{"id":2920,"date":"2018-12-11T13:46:40","date_gmt":"2018-12-11T18:46:40","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/multiply-polynomials\/"},"modified":"2018-12-11T13:46:40","modified_gmt":"2018-12-11T18:46:40","slug":"multiply-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/multiply-polynomials\/","title":{"raw":"Multiply Polynomials","rendered":"Multiply 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>Multiply monomials<\/li><li>Multiply a polynomial by a monomial<\/li><li>Multiply a binomial by a binomial<\/li><li>Multiply a polynomial by a polynomial<\/li><li>Multiply special products<\/li><li>Multiply polynomial functions<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836299778\" class=\"be-prepared\"><p id=\"fs-id1167829624120\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167833196772\" type=\"1\"><li>Distribute: \\(2\\left(x+3\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829789060\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: <span class=\"token\">\u24d0<\/span> \\({9}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({\\left(-9\\right)}^{2}\\) <span class=\"token\">\u24d2<\/span> \\(\\text{\u2212}{9}^{2}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate: \\(2{x}^{2}-5x+3\\) for \\(x=-2.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833303374\"><h3 data-type=\"title\">Multiply Monomials<\/h3><p id=\"fs-id1167833021539\">We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.<\/p><div data-type=\"example\" id=\"fs-id1167829788772\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836525743\"><div data-type=\"problem\" id=\"fs-id1167833387010\"><p id=\"fs-id1167833022571\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(3{x}^{2}\\right)\\left(-4{x}^{3}\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{5}{6}{x}^{3}y\\right)\\left(12x{y}^{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836520121\"><p id=\"fs-id1167836300221\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(3{x}^{2}\\right)\\left(-4{x}^{3}\\right)\\hfill \\\\ \\text{Use the Commutative Property to rearrange the terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}3\u00b7\\left(-4\\right)\u00b7{x}^{2}\u00b7{x}^{3}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-12{x}^{5}\\hfill \\end{array}\\)<p id=\"fs-id1167836689313\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(\\frac{5}{6}{x}^{3}y\\right)\\left(12x{y}^{2}\\right)\\hfill \\\\ \\text{Use the Commutative Property to rearrange the terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{5}{6}\u00b712\u00b7{x}^{3}\u00b7x\u00b7y\u00b7{y}^{2}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}10{x}^{4}{y}^{3}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836353225\"><div data-type=\"problem\" id=\"fs-id1167836516000\"><p id=\"fs-id1167833056558\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(5{y}^{7}\\right)\\left(-7{y}^{4}\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{2}{5}{a}^{4}{b}^{3}\\right)\\left(15a{b}^{3}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836616379\"><p id=\"fs-id1167829709229\"><span class=\"token\">\u24d0<\/span>\\(-35{y}^{11}\\)<span class=\"token\">\u24d1<\/span>\\(6{a}^{5}{b}^{6}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833050779\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829905091\"><p id=\"fs-id1167833031437\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left(-6{b}^{4}\\right)\\left(-9{b}^{5}\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{2}{3}{r}^{5}s\\right)\\left(12{r}^{6}{s}^{7}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829713143\"><p id=\"fs-id1167836538207\"><span class=\"token\">\u24d0<\/span>\\(54{b}^{9}\\)<span class=\"token\">\u24d1<\/span>\\(8{r}^{11}{s}^{8}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836502215\"><h3 data-type=\"title\">Multiply a Polynomial by a Monomial<\/h3><p>Multiplying a polynomial by a monomial is really just applying the Distributive Property.<\/p><div data-type=\"example\" id=\"fs-id1167829810683\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836699941\"><div data-type=\"problem\" id=\"fs-id1167836448130\"><p id=\"fs-id1167836665188\">Multiply: <span class=\"token\">\u24d0<\/span> \\(-2y\\left(4{y}^{2}+3y-5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(3{x}^{3}y\\left({x}^{2}-8xy+{y}^{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836293422\"><p id=\"fs-id1167836576202\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836568423\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of negative 2 y with the polynomial 4 y squared plus 3 y minus 5 in parentheses. Three arrows are drawn from the negative 2y pointing to each term in the polynomial in parentheses indicating the three multiplications. The next line shows the result when the negative 2 y is distributed: negative 2 y times 4 y squared plus negative 2 y times 3 y minus negative 2 y times 5. The simplified form is then negative 8 y to power of 3 minus 6 y to the power of 2 plus 10 y.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167829807325\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute.\\(\\phantom{\\rule{2.5em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836515792\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167836543231\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836663108\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{1.5em}{0ex}}3{x}^{3}y\\left({x}^{2}-8xy+{y}^{2}\\right)\\hfill \\\\ \\text{Distribute.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{1.5em}{0ex}}3{x}^{3}y\u00b7{x}^{2}+\\left(3{x}^{3}y\\right)\u00b7\\left(-8xy\\right)+\\left(3{x}^{3}y\\right)\u00b7{y}^{2}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{1.5em}{0ex}}3{x}^{5}y-24{x}^{4}{y}^{2}+3{x}^{3}{y}^{3}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836737859\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829720987\"><div data-type=\"problem\" id=\"fs-id1167836648572\"><p id=\"fs-id1167836512150\">Multiply: <span class=\"token\">\u24d0<\/span>\\(-3y\\left(5{y}^{2}+8y-7\\right)\\) <span class=\"token\">\u24d1<\/span> \\(4{x}^{2}{y}^{2}\\left(3{x}^{2}-5xy+3{y}^{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836791173\"><p id=\"fs-id1167836613889\"><span class=\"token\">\u24d0<\/span>\\(-15{y}^{3}-24{y}^{2}+21y\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(12{x}^{4}{y}^{2}-20{x}^{3}{y}^{3}+12{x}^{2}{y}^{4}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833350613\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829810693\"><div data-type=\"problem\" id=\"fs-id1167829689196\"><p id=\"fs-id1167829737462\">Multiply: <span class=\"token\">\u24d0<\/span>\\(4{x}^{2}\\left(2{x}^{2}-3x+5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-6{a}^{3}b\\left(3{a}^{2}-2ab+6{b}^{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829968361\"><p id=\"fs-id1167836730774\"><span class=\"token\">\u24d0<\/span>\\(8{x}^{4}-24{x}^{3}+20{x}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-18{a}^{5}b+12{a}^{4}{b}^{2}-36{a}^{3}{b}^{3}\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Multiply a Binomial by a Binomial<\/h3><p id=\"fs-id1167833349690\">Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836495233\"><div data-type=\"problem\" id=\"fs-id1167836521435\"><p id=\"fs-id1167836309885\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left(y+5\\right)\\left(y+8\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(4y+3\\right)\\left(2y-5\\right).\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836352593\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829807702\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of y plus 5 in parentheses with y plus 8 in parentheses. y plus 8 in parentheses is colored red with two red arrows drawn from y plus 8 in parentheses pointing to the y and 5 in the y plus 5 factor. The next line shows the result when the y plus 8 is distributed: y times the quantity y plus 8 in parentheses plus 5 times the quantity y plus 8 in parentheses. Then we distribute again, distributing the y to the y plus 8 and the 5 to the y plus 8. The result is y squared plus 8 y plus 5 y plus 40. Combining like terms results in the simplified form y to power of 2 plus 13 y plus 40.\" data-label=\"\"><tbody><tr><td><\/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_03_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute \\(\\left(y+8\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836511238\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute again.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829596696\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811237\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836684819\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167833274109\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of 4 y plus 3 in parentheses with 2 y minus 5 in parentheses. After distributing the quantity 2 y minus 5 in parentheses the result is 4 y times the quantity 2 y minus 5 in parentheses plus 3 times the quantity 2 y minus 5 in parentheses. Then we distribute the 4 y to the 2 y minus 5 and the 3 to the 2 y minus 5. The result is 8 y squared minus 20 y plus 6 y minus 15. Combining like terms results in the simplified form 8 y to power of 2 minus 14 y minus 15.\" data-label=\"\"><tbody><tr><td><\/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_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829879544\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute again.