently&quot;, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cdivide monomials\u201d, \u201cdivide a polynomial by using a monomial\u201d, \u201cdivide polynomials using long division\u201d, \u201cdivide polynomials using synthetic division\u201d, \u201cdivide polynomial functions\u201d, and \u201cuse the Remainder and Factor Theorem\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is &quot;confidently&quot;, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cdivide monomials\u201d, \u201cdivide a polynomial by using a monomial\u201d, \u201cdivide polynomials using long division\u201d, \u201cdivide polynomials using synthetic division\u201d, \u201cdivide polynomial functions\u201d, and \u201cuse the Remainder and Factor Theorem\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\" \/><\/span><\/p>\n<p id=\"fs-id1167835333892\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167834063173\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834063180\">\n<h4 data-type=\"title\"><a href=\"\/contents\/9e9d60d5-b373-4675-a019-2b4220bd93f4\" class=\"target-chapter\">Add and Subtract Polynomials<\/a><\/h4>\n<p id=\"fs-id1167835503854\"><strong data-effect=\"bold\">Determine the Degree of Polynomials<\/strong><\/p>\n<p id=\"fs-id1167835503860\">In the following exercises, determine the type of polynomial.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835503863\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835503865\">\n<p id=\"fs-id1167834387376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59ce818464c39d49b57c33479f1a6082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#48;&#120;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"123\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835333090\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835333092\">\n<p id=\"fs-id1167835333095\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29098117134e9502157bc1ea1dfa4c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#109;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835361921\">\n<p id=\"fs-id1167835361924\">binomial<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835361929\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835361931\">\n<p id=\"fs-id1167835361933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834185093\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834185095\">\n<p id=\"fs-id1167828409519\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72580e4a8a15e1611a4e9029ee5bda60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834438834\">\n<p id=\"fs-id1167832074368\">other polynomial<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832074373\"><strong data-effect=\"bold\">Add and Subtract Polynomials<\/strong><\/p>\n<p id=\"fs-id1167832074378\">In the following exercises, add or subtract the polynomials.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832074382\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826996499\">\n<p id=\"fs-id1167826996501\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-985b431d674c4a876f826c3231aadfe0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#112;&#43;&#49;&#49;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831949126\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831949128\">\n<p id=\"fs-id1167831949130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abef9beb387e728e9f7af0c763b0733a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831023954\">\n<p id=\"fs-id1167831023956\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d6aac413103e5f1cfb81c840bbf4499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835348487\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835348489\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e00c150a5c07c45a68c8f5df102876b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#97;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"261\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835283104\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835283106\">\n<p id=\"fs-id1167835283109\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acdabfdba27f73b504317dc4025bd1c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#109;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"277\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826986067\">\n<p id=\"fs-id1167830700148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba2dad44a8eb61dcd0e02673afcafe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#57;&#109;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834141339\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77d6a41282539baf153e7bc141a68353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#121;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"203\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832220498\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832220501\">\n<p id=\"fs-id1167832220503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8dd576aeabd63a59e34d39af0d38e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#117;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#117;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835335643\">\n<p id=\"fs-id1167835335645\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6413a24a90829f9142491fe632f230d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#117;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834376141\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834376143\">\n<p id=\"fs-id1167834376146\">Find the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40815b77663c4b9039c472d34431a13d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03c7b39b585c1a07c782fc90cef0fb62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831086752\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835281919\">\n<p id=\"fs-id1167835281921\">Find the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d43d27073cae4a5a5910e00a7935afd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04d83422fa8695511145587e2baa968b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828094974\">\n<p id=\"fs-id1167831880085\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b37793e3c6cc5e58d5c7732cc662ac9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831239103\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835498297\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835498299\">\n<p id=\"fs-id1167835498301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a93f9356f4139f5af8f8796404029342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#109;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#109;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#109;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"210\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835513167\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835513169\">\n<p id=\"fs-id1167835513171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a69101457d7fe762bc2848fcd59ab8f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#97;&#45;&#55;&#98;&#45;&#50;&#49;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"113\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834448873\">\n<p id=\"fs-id1167834448875\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44b9b72cdbef68c5e6b011c0b22b775c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#98;&#45;&#51;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834111736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831944093\">\n<p id=\"fs-id1167831944095\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7527520f9a78bb965ce2b2d5c5a371c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#112;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#112;&#45;&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827943112\">\n<p id=\"fs-id1167827943114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36c3152cb7a7138d25841b4daf622f9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"203\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345856\">\n<p id=\"fs-id1167835345858\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4efdeb4d2862c12a25a32dcb5bfac25b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826977613\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826977615\">\n<p id=\"fs-id1167826977617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a45158e915399a86648a94811d8d19a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#112;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834079481\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834079483\">\n<p id=\"fs-id1167834079485\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51800cc51ab873d34c32b9a9c5da7975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"259\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834376894\">\n<p id=\"fs-id1167834376896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fef1562cf86872d4cbd4844449b767c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#109;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"68\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835361109\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835361111\">\n<p id=\"fs-id1167826799039\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99f84e81c36b4fa946b3197e0e5e7cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#98;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"236\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834189748\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834189750\">\n<p id=\"fs-id1167831883136\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c58a337b53c34c33ab6e763e883c772_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -7px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f4a4904a8e89eacc1ad054acc26f764_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831944069\">\n<p id=\"fs-id1167831944071\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af6aabf88d10cfd104d5a7316eb351e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#121;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319081\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831887260\">\n<p id=\"fs-id1167831887263\">Find