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836484890\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513988\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836409575\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836286748\"><div data-type=\"problem\" id=\"fs-id1167836326706\"><p id=\"fs-id1167833366001\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left(x+8\\right)\\left(x+9\\right)\\) <span class=\"token\">\u24d1<\/span>\\(\\left(3c+4\\right)\\left(5c-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836364079\"><p id=\"fs-id1167833310407\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}+17x+72\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(15{c}^{2}+14c-8\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836526818\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836549431\"><div data-type=\"problem\" id=\"fs-id1167836623888\"><p id=\"fs-id1167829594717\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left(5x+9\\right)\\left(4x+3\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(5y+2\\right)\\left(6y-3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836714500\"><p id=\"fs-id1167836685227\"><span class=\"token\">\u24d0<\/span>\\(20{x}^{2}+51x+27\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(30{y}^{2}-3y-6\\)<\/div><\/div><\/div><p id=\"fs-id1167836327796\">If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the <em data-effect=\"italics\">first<\/em> terms in each binomial. The second and third terms are the product of multiplying the two <em data-effect=\"italics\">outer<\/em> terms and then the two <em data-effect=\"italics\">inner<\/em> terms. And the last term results from multiplying the two <em data-effect=\"italics\">last<\/em> terms,<\/p><p id=\"fs-id1167836597858\">We abbreviate \u201cFirst, Outer, Inner, Last\u201d as FOIL. The letters stand for \u2018First, Outer, Inner, Last\u2019. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.<\/p><p>Let\u2019s multiply \\(\\left(x+3\\right)\\left(x+7\\right)\\) using both methods.<\/p><span data-type=\"media\" id=\"fs-id1167829739282\" data-alt=\"The figure shows how four terms in the product of two binomials can be remembered according to the mnemonic acronym FOIL. The example is the quantity x plus 3 in parentheses times the quantity x plus 7 in parentheses. The expression is expanded as in the previous examples by using the distributive property twice. After distributing the quantity x plus 7 in parentheses the result is x times the quantity x plus 7 in parentheses plus 3 times the quantity x plus 7 in parentheses. Then the x is distributed the x plus 7 and the 3 is distributed to the x plus 7 to get x squared plus 7 x plus 3 x plus 21. The letter F is written under the term x squared since it was the product of the first terms in the binomials. The letter O is written under the 7 x term sine it was the product of the outer terms in the binomials. The letter I is written under the 3 x term since it was the product of the inner terms in the binomials. The letter L is written under the 21 since it was the product of the last terms in the binomial. The original expression is shown again with four arrows connecting the first, outer, inner, and last terms in the binomials showing how the four terms can be determined directly from the factored form.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how four terms in the product of two binomials can be remembered according to the mnemonic acronym FOIL. The example is the quantity x plus 3 in parentheses times the quantity x plus 7 in parentheses. The expression is expanded as in the previous examples by using the distributive property twice. After distributing the quantity x plus 7 in parentheses the result is x times the quantity x plus 7 in parentheses plus 3 times the quantity x plus 7 in parentheses. Then the x is distributed the x plus 7 and the 3 is distributed to the x plus 7 to get x squared plus 7 x plus 3 x plus 21. The letter F is written under the term x squared since it was the product of the first terms in the binomials. The letter O is written under the 7 x term sine it was the product of the outer terms in the binomials. The letter I is written under the 3 x term since it was the product of the inner terms in the binomials. The letter L is written under the 21 since it was the product of the last terms in the binomial. The original expression is shown again with four arrows connecting the first, outer, inner, and last terms in the binomials showing how the four terms can be determined directly from the factored form.\"><\/span><p id=\"fs-id1167836376675\">We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!<\/p><div data-type=\"note\" id=\"fs-id1167833019452\" class=\"howto\"><div data-type=\"title\">Use the FOIL method to multiply two binomials.<\/div><span data-type=\"media\" id=\"fs-id1167836299260\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><\/span><\/div><p id=\"fs-id1167836569080\">When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.<\/p><p id=\"fs-id1167829741782\">Now we will do an example where we use the FOIL pattern to multiply two binomials.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836626974\"><p id=\"fs-id1167836731853\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(y-7\\right)\\left(y+4\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(4x+3\\right)\\left(2x-5\\right).\\)<\/p><\/div><div data-type=\"solution\"><ol id=\"fs-id1171791694878\" type=\"1\" class=\"circled\"><li><span class=\"token\">\u24d0<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836536657\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity y minus 7 in parentheses times the quantity y plus 4 in parentheses. Step 1. Multiply the First terms. The terms y and y are colored red with an arrow connecting them. The result is y squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The terms y and 4 are colored red with an arrow connecting them. The result is 4 y and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The terms negative 7 and y are colored red with an arrow connecting them. The result is negative 7 y squared and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The terms negative 7 and 4 are colored red with an arrow connecting them. The result is negative 28 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is y squared minus 3 y minus 28.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity y minus 7 in parentheses times the quantity y plus 4 in parentheses. Step 1. Multiply the First terms. The terms y and y are colored red with an arrow connecting them. The result is y squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The terms y and 4 are colored red with an arrow connecting them. The result is 4 y and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The terms negative 7 and y are colored red with an arrow connecting them. The result is negative 7 y squared and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The terms negative 7 and 4 are colored red with an arrow connecting them. The result is negative 28 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is y squared minus 3 y minus 28.\"><\/span><\/li><li><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 4 x plus 3 in parentheses times the quantity 2 x minus 5 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms 4 x and 2 x. The product of the first terms is 8 x squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms 4 x and negative 5. The result is negative 20 x and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms 3 and 2 x. The result is 6 x and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms 3 and negative 5. The result is negative 15 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 8 y squared minus 14 x minus 15.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 4 x plus 3 in parentheses times the quantity 2 x minus 5 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms 4 x and 2 x. The product of the first terms is 8 x squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms 4 x and negative 5. The result is negative 20 x and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms 3 and 2 x. The result is 6 x and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms 3 and negative 5. The result is negative 15 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 8 y squared minus 14 x minus 15.\"><\/span><\/li><\/ol><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832999172\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829749818\"><div data-type=\"problem\" id=\"fs-id1167836548589\"><p id=\"fs-id1167836432034\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(x-7\\right)\\left(x+5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(3x+7\\right)\\left(5x-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836309368\"><p id=\"fs-id1167836513138\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}-2x-35\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(15{x}^{2}+29x-14\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836510862\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836552129\"><p>Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(b-3\\right)\\left(b+6\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(4y+5\\right)\\left(4y-10\\right).\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836327641\"><span class=\"token\">\u24d0<\/span>\\({b}^{2}+3b-18\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(16{y}^{2}-20y-50\\)<\/div><\/div><\/div><p id=\"fs-id1167836356398\">The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.<\/p><div data-type=\"example\" id=\"fs-id1167836408987\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833060099\"><div data-type=\"problem\" id=\"fs-id1167829609240\"><p id=\"fs-id1167836597450\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left({n}^{2}+4\\right)\\left(n-1\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(3pq+5\\right)\\left(6pq-11\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836550155\"><p id=\"fs-id1167829984332\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836608558\" class=\"unnumbered unstyled\" summary=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity n squared plus 4 in parentheses times the quantity n minus 1 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms. The product of the first terms is n to the power of 3 and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The result is negative n squared and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The result is 4 n and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The result is negative 4 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is n to the power of 3 minus n squared plus 4 n minus 4.