the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e1cec1656c15666100a2f400b67c840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#122;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"108\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4db0f0d8dd8be4efd76888ef487f90ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#122;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536274\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834536276\">\n<p id=\"fs-id1167834536278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c258e1759d2b3ed23ec8e7b916a5bd54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"314\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834539157\">\n<p id=\"fs-id1167835334046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d74cc8bc5616a1bb43c85c2a73ac7b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#52;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831837524\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831837526\">\n<p id=\"fs-id1167831837528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9866de46df231ac4badb423c5e77b3f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#52;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"323\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835517802\"><strong data-effect=\"bold\">Evaluate a Polynomial Function for a Given Value of the Variable<\/strong><\/p>\n<p id=\"fs-id1167835517808\">In the following exercises, find the function values for each polynomial function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835517811\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835517813\">\n<p id=\"fs-id1167834461779\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8a9c75bfbb8695d24d93a89f05d0c1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00e628545709308df2fad3ec5e77a716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dca5d7bc047aaa23a5ac85a2c257c5f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d73c012bc8b5a6f560bea30840502b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167830769619\">\n<p id=\"fs-id1167832054054\"><span class=\"token\">\u24d0<\/span> 165 <span class=\"token\">\u24d1<\/span> 39 <span class=\"token\">\u24d2<\/span> 5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832054071\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834448839\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52b912f61e2f1e4437b1901798005dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c180bd3dc0c2ee1eaf25f823f5a72c32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3ee0ed7cf1ec152b02cf6c51c179b1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b42d35fe562a099584b3845d1d6c833c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832538\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831832540\">\n<p id=\"fs-id1167831832542\">A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1da6d81e6703188ea70504dc102ba090_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -4px;\" \/> gives the height of the glasses <em data-effect=\"italics\">t<\/em> seconds after they were dropped. Find the height of the glasses when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62cb22896cec00f071e1ff1be24d270d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830703200\">\n<p id=\"fs-id1167834062151\">The height is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73f988084d5b91d9e0aa5abc2e602cee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#101;&#101;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834062168\">\n<p id=\"fs-id1167835621478\">A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> dollars each is given by the polynomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6216668447e588468d0bbabbd0ae2218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#48;&#112;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"164\" style=\"vertical-align: -4px;\" \/> Find the revenue received when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de3a9d564a6c87cb15721c1069bfdce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#49;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> dollars.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835327964\"><strong data-effect=\"bold\">Add and Subtract Polynomial Functions<\/strong><\/p>\n<p id=\"fs-id1167835327969\">In the following exercises, find <span class=\"token\">\u24d0<\/span> (<em data-effect=\"italics\">f<\/em> + <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>)\u2003<span class=\"token\">\u24d1<\/span> (<em data-effect=\"italics\">f<\/em> + <em data-effect=\"italics\">g<\/em>)(3)\u2003<span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">f<\/em> \u2212 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>)\u2003<span class=\"token\">\u24d3<\/span> (<em data-effect=\"italics\">f<\/em> \u2212 <em data-effect=\"italics\">g<\/em>)(\u22122)<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835361598\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835361600\">\n<p id=\"fs-id1167835361602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acbf6ca438f5dfd702f788104a12948d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff9c65d981a0d0d93e043fa491ede7e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834395618\">\n<p id=\"fs-id1167835318878\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f5c89ef16d2cd47a1b1384dd413d065_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#43;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca4bedb61b99d7db2c8ca1e370fa47b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#43;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-501e70e1d0eff53883d684b579c2575c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#45;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b3e47080ee835d91662d3b2544965ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#45;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826828950\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826828953\">\n<p id=\"fs-id1167826828955\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3954bfc63aff2318f231ec407ce43492_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"198\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c480631bde54ed4e73b2e6cf44cdfd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832042597\">\n<h4 data-type=\"title\"><a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb\" class=\"target-chapter\">Properties of Exponents and Scientific Notation<\/a><\/h4>\n<p id=\"fs-id1167827964080\"><strong data-effect=\"bold\">Simplify Expressions Using the Properties for Exponents<\/strong><\/p>\n<p id=\"fs-id1167827964086\">In the following exercises, simplify each expression using the properties for exponents.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826967130\">\n<div data-type=\"problem\" id=\"fs-id1167826967132\">\n<p id=\"fs-id1167826967134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5902c643bd6ba594b7568ea9c0c15083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#51;&#125;&middot;&#123;&#112;&#125;&#94;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834134987\">\n<p id=\"fs-id1167834134989\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03b426973a8166bf6128bdb8e7f8f9ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834135002\">\n<div data-type=\"problem\" id=\"fs-id1167835361742\">\n<p id=\"fs-id1167835361744\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad63af3c11debd2bbdab9ac46d4e282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#123;&#50;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828095825\">\n<div data-type=\"problem\" id=\"fs-id1167828095827\">\n<p id=\"fs-id1167828095829\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32b625e64113a0cc3820875c7784794c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&middot;&#123;&#97;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828349228\">\n<p id=\"fs-id1167834104663\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bdad4f9579f5c014c547da2e3fe127e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834104675\">\n<div data-type=\"problem\" id=\"fs-id1167834104677\">\n<p id=\"fs-id1167834104679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fd7d98d4aa0c238212ceb3966612f31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&middot;&#123;&#120;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834191995\">\n<div data-type=\"problem\" id=\"fs-id1167834191998\">\n<p id=\"fs-id1167834192000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e37e20cf097d348dbd4ea21e3a619f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#97;&#125;&middot;&#123;&#121;&#125;&#94;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826778556\">\n<p id=\"fs-id1167826778558\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1abe15e0f527ec0bc87814df755c9e88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#97;&#43;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828327170\">\n<div data-type=\"problem\" id=\"fs-id1167828327172\">\n<p id=\"fs-id1167828327174\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab312d11cf51d5980e3dc7677fa3d8b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194457\">\n<div data-type=\"problem\" id=\"fs-id1167834194459\">\n<p id=\"fs-id1167834194461\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-272445b0e381f9f7473d76623e69f15a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#54;&#125;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832226727\">\n<p id=\"fs-id1167832226729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bc4c53dcfc4f9f19b0cacff20967696_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537677\">\n<div data-type=\"problem\" id=\"fs-id1167834537679\">\n<p id=\"fs-id1167834537681\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-629798b30043d36be883c52de16f4366_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"22\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831223730\">\n<p id=\"fs-id1167831223732\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90c081e266e2b4755c3e09b89490dd48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834346682\">\n<p id=\"fs-id1167834346684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f9c29bed947a71788eb83df997df00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834346700\">\n<div data-type=\"problem\" id=\"fs-id1167830924075\">\n<p id=\"fs-id1167830924077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15f5671b3362f2067078fb12d95cd4da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830924095\">\n<div data-type=\"problem\" id=\"fs-id1167828095986\">\n<p