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547182\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/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_03_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Multiply the <em data-effect=\"italics\">First<\/em> terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836442295\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Multiply the <em data-effect=\"italics\">Outer<\/em> terms.<\/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_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Multiply the <em data-effect=\"italics\">Inner<\/em> terms.<\/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_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Multiply the <em data-effect=\"italics\">Last<\/em> terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507744\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Combine like terms\u2014there are none.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836650129\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829696676\" class=\"unnumbered unstyled\" summary=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 3 p q plus 5 in parentheses times the quantity 6 p q minus 11 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms. The product of the first terms is 18 p squared q squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The result is negative 33 p q and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The result is 30 p q and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The result is negative 55 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 18 p squared q squared minus 3 p q minus 55.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792364493\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836501741\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Multiply the <em data-effect=\"italics\">First<\/em> terms.\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836754795\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Multiply the <em data-effect=\"italics\">Outer<\/em> terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836429820\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Multiply the <em data-effect=\"italics\">Inner<\/em> terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550146\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Multiply the <em data-effect=\"italics\">Last<\/em> terms.<\/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_03_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Combine like terms.<\/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_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829879584\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836309297\"><div data-type=\"problem\"><p id=\"fs-id1167833008167\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left({x}^{2}+6\\right)\\left(x-8\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(2ab+5\\right)\\left(4ab-4\\right).\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836570171\"><span class=\"token\">\u24d0<\/span>\\({x}^{3}-8{x}^{2}+6x-48\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(8{a}^{2}{b}^{2}+12ab-20\\)<\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833046878\"><p id=\"fs-id1167836688632\">Multiply: <span class=\"token\">\u24d0<\/span>\\(\\left({y}^{2}+7\\right)\\left(y-9\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(2xy+3\\right)\\left(4xy-5\\right).\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829579784\"><span class=\"token\">\u24d0<\/span>\\({y}^{3}-9{y}^{2}+7y-63\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(8{x}^{2}{y}^{2}+2xy-15\\)<\/div><\/div><\/div><p id=\"fs-id1171791276732\">The FOIL method is usually the quickest method for multiplying two binomials, but it <em data-effect=\"italics\">only<\/em> works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.<\/p><span data-type=\"media\" id=\"fs-id1171789718031\" data-alt=\"This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says \u201cYou start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products.\u201d\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says \u201cYou start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products.\u201d\"><\/span><p id=\"fs-id1171791459374\">Now we\u2019ll apply this same method to multiply two binomials.<\/p><div data-type=\"example\" id=\"fs-id1171791417364\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1171791398514\"><div data-type=\"problem\" id=\"fs-id1171791446238\"><p id=\"fs-id1171784054248\">Multiply using the Vertical Method: \\(\\left(3y-1\\right)\\left(2y-6\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171789922301\"><p id=\"fs-id1171787442859\">It does not matter which binomial goes on the top.<\/p><p id=\"fs-id1171789582920\">\\(\\begin{array}{ccccccccccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\text{Multiply}\\phantom{\\rule{0.2em}{0ex}}3y-1\\phantom{\\rule{0.2em}{0ex}}\\text{by}\\phantom{\\rule{0.2em}{0ex}}-6.\\hfill \\\\ \\text{Multiply}\\phantom{\\rule{0.2em}{0ex}}3y-1\\phantom{\\rule{0.2em}{0ex}}\\text{by}\\phantom{\\rule{0.2em}{0ex}}2y.\\hfill \\\\ \\text{Add like terms.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\hfill 3y-1\\\\ \\hfill \\underset{___________}{\\phantom{\\rule{1.9em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2y-6}\\\\ \\hfill -18y+6\\\\ \\hfill \\underset{___________}{6{y}^{2}-\\phantom{\\rule{0.3em}{0ex}}2y\\phantom{\\rule{1.6em}{0ex}}}\\\\ \\hfill 6{y}^{2}-20y+6\\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\text{partial product}\\hfill \\\\ \\text{partial product}\\hfill \\\\ \\text{product}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1171791320793\">Notice the partial products are the same as the terms in the FOIL method.<\/p><span data-type=\"media\" id=\"fs-id1171791584225\" data-alt=\"This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1171791637725\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1171791450879\"><div data-type=\"problem\" id=\"fs-id1171789766013\"><p id=\"fs-id1171791631148\">Multiply using the Vertical Method: \\(\\left(5m-7\\right)\\left(3m-6\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171784055217\"><p id=\"fs-id1171784121997\">\\(15{m}^{2}-51m+42\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1171789589582\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1171789574277\"><div data-type=\"problem\" id=\"fs-id1171782047419\"><p id=\"fs-id1171791419018\">Multiply using the Vertical Method: \\(\\left(6b-5\\right)\\left(7b-3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791459738\"><p id=\"fs-id1171791319180\">\\(42{b}^{2}-53b+15\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1171791574620\">We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.<\/p><div data-type=\"note\" id=\"fs-id1171782147854\"><div data-type=\"title\">Multiplying Two Binomials<\/div><p id=\"fs-id1171784140960\">To multiply binomials, use the:<\/p><ul id=\"fs-id1171791449840\" data-bullet-style=\"bullet\"><li>Distributive Property<\/li><li>FOIL Method<\/li><li>Vertical Method<\/li><\/ul><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Multiply a Polynomial by a Polynomial<\/h3><p id=\"fs-id1167836571662\">We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we\u2019re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.<\/p><div data-type=\"example\" id=\"fs-id1167836660220\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829738325\"><div data-type=\"problem\"><p id=\"fs-id1167836528267\">Multiply \\(\\left(b+3\\right)\\left(2{b}^{2}-5b+8\\right)\\) using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829597387\"><p id=\"fs-id1167833349853\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167824781546\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of b plus 3 in parentheses with 2 b squared minus 5 b plus 8 in parentheses. After distributing the trinomial the result is b times the quantity 2 b squared minus 5 b plus 8 in parentheses plus 3 times the quantity 2 b squared minus 5 b plus 8 in parentheses 2 y minus 5 in parentheses. Then we distribute the b to the trinomial to get 2 b to the power of 3 minus 5 b squared plus 8 b and distribute the 3 to the trinomial to get 6 b squared minus 15 b plus 24. Combining like terms results in the simplified form 2 b to power of 3 plus b squared minus 7 b plus 24.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829711887\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829690938\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550571\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Combine like terms.\u2003\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833022850\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167833346816\"><span class=\"token\">\u24d1<\/span> It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.<\/p><div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167833377138\" class=\"unnumbered unstyled\" summary=\"This figure shows how to multiply the polynomials with the vertical method. The polynomial 2 b squared minus 5 b plus 8 is written directly over the polynomial b plus 3. The 8 is directly over the 3 and the negative 5 b is directly over the b. A horizontal line is drawn below the b plus 3. The result of multiplying 3 with the quantity 2 b squared minus 5 b plus 8 is written below the horizontal line. The result is get 6 b squared minus 15 b plus 24 with the 24 under the 3 and 8. The result of multiplying the b with the quantity 2 b squared minus 5 b plus 8 is written below the last calculation but shifted one term to the left. The result is 2 b to the power of 3 minus 5 b squared plus 8 b with the 8 b under the negative 15 b from the first multiplication. A second horizontal line is draw below the last result. The two multiplications are then added column by column. 