id=\"fs-id1167828095988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2d99e7de2ffcd1cfcf5e78b5a123c16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828095999\">\n<p id=\"fs-id1167828096001\">1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834213951\">\n<div data-type=\"problem\" id=\"fs-id1167834213953\">\n<p id=\"fs-id1167834213955\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-936e45432c8bda142efda152d07a87a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830961462\">\n<div data-type=\"problem\" id=\"fs-id1167830961464\">\n<p id=\"fs-id1167830961466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-650d55ccbaea694eca89ffaf8a890e7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#97;&#125;&#94;&#123;&#48;&#125;&#45;&#49;&#53;&#123;&#98;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831894232\">\n<p id=\"fs-id1167831894234\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830964426\"><strong data-effect=\"bold\">Use the Definition of a Negative Exponent<\/strong><\/p>\n<p id=\"fs-id1167830964431\">In the following exercises, simplify each expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830964434\">\n<div data-type=\"problem\" id=\"fs-id1167830964436\">\n<p id=\"fs-id1167830964439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c967198fe70df2cea91d05de816ec72d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834225106\">\n<div data-type=\"problem\" id=\"fs-id1167834225108\">\n<p id=\"fs-id1167834225110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d3a3e9514583c7d905b39ee455a89f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834554992\">\n<p id=\"fs-id1167834554994\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46478fe938ca02376fbdb8199deb7001_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"43\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834431376\">\n<div data-type=\"problem\" id=\"fs-id1167834431378\">\n<p id=\"fs-id1167834431380\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbbcf887a842f1552fa426cd1f1a655c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#123;&#50;&#125;&#94;&#123;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831847169\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831847173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-055ad9b41f56ed954d33195e5e7550d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835382032\">\n<p id=\"fs-id1167835382035\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ce54bccab72c1645942feeda127ed44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"16\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834282473\">\n<div data-type=\"problem\" id=\"fs-id1167834282475\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f91f094c19a3a430faa27baf2cb7906f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115481\">\n<div data-type=\"problem\" id=\"fs-id1167831115483\">\n<p id=\"fs-id1167831115486\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea810a09e75c2035d30b334ec50e8ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831896558\">\n<p id=\"fs-id1167831896560\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fea2fccb14803fabd4b58d254a01be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831896573\">\n<div data-type=\"problem\" id=\"fs-id1167831896575\">\n<p id=\"fs-id1167831896577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4c1e2158094eae15e9cd67c49d15e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#45;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"24\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826995212\">\n<div data-type=\"problem\" id=\"fs-id1167826995214\">\n<p id=\"fs-id1167826995216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bf58c6956be94b4b753308807a96a68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#54;&#125;&#94;&#123;&#45;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831832832\">\n<p>36<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832840\">\n<div data-type=\"problem\" id=\"fs-id1167831832842\">\n<p id=\"fs-id1167831832844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c64c2693290586db292db275bdd66aa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#53;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831223770\">\n<div data-type=\"problem\" id=\"fs-id1167831223772\">\n<p id=\"fs-id1167831223774\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-feb7ffb920091f80e599eb6997adb506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"57\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826987682\">\n<p id=\"fs-id1167826987684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f7b496891eb276b3992ed20de9750b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826807787\">\n<div data-type=\"problem\" id=\"fs-id1167826807789\">\n<p id=\"fs-id1167826807791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-677bed327fecfbf8b622fc23b2d73d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835338425\">\n<div data-type=\"problem\" id=\"fs-id1167832149820\">\n<p id=\"fs-id1167832149822\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-178e1a9611887eb63e1a913fd1d4ce52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832149841\">\n<p id=\"fs-id1167828401314\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1de96f31c6c2bbd001766ad4d0631ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"36\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828401329\">\n<div data-type=\"problem\" id=\"fs-id1167828401331\">\n<p id=\"fs-id1167828401333\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7d292673aa619ee2036d64513219e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835364690\">\n<div data-type=\"problem\" id=\"fs-id1167835364693\">\n<p id=\"fs-id1167835364695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10daf8e76a24c051d7c0843e35376129_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"58\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832153903\">\n<p id=\"fs-id1167835317208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea105b44aa6cc89c64ecd514ba1baf7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835317225\">In the following exercises, simplify each expression using the Product Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835317228\">\n<div data-type=\"problem\" id=\"fs-id1167834093160\">\n<p id=\"fs-id1167834093162\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84be71e83ce4933cab40f2337a99e6a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"38\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832066220\">\n<div data-type=\"problem\" id=\"fs-id1167832066222\">\n<p id=\"fs-id1167832066224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0c635221320d8a1410f650a062b4d78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"38\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831892980\">\n<p id=\"fs-id1167831892982\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa90cabf6a6131c77a7a5b0596d06a29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826801765\">\n<div data-type=\"problem\" id=\"fs-id1167826801767\">\n<p id=\"fs-id1167826801770\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef04ab4f287f0526c3efa83bc37e2c37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"45\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831969943\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831969947\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fd401d4457d2f25f02fcde97c19bff6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#45;&#51;&#125;&middot;&#123;&#120;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831228516\">\n<p id=\"fs-id1167831228519\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48541f82788d9b97a870606bd6257cd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835347942\">\n<div data-type=\"problem\" id=\"fs-id1167835347945\">\n<p id=\"fs-id1167835347947\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-740517094051ad2c43fbe3eee875f386_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#45;&#53;&#125;&middot;&#123;&#114;&#125;&#94;&#123;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832151156\">\n<div data-type=\"problem\" id=\"fs-id1167832151158\">\n<p id=\"fs-id1167832151160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c7d43fc29630b6e07d601518c65bf02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#123;&#118;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#117;&#125;&#94;&#123;&#45;&#52;&#125;&#123;&#118;&#125;&#94;&#123;&#45;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834227739\">\n<p id=\"fs-id1167828327038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ea6962625202560f0351da4698edd07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#123;&#118;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"29\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828327059\">\n<div data-type=\"problem\" id=\"fs-id1167830704814\">\n<p id=\"fs-id1167830704816\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a4a19d2d0ab525288199fe9d943e0ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"55\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987508\">\n<div data-type=\"problem\" id=\"fs-id1167826987510\">\n<p id=\"fs-id1167826987512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5383da826e6ea2a71ed96b47687c93f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#53;&#125;&middot;&#123;&#112;&#125;&#94;&#123;&#45;&#50;&#125;&middot;&#123;&#112;&#125;&#94;&#123;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831824568\">\n<p id=\"fs-id1167831824570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602a31ed140e67f6e0cbf7e0dfb4afd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"19\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831824586\">In the following exercises, simplify each expression using the Power Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831824589\">\n<div data-type=\"problem\" id=\"fs-id1167834429238\">\n<p id=\"fs-id1167834429240\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef32f79286431354b64f5af506251bf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#107;&#125;&#94;&#123;&#45;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575829\">\n<div data-type=\"problem\" id=\"fs-id1167835575831\">\n<p id=\"fs-id1167835575833\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d4c58f075ce68db25474bc407ebf28a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#113;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"20\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834523693\">\n<p