24 is brought down since nothing is below it. Negative 15 b is added to 8 b to get negative 7 b. 6 b squared is added to negative 5 b squared to get b squared. 2 b to the power of 3 is brought down since nothing is above it. The final result is 2 b to power of 3 plus b squared minus 7 b plus 24.\" data-label=\"\"><tbody><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply \\(\\left(2{b}^{2}-5b+8\\right)\\) by 3.<div data-type=\"newline\"><br><\/div>Multiply \\(\\left(2{b}^{2}-5b+8\\right)\\) by \\(b\\).<\/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_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add like terms.<\/td><td data-valign=\"top\" data-align=\"left\">\u2003\u2003<span data-type=\"media\" id=\"fs-id1167836309930\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div>\u2003\u2003<span data-type=\"media\" id=\"fs-id1167836627519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836608523\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829744225\"><div data-type=\"problem\" id=\"fs-id1167836558379\"><p id=\"fs-id1167836730622\">Multiply\\(\\left(y-3\\right)\\left({y}^{2}-5y+2\\right)\\) using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836520603\"><p id=\"fs-id1167833008070\"><span class=\"token\">\u24d0<\/span>\\({y}^{3}-8{y}^{2}+17y-6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({y}^{3}-8{y}^{2}+17y-6\\)<\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829790877\"><div data-type=\"problem\"><p id=\"fs-id1167833142588\">Multiply \\(\\left(x+4\\right)\\left(2{x}^{2}-3x+5\\right)\\) using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> The Vertical Method.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829880204\"><p id=\"fs-id1167824701386\"><span class=\"token\">\u24d0<\/span>\\(2{x}^{3}+5{x}^{2}-7x+20\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({y}^{3}-8{y}^{2}+17y-6\\)<\/div><\/div><\/div><p id=\"fs-id1167836549264\">We have now seen two methods you can use to multiply a polynomial by a polynomial. After you practice each method, you\u2019ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.<\/p><div data-type=\"note\" id=\"fs-id1167836305065\"><div data-type=\"title\">Multiplying a Polynomial by a Polynomial<\/div><p id=\"fs-id1167829594694\">To multiply a trinomial by a binomial, use the:<\/p><ul id=\"fs-id1167829807211\" data-bullet-style=\"bullet\"><li>Distributive Property<\/li><li>Vertical Method<\/li><\/ul><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830123961\"><h3 data-type=\"title\">Multiply Special Products<\/h3><p id=\"fs-id1167833196636\">Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let\u2019s start by looking at three examples and look for a pattern.<\/p><p id=\"fs-id1167829786669\">Look at these results. Do you see any patterns?<\/p><span data-type=\"media\" id=\"fs-id1167833240126\" data-alt=\"The figure shows three examples of squaring a binomial. In the first example x plus 9 is squared to get x plus 9 times x plus 9 which is x squared plus 9 x plus 9 x plus 81 which simplifies to x squared plus 18 x plus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 7 is squared to get y minus y times y minus 7 which is y squared minus 7 y minus 7 y plus 49 which simplifies to y squared minus 14 y plus 49. Colors show that y squared comes from the square of the y in the original binomial and 49 comes from the square of the negative 7 in the original binomial. In the third example 2 x plus 3 is squared to get 2 x plus 3 times 2 x plus 3 which is 4 x squared plus 6 x plus 6 x plus 9 which simplifies to 4 x squared plus 12 x plus 9. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 9 comes from the square of the 3 in the original binomial.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three examples of squaring a binomial. In the first example x plus 9 is squared to get x plus 9 times x plus 9 which is x squared plus 9 x plus 9 x plus 81 which simplifies to x squared plus 18 x plus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 7 is squared to get y minus y times y minus 7 which is y squared minus 7 y minus 7 y plus 49 which simplifies to y squared minus 14 y plus 49. Colors show that y squared comes from the square of the y in the original binomial and 49 comes from the square of the negative 7 in the original binomial. In the third example 2 x plus 3 is squared to get 2 x plus 3 times 2 x plus 3 which is 4 x squared plus 6 x plus 6 x plus 9 which simplifies to 4 x squared plus 12 x plus 9. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 9 comes from the square of the 3 in the original binomial.\"><\/span><p id=\"fs-id1167836628650\">What about the number of terms? In each example we squared a binomial and the result was a trinomial.<\/p><div data-type=\"equation\" id=\"fs-id1167836698747\" class=\"unnumbered\" data-label=\"\">\\({\\left(a+b\\right)}^{2}=___+___+___\\)<\/div><p id=\"fs-id1167829609160\">Now look at the <em data-effect=\"italics\">first term<\/em> in each result. Where did it come from?<\/p><p id=\"fs-id1167829947837\">The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!<\/p><div data-type=\"equation\" id=\"fs-id1167830093894\" class=\"unnumbered\" data-label=\"\">\\({\\left(a+b\\right)}^{2}={a}^{2}+___+___\\)<\/div><p id=\"fs-id1167829715919\">\u2003\u2003<em data-effect=\"italics\">To get the first term of the product, square the first term.<\/em><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><p>Where did the <em data-effect=\"italics\">last term<\/em> come from? Look at the examples and find the pattern.<\/p><p id=\"fs-id1167836325829\">The last term is the product of the last terms, which is the square of the last term.<\/p><div data-type=\"equation\" id=\"fs-id1167836547530\" class=\"unnumbered\" data-label=\"\">\\({\\left(a+b\\right)}^{2}=___+___+{b}^{2}\\)<\/div><p id=\"fs-id1167829738246\">\u2003\u2003<em data-effect=\"italics\">To get the last term of the product, square the last term.<\/em><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><p id=\"fs-id1167833053517\">Finally, look at the <em data-effect=\"italics\">middle term<\/em>. Notice it came from adding the \u201couter\u201d and the \u201cinner\u201d terms\u2014which are both the same! So the middle term is double the product of the two terms of the binomial.<\/p><div data-type=\"equation\" id=\"fs-id1167832951176\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}{\\left(a+b\\right)}^{2}=___+2ab+___\\hfill \\\\ {\\left(a-b\\right)}^{2}=___-2ab+___\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167829879462\">\u2003\u2003<em data-effect=\"italics\">To get the middle term of the product, multiply the terms and double their product.<\/em><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><p id=\"fs-id1167829694908\">Putting it all together:<\/p><div data-type=\"note\" id=\"fs-id1167836524528\"><div data-type=\"title\">Binomial Squares Pattern<\/div><p id=\"fs-id1167826188116\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p><span data-type=\"media\" id=\"fs-id1167829905170\" data-alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><\/span><p id=\"fs-id1171791310441\">To square a binomial, square the first term, square the last term , double their product.<\/p><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829711970\"><p id=\"fs-id1167836729154\">Multiply: <span class=\"token\">\u24d0<\/span> \\({\\left(x+5\\right)}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({\\left(2x-3y\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836713911\"><p id=\"fs-id1167829713617\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167833338740\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply x plus 5 squared using the formula a plus b squared equals a squared plus 2 a b plus b squared. Squaring the first term results in x squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times x times 5 which matches up with 2 a b in the formula. The simplified version is x squared plus 10 x plus 25.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836511568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Square the first term.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694664\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Square the last term.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829689053\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Double their product.\u2003\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836534247\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836386651\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836481274\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x minus 3 y squared using the formula a minus b squared equals a squared minus 2 a b plus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 3 y squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 2 x times 3 y which matches up with 2 a b in the formula. The simplified version is 4 x squared minus 12 x y plus 9 y squared.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836293412\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833057054\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829715332\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836519176\"><div data-type=\"problem\" id=\"fs-id1167833066308\"><p>Multiply: <span class=\"token\">\u24d0<\/span>\\({\\left(x+9\\right)}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({\\left(2c-d\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833365593\"><p id=\"fs-id1167833018052\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}+18x+81\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(4{c}^{2}-4cd+{d}^{2}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836667065\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836543225\"><div data-type=\"problem\" id=\"fs-id1167833021492\"><p id=\"fs-id1167836697177\">Multiply: <span class=\"token\">\u24d0<\/span>\\({\\left(y+11\\right)}^{2}\\) <span class=\"token\">\u24d1<\/span> \\({\\left(4x-5y\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836367497\"><p id=\"fs-id1167833328786\"><span class=\"token\">\u24d0<\/span>\\({y}^{2}+22y+121\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(16{x}^{2}-40xy+25{y}^{2}\\)<\/div><\/div><\/div><p id=\"fs-id1167836686187\">We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.<\/p><p id=\"fs-id1167824617566\">A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a <strong data-effect=\"bold\">conjugate pair<\/strong> and is of the form \\(\\left(a-b\\right),\\left(a+b\\right).\\)<\/p><div data-type=\"note\" id=\"fs-id1167833345895\"><div data-type=\"title\">Conjugate Pair<\/div><p id=\"fs-id1167836716282\">A <span data-type=\"term\" class=\"no-emphasis\">conjugate pair<\/span> is two binomials of the form<\/p><div data-type=\"equation\" id=\"fs-id1167825835948\" class=\"unnumbered\" data-label=\"\">\\(\\left(a-b\\right),\\left(a+b\\right).\\)<\/div><p id=\"fs-id1167836485185\">The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/p><\/div><p id=\"fs-id1167836408384\">There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let\u2019s look for the pattern by using FOIL to multiply some conjugate pairs.<\/p><span data-type=\"media\" id=\"fs-id1167824617552\" data-alt=\"The figure shows three examples of multiplying a binomial with its conjugate. In the first example x plus 9 is multiplied with x minus 9 to get x squared minus 9 x plus 9 x minus 81 which simplifies to x squared minus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 8 is multiplied with y plus 8 to get y squared plus 8 y minus 8 y minus 64 which simplifies to y squared minus 64. Colors show that y squared comes from the square of the y in the original binomial and 64 comes from the square of the 8 in the original binomial. In the third example 2 x minus 5 is multiplied with 2 x plus 5 to get 4 x squared plus 10 x minus 10 x minus 25 which simplifies to 4 x squared minus 25. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 25 comes from the square of the 5 in the original binomial.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three examples of multiplying a binomial with its conjugate. In the first example x plus 9 is multiplied with x minus 9 to get x squared minus 9 x plus 9 x minus 81 which simplifies to x squared minus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 8 is multiplied with y plus 8 to get y squared plus 8 y minus 8 y minus 64 which simplifies to y squared minus 64. Colors show that y squared comes from the square of the y in the original binomial and 64 comes from the square of the 8 in the original binomial. In the third example 2 x minus 5 is multiplied with 2 x plus 5 to get 4 x squared plus 10 x minus 10 x minus 25 which simplifies to 4 x squared minus 25. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 25 comes from the square of the 5 in the original binomial.\"><\/span><p id=\"fs-id1167829750123\">What do you observe about the products?<\/p><p id=\"fs-id1167825702482\">The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.<\/p><p id=\"fs-id1167829748237\">Each <em data-effect=\"italics\">first term<\/em> is the product of the first terms of the binomials, and since they are identical it is the square of the first term.<\/p><div data-type=\"equation\" id=\"fs-id1167833135469\" class=\"unnumbered\" data-label=\"\">\\(\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-___\\)<\/div><p id=\"fs-id1167836664763\">\u2003\u2003\u2003<em data-effect=\"italics\">To get the first term, square the first term.<\/em><\/p><p id=\"fs-id1167826172060\">The <em data-effect=\"italics\">last term<\/em> came from multiplying the last terms, the square of the last term.<\/p><div data-type=\"equation\" id=\"fs-id1167832935990\" class=\"unnumbered\" data-label=\"\">\\(\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}\\)<\/div><p id=\"fs-id1167836606241\">\u2003\u2003\u2003<em data-effect=\"italics\">To get the last term, square the last term<\/em>.<\/p><p id=\"fs-id1167829783760\">Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.<\/p><p id=\"fs-id1167833139755\">The product of conjugates is always of the form \\({a}^{2}-{b}^{2}.\\) This is called a <strong data-effect=\"bold\">difference of squares<\/strong>.<\/p><p id=\"fs-id1167836360699\">This leads to the pattern:<\/p><div data-type=\"note\" id=\"fs-id1167833349943\"><div data-type=\"title\">Product of Conjugates Pattern<\/div><p id=\"fs-id1167833349948\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p><span data-type=\"media\" id=\"fs-id1167825918985\" data-alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><\/span><p id=\"fs-id1167824578716\">The product is called a difference of squares.<\/p><p id=\"fs-id1167836717037\">To multiply conjugates, square the first term, square the last term, write it as a difference of squares.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167836717042\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833020420\"><div data-type=\"problem\" id=\"fs-id1167833020422\"><p id=\"fs-id1167833020424\">Multiply using the product of conjugates pattern: <span class=\"token\">\u24d0<\/span> \\(\\left(2x+5\\right)\\left(2x-5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(5m-9n\\right)\\left(5m+9n\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836722468\"><p id=\"fs-id1167836722470\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829694181\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x plus 5 times 2 x minus 5 using the formula a plus b times a minus b equals a squared minus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Simplifying the product results in 4 x squared minus 25.\" data-label=\"\"><tbody><tr><td data-valign=\"top\" data-align=\"left\">Are the binomials conjugates?<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836525165\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">It is the product of conjugates.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836648845\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Square the first term, \\(2x.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833051940\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Square the last term, \\(5.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829739552\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify. The product is a difference of squares.\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833175364\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167830121978\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829689596\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 5 m minus 9 n times 5 m plus 9 n using the formula a minus b times a plus b equals a squared minus b squared. Squaring the first term results in 5 m squared which matches up with the term a squared in the formula. Squaring the last term results in 9 n squared which matches up with the term b squared in the formula. Simplifying the product results in 25 m squared minus 81 n squared.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824733940\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">This fits the pattern.\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829739570\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the pattern.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167825836088\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833186407\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836429507\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836429510\"><div data-type=\"problem\" id=\"fs-id1167836429512\"><p id=\"fs-id1167836429514\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(6x+5\\right)\\left(6x-5\\right)\\) <span class=\"token\">\u24d1<\/span>\\(\\left(4p-7q\\right)\\left(4p+7q\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824733730\"><p id=\"fs-id1167824733732\"><span class=\"token\">\u24d0<\/span>\\(36{x}^{2}-25\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(16{p}^{2}-49{q}^{2}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836791451\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836791454\"><div data-type=\"problem\" id=\"fs-id1167836791456\"><p id=\"fs-id1167836791458\">Multiply: <span class=\"token\">\u24d0<\/span> \\(\\left(2x+7\\right)\\left(2x-7\\right)\\) <span class=\"token\">\u24d1<\/span>\\(\\left(3x-y\\right)\\left(3x+y\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829860523\"><p id=\"fs-id1167833396771\"><span class=\"token\">\u24d0<\/span>\\(4{x}^{2}-49\\)<span class=\"token\">\u24d1<\/span>\\(9{x}^{2}-{y}^{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833202398\">We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.<\/p><div data-type=\"note\" id=\"fs-id1167836399600\"><div data-type=\"title\">Comparing the Special Product Patterns<\/div><table id=\"fs-id1167836399606\" summary=\"Table has two columns. The left column lists binomial squares and shows two equations: a plus b in parentheses square equals a square plus 2 a b plus b squared and a minus b in parentheses squared equals a squared minus 2 ab plus b squared. Below the equations the text states squaring a binomial, product is a trinomial, inner and outer terms with foil are the same, and middle term is double the product of the terms. The right column lists product of conjugates with the equation a minus b times a plus b equals a squared minus b squared, Below the equation the test states multiplying conjugates, product is a binomial, inner and outer terms with foil are opposites, and there is no middle term.