id=\"fs-id1167834523695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e79ddeaf9ce361c85d439bae051cf98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"20\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966574\">\n<div data-type=\"problem\" id=\"fs-id1167827966577\">\n<p id=\"fs-id1167827966579\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c667b3b0664f353ddb0142c146c8382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#45;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"22\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834099008\">\n<div data-type=\"problem\" id=\"fs-id1167834099010\">\n<p id=\"fs-id1167834099012\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9ab6b2f1a2eece9d3763a51ba22a8e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#45;&#51;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#45;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832185134\">\n<p id=\"fs-id1167832185137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-444500c4e43e38956dd248f19f2f7129_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832185148\">In the following exercises, simplify each expression using the Product to a Power Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834472872\">\n<div data-type=\"problem\" id=\"fs-id1167834472874\">\n<p id=\"fs-id1167834472876\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc8bb768eb7e0ba6de33e2c7b2e512e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#97;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537063\">\n<div data-type=\"problem\" id=\"fs-id1167834537065\">\n<p id=\"fs-id1167834537067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfa2edb35aeb35e64d9db3849602bfda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#112;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831833314\">\n<p id=\"fs-id1167831833316\">1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831833321\">\n<div data-type=\"problem\" id=\"fs-id1167831833323\">\n<p id=\"fs-id1167831833325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de1e762e5f3505bf0e95ff7814140291_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834402901\">\n<div data-type=\"problem\" id=\"fs-id1167834402903\">\n<p id=\"fs-id1167834402905\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-050311fe1eb919a37ad0fb44bcf92176_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#121;&#125;&#94;&#123;&#45;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"58\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834423591\">\n<p id=\"fs-id1167834423593\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-140bb182a2240caa3937e27d4bd2c380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"15\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834526657\">In the following exercises, simplify each expression using the Quotient to a Power Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834526660\">\n<div data-type=\"problem\" id=\"fs-id1167834526662\">\n<p id=\"fs-id1167834526664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd4a21b51bd81b57d56b2ee28f094f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"51\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533423\">\n<div data-type=\"problem\" id=\"fs-id1167834533425\">\n<p id=\"fs-id1167835311756\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cf243bd854eb74f98d8b7b0872d0fd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#122;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"58\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834527540\">\n<p id=\"fs-id1167834527543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a80b60bcfccb9582a6128bc432eec5c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#49;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835281638\">\n<div data-type=\"problem\" id=\"fs-id1167835281640\">\n<p id=\"fs-id1167835281642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20119b6b05890f2e8732a7a870ea1331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#112;&#125;&#94;&#123;&#45;&#51;&#125;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"59\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832092251\">In the following exercises, simplify each expression by applying several properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832092254\">\n<div data-type=\"problem\" id=\"fs-id1167832092256\">\n<p id=\"fs-id1167832092258\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02d002acd980436bd7ec24ec4d621566_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"108\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834525167\">\n<p id=\"fs-id1167834525170\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c28bea3055a755539f23049371643d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#55;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826828297\">\n<div data-type=\"problem\" id=\"fs-id1167826828299\">\n<p id=\"fs-id1167826828301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a49e441687f402f675712f200d5a462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#123;&#97;&#125;&#94;&#123;&#45;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"103\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830698714\">\n<div data-type=\"problem\" id=\"fs-id1167830698716\">\n<p id=\"fs-id1167830698718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f51a99815964fee8408ba460463bf26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#120;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"161\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834226021\">\n<p id=\"fs-id1167834226023\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23487bbad0f14910e77477ce15ad754f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"22\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831921311\">In the following exercises, write each number in scientific notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831921315\">\n<div data-type=\"problem\" id=\"fs-id1167831921317\">\n<p id=\"fs-id1167831921319\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2b83530180caa4db5ae20c8027fb8ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#54;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831910363\">\n<div data-type=\"problem\" id=\"fs-id1167831910365\">\n<p id=\"fs-id1167831910367\">5,300,000<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832150134\">\n<p id=\"fs-id1167832150136\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a94433569b3e05cd0d9a094e9face0a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832150158\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832153411\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-715b9b55b74d4062267babdd24cf0501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#48;&#56;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826926708\">In the following exercises, convert each number to decimal form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826926712\">\n<div data-type=\"problem\" id=\"fs-id1167826926714\">\n<p id=\"fs-id1167826926716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8273bd04e6aeadde4388f7ece560a94e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834422979\">\n<p id=\"fs-id1167834422981\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44a5f73682da2413bec4cdc8881fa9e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#57;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834462078\">\n<div data-type=\"problem\" id=\"fs-id1167834462080\">\n<p id=\"fs-id1167834462082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c6232b37e23da9e6ac6f7b91ba5099_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830963329\">\n<div data-type=\"problem\" id=\"fs-id1167830963331\">\n<p id=\"fs-id1167830963333\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52bf7a5e880c8902b7c376f9ab5d894e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#46;&#52;&#49;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834060992\">\n<p id=\"fs-id1167834060994\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5db7e8b6ccaf8276084631ce3cd11c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#48;&#48;&#48;&#57;&#52;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"86\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832057327\">In the following exercises, multiply or divide as indicated. Write your answer in decimal form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832057332\">\n<div data-type=\"problem\" id=\"fs-id1167832057334\">\n<p id=\"fs-id1167832057336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f22d04eaa0f49faccf546bac640abe54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830703370\">\n<div data-type=\"problem\" id=\"fs-id1167828094930\">\n<p id=\"fs-id1167828094932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfd8d1741c69ba15c963e9c516e7e022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826932451\">\n<p id=\"fs-id1167826932453\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-970e9719da2907c02a944585398adb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#48;&#48;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826932461\">\n<div data-type=\"problem\" id=\"fs-id1167826932464\">\n<p id=\"fs-id1167826932466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fec4e36b7c0c4c1f094e5e711532e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#57;&#125;&#125;&#123;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"44\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826804661\">\n<div data-type=\"problem\" id=\"fs-id1167826804663\">\n<p id=\"fs-id1167826804665\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0dd95b30cf02b39331d52b9f2d1b453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#51;&#125;&#125;&#123;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"44\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831909571\">\n<p id=\"fs-id1167831909573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcf2d77a88b5b3c75869db32b4929b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831909582\">\n<h4 data-type=\"title\"><a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c\" class=\"target-chapter\">Multiply Polynomials<\/a><\/h4>\n<p><strong data-effect=\"bold\">Multiply Monomials<\/strong><\/p>\n<p id=\"fs-id1167835281070\">In the following exercises, multiply the monomials.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835281074\">\n<div data-type=\"problem\" id=\"fs-id1167835281076\">\n<p id=\"fs-id1167835281078\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edbfcb7ede34bbebe79da95f84c33a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#112;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#112;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834309057\">\n<div data-type=\"problem\" id=\"fs-id1167834309059\">\n<p id=\"fs-id1167834309061\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-807a6a5e055b73738942817d3a8e6938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#48;&#123;&#99;&#125;&#94;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834372718\">\n<p id=\"fs-id1167820628265\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f06ed6366a0e197b925e9855fe0fdcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#99;&#125;&#94;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167820628280\">\n<div data-type=\"problem\" id=\"fs-id1167820628282\">\n<p id=\"fs-id1167820628284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93af6ea2d5329f5746c2ae43cbf499dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#120;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835281800\">\n<div data-type=\"problem\" id=\"fs-id1167835281802\">\n<p id=\"fs-id1167835281804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdbc13fcb10415180c19592564495468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831882523\">\n<p id=\"fs-id1167834440810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44fd9a7731ba157c53599c3d1073fbd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#109;&#125;&#94;&#123;&#55;&#125;&#123;&#110;&#125;&#94;&#123;&#49;&#48;&#125;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834440833\"><strong data-effect=\"bold\">Multiply a Polynomial by a Monomial<\/strong><\/p>\n<p id=\"fs-id1167835510599\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835510602\">\n<div data-type=\"problem\" id=\"fs-id1167835510604\">\n<p id=\"fs-id1167835510606\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db62802d0b7c407f061f1fbcb72ebc1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997544\">\n<div data-type=\"problem\" id=\"fs-id1167826997546\">\n<p id=\"fs-id1167826997548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a085c43d5b8c00cd1c11d43cfa3fd37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#97;&#45;&#51;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"131\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228231\">\n<p id=\"fs-id1167834228233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e916d3edb10be1ea5deaea47b6d77ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"120\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832128806\">\n<div data-type=\"problem\" id=\"fs-id1167832128808\">\n<p id=\"fs-id1167832128810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc64c21974ab3dc6fe98e9106f418a1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"122\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835352743\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0a86fa5933d409f902077f6a1cf20d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#110;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826870791\">\n<p id=\"fs-id1167826870793\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39b462a0606a411f7a4f8d87b456ba03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#48;&#123;&#110;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p><strong data-effect=\"bold\">Multiply a Binomial by a Binomial<\/strong><\/p>\n<p id=\"fs-id1167826967168\">In the following exercises, multiply the binomials using:<\/p>\n<p id=\"fs-id1167826967171\"><span class=\"token\">\u24d0<\/span> the Distributive Property <span class=\"token\">\u24d1<\/span> the FOIL method <span class=\"token\">\u24d2<\/span> the Vertical Method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826779585\">\n<div data-type=\"problem\" id=\"fs-id1167826779587\">\n<p id=\"fs-id1167826779589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da21a549fde5860ab2e40f60524ace2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922866\">\n<div data-type=\"problem\" id=\"fs-id1167831922868\">\n<p id=\"fs-id1167831922870\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b1502f358794c5dd36089cb111e1e32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834422466\">\n<p id=\"fs-id1167834422468\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7acfca2b8cec0636b4a23bee27d854ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#45;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831071328\">\n<div data-type=\"problem\" id=\"fs-id1167831071330\">\n<p id=\"fs-id1167831071332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efce70e8962d9414d54e67fd975f3e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834132827\">\n<div data-type=\"problem\" id=\"fs-id1167834132829\">\n<p id=\"fs-id1167834132831\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f7cada8f241c6e3ddf58844759cabb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#112;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831928882\">\n<p id=\"fs-id1167831040265\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c2093cef0ee4b97591ac790df1390a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#51;&#112;&#43;&#49;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831040290\">In the following exercises, multiply the binomials. Use any method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835281740\">\n<div data-type=\"problem\" id=\"fs-id1167835281742\">\n<p id=\"fs-id1167835281745\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b6b05c992b319a0e31085f47f54be5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834593082\">\n<div data-type=\"problem\" id=\"fs-id1167834593084\">\n<p id=\"fs-id1167834593086\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88ba90571718eeab48eed304f7a445a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826804090\">\n<p id=\"fs-id1167826804092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c60c0638b93f27a41bdc11e83413b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#107;&#45;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831040418\">\n<div data-type=\"problem\" id=\"fs-id1167831040421\">\n<p id=\"fs-id1167831040423\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04035792dfe64463ddd8d477971daa6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#117;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064164\">\n<div data-type=\"problem\" id=\"fs-id1167834064166\">\n<p id=\"fs-id1167834064168\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7a7b4faf3b4fe71a6b5a6d0a849995c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834439047\">\n<p id=\"fs-id1167834439049\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beb58da7424318bc9bed8ee56b6edc38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#57;&#121;&#43;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826780810\">\n<div data-type=\"problem\" id=\"fs-id1167826780812\">\n<p id=\"fs-id1167826780814\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-988366a49da22508e85b8370b6ba2db0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830838103\">\n<div data-type=\"problem\" id=\"fs-id1167830838105\">\n<p id=\"fs-id1167830838107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-161c8cdecc50228d64da1c6868c77da4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834229179\">\n<p id=\"fs-id1167834229181\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7022637c1ab8fd1dddaf7373890846e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832150075\">\n<div data-type=\"problem\" id=\"fs-id1167832150077\">\n<p id=\"fs-id1167832150079\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40a764691ac167bd9dad0cde9efdb335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#99;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834094072\">\n<div data-type=\"problem\" id=\"fs-id1167834094074\">\n<p id=\"fs-id1167834094077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f29aa59bd2b4a875c7f418323caaa7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#97;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832076252\">\n<p id=\"fs-id1167832076254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-153dd608cda398da6b8acd4328bc4df7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#51;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835517887\"><strong data-effect=\"bold\">Multiply a Polynomial by a Polynomial<\/strong><\/p>\n<p id=\"fs-id1167835517893\">In the following exercises, multiply using <span class=\"token\">\u24d0<\/span> the Distributive Property <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835517904\">\n<div data-type=\"problem\" id=\"fs-id1167835517906\">\n<p id=\"fs-id1167835517909\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2d7dfb8fae6b7340057d2bffb90344_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115525\">\n<div data-type=\"problem\" id=\"fs-id1167831115527\">\n<p id=\"fs-id1167831115529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fcb74e5d7e2fb13afb6971669df5085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#98;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834463251\">\n<p id=\"fs-id1167834463253\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a65cfdbad574c94caba22d5f3f6e0ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#55;&#98;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831835374\">In the following exercises, multiply. Use either method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835413972\">\n<div data-type=\"problem\" id=\"fs-id1167835413974\">\n<p id=\"fs-id1167835413976\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb4c5a929ce65c1d1f8b1d13225764c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#109;&#45;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"185\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834527258\">\n<div data-type=\"problem\" id=\"fs-id1167834527260\">\n<p id=\"fs-id1167834527263\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ff943513d3e5d9772d35f69d2321509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"184\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834058955\">\n<p