\" class=\"unnumbered\" data-label=\"\"><thead><tr><th data-valign=\"top\" data-align=\"center\">Binomial Squares<\/th><th data-valign=\"top\" data-align=\"center\">Product of Conjugates<\/th><\/tr><\/thead><tbody><tr><td data-valign=\"top\" data-align=\"center\">\\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"center\">\\({\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Squaring a binomial<\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Multiplying conjugates<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">trinomial<\/strong><\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">binomial.<\/strong><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">the same.<\/strong><\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">opposites.<\/strong><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Middle term is <strong data-effect=\"bold\">double the product<\/strong> of the terms<\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003There is <strong data-effect=\"bold\">no<\/strong> middle term.<\/td><\/tr><\/tbody><\/table><\/div><div data-type=\"example\" id=\"fs-id1167829748535\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829748537\"><div data-type=\"problem\" id=\"fs-id1167829748539\"><p id=\"fs-id1167829748541\">Choose the appropriate pattern and use it to find the product:<\/p><p id=\"fs-id1167829748544\"><span class=\"token\">\u24d0<\/span>\\(\\left(2x-3\\right)\\left(2x+3\\right)\\)<span class=\"token\">\u24d1<\/span>\\({\\left(5x-8\\right)}^{2}\\)<span class=\"token\">\u24d2<\/span>\\({\\left(6m+7\\right)}^{2}\\)<span class=\"token\">\u24d3<\/span>\\(\\left(5x-6\\right)\\left(6x+5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826172057\"><p id=\"fs-id1167824733785\"><span class=\"token\">\u24d0<\/span>\\(\\left(2x-3\\right)\\left(2x+3\\right)\\)<\/p><p id=\"fs-id1167826172195\">These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.<\/p><table id=\"fs-id1167826172199\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x minus 3 times 2 x plus 3 using the formula a minus b times a plus b equals a squared minus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 3 squared which matches up with the term b squared in the formula. Simplifying the product results in 4 x squared minus 9.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836791299\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836791320\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824733756\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167824733770\"><span class=\"token\">\u24d1<\/span>\\({\\left(8x-5\\right)}^{2}\\)<\/p><p id=\"fs-id1167825987072\">We are asked to square a binomial. It fits the binomial squares pattern.<\/p><table id=\"fs-id1167825987075\" class=\"unnumbered unstyled\" summary=\"The example shows how to square 8 x minus 5 using the formula a minus b squared equals a squared minus 2 a b plus b squared. Squaring the first term results in the quantity 8 x in parentheses squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 8 x times 5 which matches up with 2 a b in the formula. The simplified version is 64 x squared minus 80 x plus 25.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829738827\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829738848\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836713594\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836713608\"><span class=\"token\">\u24d2<\/span>\\({\\left(6m+7\\right)}^{2}\\)<\/p><p id=\"fs-id1167836700258\">Again, we will square a binomial so we use the binomial squares pattern.<\/p><table id=\"fs-id1167836700261\" class=\"unnumbered unstyled\" summary=\"The example shows how to square 6 m plus 7 using the formula a plus b squared equals a squared plus 2 a b plus b squared. Squaring the first term results in the quantity 6 m in parentheses squared which matches up with the term a squared in the formula. Squaring the last term results in 7 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 6 m times 7 which matches up with 2 a b in the formula. The simplified version is 36 m squared plus 84 m plus 49.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829860644\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829860665\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833349602\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167833349616\"><span class=\"token\">\u24d3<\/span>\\(\\left(5x-6\\right)\\left(6x+5\\right)\\)<\/p><p id=\"fs-id1167829715204\">This product does not fit the patterns, so we will use FOIL.<\/p><p id=\"fs-id1167829715207\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}\\left(5x-6\\right)\\left(6x+5\\right)\\hfill \\\\ \\text{Use FOIL.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}30{x}^{2}+25x-36x-30\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}30{x}^{2}-11x-30\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833051916\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833051919\"><div data-type=\"problem\" id=\"fs-id1167833051922\"><p id=\"fs-id1167833051924\">Choose the appropriate pattern and use it to find the product:<\/p><p id=\"fs-id1167833051927\"><span class=\"token\">\u24d0<\/span>\\(\\left(9b-2\\right)\\left(2b+9\\right)\\)<span class=\"token\">\u24d1<\/span>\\({\\left(9p-4\\right)}^{2}\\)<span class=\"token\">\u24d2<\/span>\\({\\left(7y+1\\right)}^{2}\\)<span class=\"token\">\u24d3<\/span>\\(\\left(4r-3\\right)\\left(4r+3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836690445\"><p id=\"fs-id1167836690447\"><span class=\"token\">\u24d0<\/span> FOIL; \\(18{b}^{2}+77b-18\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Binomial Squares; \\(81{p}^{2}-72p+16\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Binomial Squares; \\(49{y}^{2}+14y+1\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> Product of Conjugates; \\(16{r}^{2}-9\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825836245\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167825836248\"><div data-type=\"problem\" id=\"fs-id1167825836250\"><p id=\"fs-id1167825836252\">Choose the appropriate pattern and use it to find the product:<\/p><p id=\"fs-id1167825836256\"><span class=\"token\">\u24d0<\/span>\\({\\left(6x+7\\right)}^{2}\\)<span class=\"token\">\u24d1<\/span>\\(\\left(3x-4\\right)\\left(3x+4\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(2x-5\\right)\\left(5x-2\\right)\\)<span class=\"token\">\u24d3<\/span>\\({\\left(6n-1\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825987123\"><p id=\"fs-id1167825987125\"><span class=\"token\">\u24d0<\/span> Binomial Squares; \\(36{x}^{2}+84x+49\\) <span class=\"token\">\u24d1<\/span> Product of Conjugates; \\(9{x}^{2}-16\\) <span class=\"token\">\u24d2<\/span> FOIL; \\(10{x}^{2}-29x+10\\) <span class=\"token\">\u24d3<\/span> Binomial Squares; \\(36{n}^{2}-12n+1\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832936015\"><h3 data-type=\"title\">Multiply Polynomial Functions<\/h3><p id=\"fs-id1167832936020\">Just as polynomials can be multiplied, polynomial functions can also be multiplied.<\/p><div data-type=\"note\" id=\"fs-id1167832936023\"><div data-type=\"title\">Multiplication of Polynomial Functions<\/div><p id=\"fs-id1167832936029\">For functions \\(f\\left(x\\right)\\) and \\(g\\left(x\\right),\\)<\/p><div data-type=\"equation\" id=\"fs-id1167829878976\" class=\"unnumbered\" data-label=\"\">\\(\\left(f\u00b7g\\right)\\left(x\\right)=f\\left(x\\right)\u00b7g\\left(x\\right)\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1167829621729\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829621731\"><div data-type=\"problem\" id=\"fs-id1167829621733\"><p id=\"fs-id1167829621735\">For functions \\(f\\left(x\\right)=x+2\\) and \\(g\\left(x\\right)={x}^{2}-3x-4,\\) find: <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826172148\"><p id=\"fs-id1167826172150\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; f\\left(x\\right)\u00b7g\\left(x\\right)\\hfill \\\\ \\text{Substitute for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)\\text{and}\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right).\\hfill &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; \\left(x+2\\right)\\left({x}^{2}-3x-4\\right)\\hfill \\\\ \\text{Multiply the polynomials.}\\hfill &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; x\\left({x}^{2}-3x-4\\right)+2\\left({x}^{2}-3x-4\\right)\\hfill \\\\ \\text{Distribute.}\\hfill &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; {x}^{3}-3{x}^{2}-4x+2{x}^{2}-6x-8\\hfill \\\\ \\text{Combine like terms.}\\hfill &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; {x}^{3}-{x}^{2}-10x-8\\hfill \\end{array}\\)<p id=\"fs-id1167829739447\"><span class=\"token\">\u24d1<\/span> In part <span class=\"token\">\u24d0<\/span> we found \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) and now are asked to find \\(\\left(f\u00b7g\\right)\\left(2\\right).\\)<\/p><p id=\"fs-id1167836554557\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(x\\right)&amp; =\\hfill &amp; {x}^{3}-{x}^{2}-10x-8\\hfill \\\\ \\text{To find}\\phantom{\\rule{0.2em}{0ex}}\\left(f\u00b7g\\right)\\left(2\\right),\\phantom{\\rule{0.2em}{0ex}}\\text{substitute}\\phantom{\\rule{0.2em}{0ex}}x=2.