id=\"fs-id1167834058957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5575cfc637bf6cf10fd66df1782f94bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#121;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832185177\"><strong data-effect=\"bold\">Multiply Special Products<\/strong><\/p>\n<p id=\"fs-id1167832185182\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832185185\">\n<div data-type=\"problem\" id=\"fs-id1167832185187\">\n<p id=\"fs-id1167832185190\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1811a7f265fa80465a90efe826327f2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834137607\">\n<div data-type=\"problem\" id=\"fs-id1167834137609\">\n<p id=\"fs-id1167834137611\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e4b1145af70ca10b0ca16b762b019de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830769593\">\n<p id=\"fs-id1167830769596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a0d4f9db06649daf0a5f74a3897c38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828096031\">\n<div data-type=\"problem\" id=\"fs-id1167828096033\">\n<p id=\"fs-id1167828096035\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16d7dd393da3a4820532fe51ba3f69fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"77\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834373426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-591dcd5e269207315cc651dacb880266_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#112;&#43;&#55;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835410079\">\n<p id=\"fs-id1167835410082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1eb9bb9c3ece0e50e8a4ad7874de083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#48;&#112;&#113;&#43;&#52;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834120554\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834535761\">\n<div data-type=\"problem\" id=\"fs-id1167834535764\">\n<p id=\"fs-id1167834535766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1e69f5ec76facfafa059202366be77f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834141253\">\n<div data-type=\"problem\" id=\"fs-id1167834141255\">\n<p id=\"fs-id1167834141257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adfb84373dd83fe6703f67a612e6dc70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835311479\">\n<p id=\"fs-id1167835311481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8c1949847d973e17f09afc7afbb5851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834515452\">\n<div data-type=\"problem\" id=\"fs-id1167834515454\">\n<p id=\"fs-id1167834515456\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2627f2094c88cfa701130eba74ddf6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"131\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301973\">\n<div data-type=\"problem\" id=\"fs-id1167835301976\">\n<p id=\"fs-id1167835301978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeec3de5c1949db45e0081ff3964f745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"198\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828095962\">\n<p id=\"fs-id1167828095964\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bba2eaa162d6ff3b566dd52e2044bb5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#54;&#125;&#45;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834461813\">\n<div data-type=\"problem\" id=\"fs-id1167834461815\">\n<p id=\"fs-id1167834461817\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a2fc95a2bc5bed013370883806aed7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"193\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834462484\">\n<h4 data-type=\"title\"><a href=\"\/contents\/4102cdd7-8da4-4802-83c7-342d2f3cd3f5\" class=\"target-chapter\">Divide Monomials<\/a><\/h4>\n<p id=\"fs-id1167831919324\"><strong data-effect=\"bold\">Divide Monomials<\/strong><\/p>\n<p id=\"fs-id1167831919330\">In the following exercises, divide the monomials.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831919334\">\n<div data-type=\"problem\" id=\"fs-id1167831919336\">\n<p id=\"fs-id1167831919338\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f06012cdd3dfa1d9ac1c43249669b173_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#123;&#112;&#125;&#94;&#123;&#49;&#50;&#125;&divide;&#56;&#123;&#112;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835416860\">\n<p id=\"fs-id1167835416862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58c1af4d84eb6759c06597f9860b0fea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#112;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835416876\">\n<div data-type=\"problem\" id=\"fs-id1167835416878\">\n<p id=\"fs-id1167835416880\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f06c946c7bcce9c2f8b60d919592e690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#54;&#123;&#97;&#125;&#94;&#123;&#56;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799010\">\n<div data-type=\"problem\" id=\"fs-id1167826799012\">\n<p id=\"fs-id1167826799014\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e1e84ee305db5f37dac333a70d1fbbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#53;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#123;&#45;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"45\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835389999\">\n<p id=\"fs-id1167835390001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506b19ab825fef189164b182f96654eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"30\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835390019\">\n<div data-type=\"problem\" id=\"fs-id1167835390021\">\n<p id=\"fs-id1167835390023\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea61c46e489f843d4040d85ce75e0052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#45;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"39\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832076183\">\n<div data-type=\"problem\" id=\"fs-id1167832076185\">\n<p id=\"fs-id1167832076187\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baec1120592574fa2afa66fdc37d4272_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#56;&#123;&#97;&#125;&#94;&#123;&#57;&#125;&#98;&#125;&#123;&#55;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"34\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831824823\">\n<p id=\"fs-id1167831824826\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12f7685e2e4f6d94849c66346943dff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831824848\">\n<div data-type=\"problem\" id=\"fs-id1167831824850\">\n<p id=\"fs-id1167831824852\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f7e865f934e84b7f39413f356901863_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#123;&#117;&#125;&#94;&#123;&#54;&#125;&#123;&#118;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#53;&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"43\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828096067\">\n<div data-type=\"problem\" id=\"fs-id1167828096069\">\n<p id=\"fs-id1167828096071\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d73554e0121e2a346c2405d433e49d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#109;&#125;&#94;&#123;&#57;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#109;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"110\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834469637\">\n<p id=\"fs-id1167834469639\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f36f74b9e245af01e25d09c55532222e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#109;&#125;&#94;&#123;&#57;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834469661\">\n<div data-type=\"problem\" id=\"fs-id1167834469663\">\n<p id=\"fs-id1167835517726\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-151ada7de195c221a1906423584a934a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#50;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#52;&#114;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#114;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"102\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831969743\"><strong data-effect=\"bold\">Divide a Polynomial by a Monomial<\/strong><\/p>\n<p id=\"fs-id1167831890813\">In the following exercises, divide each polynomial by the monomial<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831890816\">\n<div data-type=\"problem\" id=\"fs-id1167831890819\">\n<p id=\"fs-id1167831890821\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5c47c73e135fbfc670cb98ed1917f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#52;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#50;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828095649\">\n<p id=\"fs-id1167828095651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6be24e973770d972cf4561302edd3d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832052514\">\n<div data-type=\"problem\" id=\"fs-id1167832052516\">\n<p id=\"fs-id1167832052518\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3bfd78d4693fbfe4840066d734f1a3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#53;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"149\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832042166\">\n<div data-type=\"problem\" id=\"fs-id1167832042168\">\n<p id=\"fs-id1167832042170\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b51608cf8d76f50fe7162b758cc6ffd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#51;&#125;&#123;&#45;&#52;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832042416\">\n<p id=\"fs-id1167832042419\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96da264caa134b83f50a117d25059b17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#120;&#45;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"101\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832042444\"><strong data-effect=\"bold\">Divide Polynomials using Long Division<\/strong><\/p>\n<p id=\"fs-id1167832042450\">In the following exercises, divide each polynomial by the binomial.