\\hfill &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(2\\right)&amp; =\\hfill &amp; {2}^{3}-{2}^{2}-10\u00b72-8\\hfill \\\\ &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(2\\right)&amp; =\\hfill &amp; 8-4-20-8\\hfill \\\\ &amp; &amp; &amp; \\hfill \\left(f\u00b7g\\right)\\left(2\\right)&amp; =\\hfill &amp; -24\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829718351\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829718355\"><div data-type=\"problem\" id=\"fs-id1167829718357\"><p id=\"fs-id1167829718360\">For functions \\(f\\left(x\\right)=x-5\\) and \\(g\\left(x\\right)={x}^{2}-2x+3,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829689565\"><p id=\"fs-id1167829689567\"><span class=\"token\">\u24d0<\/span>\\(\\left(f\u00b7g\\right)\\left(x\\right)={x}^{3}-7{x}^{2}+13x-15\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(f\u00b7g\\right)\\left(2\\right)=-9\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836648781\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836648785\"><div data-type=\"problem\" id=\"fs-id1167836648787\"><p id=\"fs-id1167836648789\">For functions \\(f\\left(x\\right)=x-7\\) and \\(g\\left(x\\right)={x}^{2}+8x+4,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829614450\"><p id=\"fs-id1167829614452\"><span class=\"token\">\u24d0<\/span>\\(\\left(f\u00b7g\\right)\\left(x\\right)={x}^{3}+{x}^{2}-52x-28\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(f\u00b7g\\right)\\left(2\\right)=-120\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825681186\" class=\"media-2\"><p id=\"fs-id1167825681190\">Access this online resource for additional instruction and practice with multiplying polynomials.<\/p><ul id=\"fs-id1167825681194\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37Introspecprod\">Introduction to special products of binomials<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167825681208\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167825681215\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to use the FOIL method to multiply two binomials.<\/strong><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836409131\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><\/span><\/li><li><strong data-effect=\"bold\">Multiplying Two Binomials:<\/strong> To multiply binomials, use the: <ul id=\"fs-id1167836409155\" data-bullet-style=\"open-circle\"><li>Distributive Property<\/li><li>FOIL Method<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Multiplying a Polynomial by a Polynomial:<\/strong> To multiply a trinomial by a binomial, use the: <ul id=\"fs-id1167836409175\" data-bullet-style=\"open-circle\"><li>Distributive Property<\/li><li>Vertical Method<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Binomial Squares Pattern<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers, <span data-type=\"media\" id=\"fs-id1167836536277\" data-alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><\/span><\/li><li><strong data-effect=\"bold\">Product of Conjugates Pattern<\/strong><div data-type=\"newline\"><br><\/div> If \\(a,b\\) are real numbers<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167836536314\" data-alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><\/span><div data-type=\"newline\"><br><\/div> The product is called a difference of squares.<div data-type=\"newline\"><br><\/div> To multiply conjugates, square the first term, square the last term, write it as a difference of squares.<\/li><li><strong data-effect=\"bold\">Comparing the Special Product Patterns<\/strong><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167833346332\" class=\"unnumbered\" summary=\"Table has two columns. The left column lists binomial squares and shows two equations: a plus b in parentheses square equals a square plus 2 a b plus b squared and a minus b in parentheses squared equals a squared minus 2 ab plus b squared. Below the equations the text states squaring a binomial, product is a trinomial, inner and outer terms with foil are the same, and middle term is double the product of the terms. The right column lists product of conjugates with the equation a minus b times a plus b equals a squared minus b squared, Below the equation the test states multiplying conjugates, product is a binomial, inner and outer terms with foil are opposites, and there is no middle term.\" data-label=\"\"><thead><tr><th data-valign=\"top\" data-align=\"center\">Binomial Squares<\/th><th data-valign=\"top\" data-align=\"center\">Product of Conjugates<\/th><\/tr><\/thead><tbody><tr><td data-valign=\"top\" data-align=\"center\">\\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\({\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"center\">\\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Squaring a binomial<\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Multiplying conjugates<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">trinomial<\/strong><\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">binomial.<\/strong><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">the same.<\/strong><\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">opposites.<\/strong><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Middle term is <strong data-effect=\"bold\">double the product<\/strong> of the terms<\/td><td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003There is <strong data-effect=\"bold\">no<\/strong> middle term.<\/td><\/tr><\/tbody><\/table><\/li><li><strong data-effect=\"bold\">Multiplication of Polynomial Functions:<\/strong><ul id=\"fs-id1167829828527\" data-bullet-style=\"open-circle\"><li>For functions \\(f\\left(x\\right)\\) and \\(g\\left(x\\right),\\)<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167829828567\" class=\"unnumbered\" data-label=\"\">\\(\\left(f\u00b7g\\right)\\left(x\\right)=f\\left(x\\right)\u00b7g\\left(x\\right)\\)<\/div><\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836788245\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836788250\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836788257\"><strong data-effect=\"bold\">Multiply Monomials<\/strong><\/p><p id=\"fs-id1167836788263\">In the following exercises, multiply the monomials.<\/p><div data-type=\"exercise\" id=\"fs-id1167829833532\"><div data-type=\"problem\" id=\"fs-id1167829833535\"><p id=\"fs-id1167829833537\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(6{y}^{7}\\right)\\left(-3{y}^{4}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{4}{7}r{s}^{2}\\right)\\left(14r{s}^{3}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826024750\"><div data-type=\"problem\" id=\"fs-id1167833021418\"><p id=\"fs-id1167833021421\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(-10{x}^{5}\\right)\\left(-3{x}^{3}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{5}{8}{x}^{3}y\\right)\\left(24{x}^{5}y\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836664315\"><p id=\"fs-id1167836664317\"><span class=\"token\">\u24d0<\/span>\\(30{x}^{8}\\)<span class=\"token\">\u24d1<\/span>\\(15{x}^{8}{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836664356\"><div data-type=\"problem\" id=\"fs-id1167836664358\"><p id=\"fs-id1167836738069\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(-8{u}^{6}\\right)\\left(-9u\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{2}{3}{x}^{2}y\\right)\\left(\\frac{3}{4}x{y}^{2}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548299\"><div data-type=\"problem\" id=\"fs-id1167829688071\"><p id=\"fs-id1167829688073\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(-6{c}^{4}\\right)\\left(-12c\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\frac{3}{5}{m}^{3}{n}^{2}\\right)\\left(\\frac{5}{9}{m}^{2}{n}^{3}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836440683\"><p id=\"fs-id1167836440685\"><span class=\"token\">\u24d0<\/span>\\(72{c}^{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{1}{3}{m}^{5}{n}^{5}\\)<\/p><\/div><\/div><p id=\"fs-id1167826172074\"><strong data-effect=\"bold\">Multiply a Polynomial by a Monomial<\/strong><\/p><p id=\"fs-id1167826172080\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1167826172083\"><div data-type=\"problem\" id=\"fs-id1167826172085\"><p id=\"fs-id1167826172087\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-8x\\left({x}^{2}+2x-15\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(5p{q}^{3}\\left({p}^{2}-2pq+6{q}^{2}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829732189\"><div data-type=\"problem\" id=\"fs-id1167829732191\"><p id=\"fs-id1167829732193\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-5t\\left({t}^{2}+3t-18\\right);\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(9{r}^{3}s\\left({r}^{2}-3rs+5{s}^{2}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167826171307\"><p id=\"fs-id1167826171310\"><span class=\"token\">\u24d0<\/span>\\(-5{t}^{3}-15{t}^{2}+90t\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(9s{r}^{5}-27{s}^{2}{r}^{4}+45{s}^{3}{r}^{3}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836400053\"><div data-type=\"problem\" id=\"fs-id1167836400055\"><p id=\"fs-id1167836400057\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-8y\\left({y}^{2}+2y-15\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-4{y}^{2}{z}^{2}\\left(3{y}^{2}+12yz-{z}^{2}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836429551\"><div data-type=\"problem\" id=\"fs-id1167836429554\"><p id=\"fs-id1167836429556\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-5m\\left({m}^{2}+3m-18\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-3{x}^{2}{y}^{2}\\left(7{x}^{2}+10xy-{y}^{2}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833232422\"><p id=\"fs-id1167833232424\"><span class=\"token\">\u24d0<\/span>\\(-5{m}^{3}-15{m}^{2}+90m\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-21{x}^{4}{y}^{2}-30{x}^{3}{y}^{3}+3{x}^{2}{y}^{4}\\)<\/div><\/div><p id=\"fs-id1167836573095\"><strong data-effect=\"bold\">Multiply a Binomial by a Binomial<\/strong><\/p><p id=\"fs-id1167836573100\">In the following exercises, multiply the binomials using <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-id1167836573104\"><div data-type=\"problem\" id=\"fs-id1167836573106\"><p id=\"fs-id1167836573108\">\\(\\left(w+5\\right)\\left(w+7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829809026\"><div data-type=\"problem\" id=\"fs-id1167829809029\"><p id=\"fs-id1167833020625\">\\(\\left(y+9\\right)\\left(y+3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829809070\"><p id=\"fs-id1167829809072\">\\({y}^{2}+12y+27\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833020623\"><p id=\"fs-id1167829809031\">\\(\\left(4p+11\\right)\\left(5p-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826172576\"><div data-type=\"problem\" id=\"fs-id1167826172578\"><p id=\"fs-id1167826172580\">\\(\\left(7q+4\\right)\\left(3q-8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826172619\"><p id=\"fs-id1167826172621\">\\(21{q}^{2}-44q-32\\)<\/p><\/div><\/div><p id=\"fs-id1167829614536\">In the following exercises, multiply the binomials. Use any method.