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167828095056\">\n<div data-type=\"problem\" id=\"fs-id1167828095058\">\n<p id=\"fs-id1167828095060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23e8472260feee80187b062ee241011f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#49;&#120;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"189\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828095890\">\n<div data-type=\"problem\" id=\"fs-id1167828095892\">\n<p id=\"fs-id1167828095894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cf7711b5ad6638e4fa826bf8b8e8a31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#43;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832144287\">\n<p id=\"fs-id1167832144289\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c456893edf44eb98a38c6f3635f84b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#45;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#113;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832144316\">\n<div data-type=\"problem\" id=\"fs-id1167831023258\">\n<p id=\"fs-id1167831023260\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af6565dc9aedcaa7e74490dade25681b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#110;&#43;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"222\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067631\">\n<div data-type=\"problem\" id=\"fs-id1167832067633\">\n<p id=\"fs-id1167832067635\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d109325e26cf1f3419ba38bf46c1b747_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505002\">\n<p id=\"fs-id1167834505005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-828f71e4ea5ab23e6942de3c86e91a68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834505025\"><strong data-effect=\"bold\">Divide Polynomials using Synthetic Division<\/strong><\/p>\n<p id=\"fs-id1167834505030\"><em data-effect=\"italics\">In the following exercises<\/em>, use synthetic Division to find the quotient and remainder.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834535981\">\n<div data-type=\"problem\" id=\"fs-id1167834535983\">\n<p id=\"fs-id1167834535985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b619c20476e1d34958da2cdebb360cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826828663\">\n<div data-type=\"problem\" id=\"fs-id1167826828665\">\n<p id=\"fs-id1167826828667\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c1e444fd47c0efa07e682eed398b9b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"172\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828409726\">\n<p id=\"fs-id1167828409728\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78edc652e93200bf898de6a115e78a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830697224\">\n<div data-type=\"problem\" id=\"fs-id1167830697226\">\n<p id=\"fs-id1167830697228\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f43ed3159f806c9458dc22259d054d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167827966958\"><strong data-effect=\"bold\">Divide Polynomial Functions<\/strong><\/p>\n<p id=\"fs-id1167827966964\">In the following exercises, divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167827966968\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827966970\">\n<p id=\"fs-id1167827966972\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe27bcf6f6b977e4cd5c96c11d48a659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#120;&#43;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e76e2f980c4c84f953587ba94140a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-019d10c37202237db99ce34c0f070e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167826829004\">\n<p id=\"fs-id1167826829006\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ec3166c7642e64f4521a246862addb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"121\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f6d31cb175d580a41567e23c603f101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"116\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835498376\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835498378\">\n<p id=\"fs-id1167835498380\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d4b3c18213548eb1a6f9fe942975e28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"189\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-def2c9f1007422c50cedbae34f85a27f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce350cf172ba8cfc9e378ea528592ed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835526589\"><strong data-effect=\"bold\">Use the Remainder and Factor Theorem<\/strong><\/p>\n<p id=\"fs-id1167835526596\">In the following exercises, use the Remainder Theorem to find the remainder.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835526599\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835526601\">\n<p id=\"fs-id1167835526603\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9b9f262a020c8f5b32ee5a347cb197c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/> is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834555205\">\n<p id=\"fs-id1167834555207\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835468472\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835468474\">\n<p id=\"fs-id1167835468476\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7767e4ab46fad8049c168a00a7515729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#120;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/> divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167828240690\">In the following exercises, use the Factor Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e8142979189afc6c0de298a8169ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of the polynomial function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167828240703\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828240705\">\n<p id=\"fs-id1167828240707\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79c51dce88ed8deaaab9733f311dd7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a8d1d3329a1f9a77998051fa7fb11ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835530790\">\n<p id=\"fs-id1167835530792\">no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835530798\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835530800\">\n<p id=\"fs-id1167835530802\">Determine whether <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-193c295d8fc62eac0984aa4fc668bf9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> is a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af091a1d73057f65807d15f36f171843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167828325874\">\n<h3 data-type=\"title\">Chapter Practice Test<\/h3>\n<div data-type=\"exercise\" id=\"fs-id1167828325880\">\n<div data-type=\"problem\" id=\"fs-id1167828325882\">\n<p id=\"fs-id1167828325884\">For the polynomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd0ea3c1669cec6b0f1849374258b445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167828325911\"><span class=\"token\">\u24d0<\/span> Is it a monomial, binomial, or trinomial? <span class=\"token\">\u24d1<\/span> What is its degree?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834461865\">\n<p id=\"fs-id1167834461868\"><span class=\"token\">\u24d0<\/span> trinomial <span class=\"token\">\u24d1<\/span> 4<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834461881\">\n<div data-type=\"problem\" id=\"fs-id1167834461883\">\n<p id=\"fs-id1167834461886\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80244279a8116cb5cf0f5773ea91ee82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#97;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"233\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827957306\">\n<div data-type=\"problem\" id=\"fs-id1167827957308\">\n<p id=\"fs-id1167827957310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff83b074a73dec7022fd16dc4126d9e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"214\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834152035\">\n<p id=\"fs-id1167834152037\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f4919a5a2aa1731f2cbe2ed5abc11c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830959530\">\n<div data-type=\"problem\" id=\"fs-id1167830959533\">\n<p id=\"fs-id1167830959535\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81d9ad5205722420cf3d85d5ded8db0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830757432\">\n<div data-type=\"problem\" id=\"fs-id1167830757435\">\n<p id=\"fs-id1167830757437\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-647de174059532dcbcbc42788a1aa454_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#45;&#51;&#125;&#123;&#120;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830757454\">\n<p id=\"fs-id1167830757456\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834494725\">\n<div data-type=\"problem\" id=\"fs-id1167834494727\">\n<p id=\"fs-id1167834494729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f25f85373ccb7441abdb02a8c69260ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#53;&#125;&#94;&#123;&#54;&#125;&#125;&#123;&#123;&#53;&#125;&#94;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834494763\">\n<div data-type=\"problem\" id=\"fs-id1167834494766\">\n<p id=\"fs-id1167834494768\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae8da867d156e7c8096d56deed2f633b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#55;&#123;&#97;&#125;&#94;&#123;&#49;&#56;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#51;&#125;&#123;&#99;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834527804\">\n<p id=\"fs-id1167834527806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834527813\">\n<div data-type=\"problem\" id=\"fs-id1167835609183\">\n<p id=\"fs-id1167835609185\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57506436cedcee120ececd109bc26d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835609210\">\n<div data-type=\"problem\" id=\"fs-id1167835609212\">\n<p id=\"fs-id1167835609215\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a213a568d9c3139855d2ab79b5dc2732_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827897706\">\n<p id=\"fs-id1167827897708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55cd1c69e9cc2afa30da5887bfd64adb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"21\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827897726\">\n<div data-type=\"problem\" id=\"fs-id1167827897728\">\n<p id=\"fs-id1167827897730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b178ac41cc253c0b33ecb3d76f8696f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#45;&#51;&#125;&middot;&#123;&#112;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831891050\">\n<div data-type=\"problem\" id=\"fs-id1167831891052\">\n<p id=\"fs-id1167831891055\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1177d4af85d130935d129e5c615d1e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#45;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"24\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831224091\">\n<p id=\"fs-id1167831224093\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49858b69835c5f5c30348b8212823170_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831224105\">\n<div data-type=\"problem\" id=\"fs-id1167831224107\">\n<p id=\"fs-id1167831224109\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-481a9324bfb7dab1e7886ead3a8ebca5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"59\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834517697\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834517702\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cba73399bd80f148788ee18d6e9ef5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#115;&#125;&#123;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"34\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834097856\">\n<p id=\"fs-id1167834097859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6753b931c3ab325d52f28343ce3b1298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#114;&#125;&#123;&#123;&#115;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834097878\">\n<div data-type=\"problem\" id=\"fs-id1167834097880\">\n<p id=\"fs-id1167834097882\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6566681bc510a2836b0450e90ee3854c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#57;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#45;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"58\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834461621\">\n<div data-type=\"problem\" id=\"fs-id1167834461623\">\n<p id=\"fs-id1167835416122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1a892973fc8f39582b4d71eea03e63b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835416162\">\n<p id=\"fs-id1167835416165\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1feb4f114d3adc73b37ee5805f9776d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831023446\">\n<div data-type=\"problem\" id=\"fs-id1167831023448\">\n<p id=\"fs-id1167831023450\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84eb5903246b7d1b8f396bdb1ea73c7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#117;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#117;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828409666\">\n<div data-type=\"problem\" id=\"fs-id1167828409668\">\n<p id=\"fs-id1167828409670\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cd30eb754b6517b6f8d4853dda95baa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#109;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826813806\">\n<p id=\"fs-id1167826813809\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78fbe8baa7731cf88aed2de24753e7e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#109;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835498878\">\n<div data-type=\"problem\" id=\"fs-id1167835498880\">\n<p id=\"fs-id1167835498882\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cbc1749cab8543518543c8af315221f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#110;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"170\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828095700\">\n<div data-type=\"problem\" id=\"fs-id1167828095703\">\n<p id=\"fs-id1167828095705\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92a4df34bf1588be071b74136895c1dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830960556\">\n<p id=\"fs-id1167830960558\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3977be9b190e54123a631c9ef2f9813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"115\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830960583\">\n<div data-type=\"problem\" id=\"fs-id1167830960585\">\n<p id=\"fs-id1167830960587\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-692378f11a5f458919e31ed00812db55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831909422\">\n<div data-type=\"problem\" id=\"fs-id1167831909424\">\n<p id=\"fs-id1167831909426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b774b0f123e7b4023c11b1a2149c88ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#53;&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832133966\">\n<p id=\"fs-id1167832133968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1323fac961275cacbdd62b6c50664e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832133988\">\n<div data-type=\"problem\" id=\"fs-id1167832133990\">\n<p id=\"fs-id1167832133993\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b1104f32255fd999c79a9bcf492fc49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"247\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826879986\">\n<div data-type=\"problem\" id=\"fs-id1167826879989\">\n<p id=\"fs-id1167826879991\">Use the Factor Theorem to determine if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46f0e3d6dc9b5acf33c3a986b9d92d4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#120;&#43;&#49;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"159\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831224786\">\n<p id=\"fs-id1167831224788\">yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831224794\">\n<div data-type=\"problem\" id=\"fs-id1167831224796\">\n<p id=\"fs-id1167831224798\"><span class=\"token\">\u24d0<\/span> Convert 112,000 to scientific notation. <span class=\"token\">\u24d1<\/span> Convert <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22a969c037d6ca78d1e5ba2bee616c79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#50;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/> to decimal form.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831882323\">In the following exercises, simplify and write your answer in decimal form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831882326\">\n<div data-type=\"problem\" id=\"fs-id1167831882329\">\n<p id=\"fs-id1167831882331\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82fff7bd9c37e6b7ca61b42e8352bff8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228714\">\n<p id=\"fs-id1167834228716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ecc7d5afa236d000a621f813394256f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834228739\">\n<div data-type=\"problem\" id=\"fs-id1167834228741\">\n<p id=\"fs-id1167834228743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-467847c2c7dec9ef0f5815f0c6488332_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"44\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835496126\">\n<div data-type=\"problem\" id=\"fs-id1167835496128\">\n<p id=\"fs-id1167835496130\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25369390ece0884848056c09cfcb9887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7454c1320ac593411a470ad12380405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dca5d7bc047aaa23a5ac85a2c257c5f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d73c012bc8b5a6f560bea30840502b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834525293\">\n<p id=\"fs-id1167834525295\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dce5a8d8a80b6aa9c2a9f6810851e582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b44aa122f73c22d86f2d764af01f919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834525318\">\n<div data-type=\"problem\" id=\"fs-id1167834525320\">\n<p id=\"fs-id1167834525322\">For <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f310ef35811740083935c8f0e8aa0f77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78f40c57bd3d3648de760f91c5e59186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1f7743bc12c43eb246486a8c39b5fbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#43;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cda3ec5fbc469c83c21b76f544d99e7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#43;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f66316491cb564a03a7edaa9e6d0419a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#45;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd8c6252be4b1680201a71f7d1706b2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#45;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831893243\">\n<div data-type=\"problem\" id=\"fs-id1167831893245\">\n<p id=\"fs-id1167831893247\">For functions<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f11446d4d3c244e5a23b5b747f1cbbce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#51;&#120;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e76e2f980c4c84f953587ba94140a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e6f0b73b90c7be62901c2c99c864d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"56\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce350cf172ba8cfc9e378ea528592ed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167828094875\">\n<p id=\"fs-id1167828094877\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31e78636294c26ae0123660de4fb1084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"130\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0e2b984169e9c39a1d657e8154c173f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"97\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834505309\">\n<div data-type=\"problem\" id=\"fs-id1167834505311\">\n<p id=\"fs-id1167834505313\">A hiker drops a pebble from a bridge 240 feet above a canyon. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-047e5c7662f02305fd1b774edffb46b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -4px;\" \/> gives the height of the pebble <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> seconds after it was dropped. Find the height when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-503c63895ae205ca9f57cca3112d8218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2961","chapter","type-chapter","status-publish","hentry"],"part":2727,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2961","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2961\/revisions"}],"predecessor-version":[{"id":15244,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2961\/revisions\/15244"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2727"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2961\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2961"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2961"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2961"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}