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829614542\"><p id=\"fs-id1167829614544\">\\(\\left(x+8\\right)\\left(x+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833224114\"><div data-type=\"problem\" id=\"fs-id1167833224116\"><p id=\"fs-id1167833224118\">\\(\\left(y-6\\right)\\left(y-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832980635\"><p id=\"fs-id1167832980637\">\\({y}^{2}-8y+12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832980592\"><div data-type=\"problem\" id=\"fs-id1167832980594\"><p id=\"fs-id1167832980596\">\\(\\left(2t-9\\right)\\left(10t+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836615837\"><div data-type=\"problem\" id=\"fs-id1167836615839\"><p id=\"fs-id1167836615842\">\\(\\left(6p+5\\right)\\left(p+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836615878\"><p id=\"fs-id1167836615880\">\\(6{p}^{2}+11p+5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836498060\"><div data-type=\"problem\" id=\"fs-id1167836498062\"><p id=\"fs-id1167836498064\">\\(\\left(q-5\\right)\\left(q+8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833007565\"><div data-type=\"problem\" id=\"fs-id1167833007567\"><p id=\"fs-id1167833007569\">\\(\\left(m+11\\right)\\left(m-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836536203\"><p id=\"fs-id1167836536205\">\\({m}^{2}+7m-44\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836536162\"><div data-type=\"problem\" id=\"fs-id1167836536164\"><p id=\"fs-id1167836536166\">\\(\\left(7m+1\\right)\\left(m-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833327993\"><div data-type=\"problem\" id=\"fs-id1167833327995\"><p id=\"fs-id1167833327997\">\\(\\left(3r-8\\right)\\left(11r+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833328036\"><p id=\"fs-id1167833328038\">\\(33{r}^{2}-85r-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829849451\"><div data-type=\"problem\" id=\"fs-id1167829849453\"><p id=\"fs-id1167829849455\">\\(\\left({x}^{2}+3\\right)\\left(x+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833407519\"><div data-type=\"problem\" id=\"fs-id1167833407521\"><p id=\"fs-id1167833407523\">\\(\\left({y}^{2}-4\\right)\\left(y+3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833407485\"><p id=\"fs-id1167833407487\">\\({y}^{3}+3{y}^{2}-4y-12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829713072\"><div data-type=\"problem\" id=\"fs-id1167829713074\"><p id=\"fs-id1167829713076\">\\(\\left(5ab-1\\right)\\left(2ab+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833274042\"><div data-type=\"problem\" id=\"fs-id1167833274044\"><p id=\"fs-id1167833274046\">\\(\\left(2xy+3\\right)\\left(3xy+2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836714000\"><p id=\"fs-id1167836714002\">\\(6{x}^{2}{y}^{2}+13xy+6\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836714036\"><p id=\"fs-id1167836714038\">\\(\\left({x}^{2}+8\\right)\\left({x}^{2}-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833202435\"><div data-type=\"problem\" id=\"fs-id1167833202437\"><p id=\"fs-id1167833202439\">\\(\\left({y}^{2}-7\\right)\\left({y}^{2}-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836596854\"><p id=\"fs-id1167836596856\">\\({y}^{4}-11{y}^{2}+28\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836649660\"><div data-type=\"problem\" id=\"fs-id1167836649662\"><p id=\"fs-id1167836649665\">\\(\\left(6pq-3\\right)\\left(4pq-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738940\"><div data-type=\"problem\" id=\"fs-id1167829738942\"><p id=\"fs-id1167829738944\">\\(\\left(3rs-7\\right)\\left(3rs-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829738987\"><p id=\"fs-id1167829738990\">\\(9{r}^{2}{s}^{2}-33rs+28\\)<\/p><\/div><\/div><p id=\"fs-id1167833316662\"><strong data-effect=\"bold\">Multiply a Polynomial by a Polynomial<\/strong><\/p><p id=\"fs-id1167833316668\">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-id1167833316682\"><div data-type=\"problem\" id=\"fs-id1167833316684\"><p id=\"fs-id1167833316686\">\\(\\left(x+5\\right)\\left({x}^{2}+4x+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829908830\"><div data-type=\"problem\" id=\"fs-id1167829908832\"><p id=\"fs-id1167829908834\">\\(\\left(u+4\\right)\\left({u}^{2}+3u+2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833346192\"><p id=\"fs-id1167833346194\">\\({u}^{3}+7{u}^{2}+14u+8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171782121318\"><div data-type=\"problem\" id=\"fs-id1171782121321\"><p id=\"fs-id1171791619225\">\\(\\left(y+8\\right)\\left(4{y}^{2}+y-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791696924\"><div data-type=\"problem\" id=\"fs-id1171789580637\"><p id=\"fs-id1171789580639\">\\(\\left(a+10\\right)\\left(3{a}^{2}+a-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171782121850\"><p id=\"fs-id1171782121852\">\\(3{a}^{3}+31{a}^{2}+5a-50\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833369849\"><div data-type=\"problem\" id=\"fs-id1167833369851\"><p id=\"fs-id1167833369853\">\\(\\left({y}^{2}-3y+8\\right)\\left(4{y}^{2}+y-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833310592\"><div data-type=\"problem\" id=\"fs-id1167833310594\"><p id=\"fs-id1167833310596\">\\(\\left(2{a}^{2}-5a+10\\right)\\left(3{a}^{2}+a-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829748686\"><p id=\"fs-id1167829748688\">\\(6{a}^{4}-13{a}^{3}+15{a}^{2}+35a-50\\)<\/p><\/div><\/div><p id=\"fs-id1167829748732\"><strong data-effect=\"bold\">Multiply Special Products<\/strong><\/p><p id=\"fs-id1171789696083\">In the following exercises, multiply. Use either method.<\/p><div data-type=\"exercise\" id=\"fs-id1171789696086\"><div data-type=\"problem\" id=\"fs-id1171791619235\"><p id=\"fs-id1171791619238\">\\(\\left(w-7\\right)\\left({w}^{2}-9w+10\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791410355\"><div data-type=\"problem\" id=\"fs-id1171791410357\"><p id=\"fs-id1171791410359\">\\(\\left(p-4\\right)\\left({p}^{2}-6p+9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791605167\"><p id=\"fs-id1171791605169\">\\({p}^{3}-10{p}^{2}+33p-36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791603827\"><div data-type=\"problem\" id=\"fs-id1171791603829\"><p id=\"fs-id1171791603832\">\\(\\left(3q+1\\right)\\left({q}^{2}-4q-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791508624\"><div data-type=\"problem\" id=\"fs-id1171791508626\"><p id=\"fs-id1171791508628\">\\(\\left(6r+1\\right)\\left({r}^{2}-7r-9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791466488\"><p id=\"fs-id1171791466490\">\\(6{r}^{3}-41{r}^{2}-61r-9\\)<\/p><\/div><\/div><p id=\"fs-id1167829748738\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1167836694041\"><div data-type=\"problem\" id=\"fs-id1167836694043\"><p id=\"fs-id1167836694046\">\\({\\left(w+4\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836556957\"><div data-type=\"problem\" id=\"fs-id1167836556960\"><p id=\"fs-id1167836556962\">\\({\\left(q+12\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836556925\"><p id=\"fs-id1167836556928\">\\({q}^{2}+24q+144\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836694094\"><div data-type=\"problem\" id=\"fs-id1167836694096\"><p id=\"fs-id1167836694099\">\\({\\left(3x-y\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833046991\"><div data-type=\"problem\" id=\"fs-id1167833046994\"><p id=\"fs-id1167833046996\">\\({\\left(2y-3z\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833047022\"><p id=\"fs-id1167833047024\">\\(4{y}^{2}-12yz+9{z}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833053637\"><div data-type=\"problem\" id=\"fs-id1167833053639\"><p id=\"fs-id1167833053642\">\\({\\left(y+\\frac{1}{4}\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024831\"><div data-type=\"problem\" id=\"fs-id1167833024833\"><p id=\"fs-id1167833024836\">\\({\\left(x+\\frac{2}{3}\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720596\"><p id=\"fs-id1167829720598\">\\({x}^{2}+\\frac{4}{3}x+\\frac{4}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829720559\"><div data-type=\"problem\" id=\"fs-id1167829720561\"><p id=\"fs-id1167829720563\">\\({\\left(\\frac{1}{5}x-\\frac{1}{7}y\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829651524\"><div data-type=\"problem\" id=\"fs-id1167829651526\"><p id=\"fs-id1167829651528\">\\({\\left(\\frac{1}{8}x-\\frac{1}{9}y\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829651561\"><p id=\"fs-id1167829651563\">\\(\\frac{1}{64}{x}^{2}-\\frac{1}{36}xy+\\frac{1}{81}{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829620767\"><div data-type=\"problem\" id=\"fs-id1167829620769\"><p id=\"fs-id1167829620771\">\\({\\left(3{x}^{2}+2\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791328727\"><div data-type=\"problem\" id=\"fs-id1171791328729\"><p id=\"fs-id1171791591402\">\\({\\left(5{u}^{2}+9\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836685510\"><p>\\(25{u}^{4}+90{u}^{2}+81\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824767168\"><div data-type=\"problem\" id=\"fs-id1167824767170\"><p id=\"fs-id1167824767172\">\\({\\left(4{y}^{3}-2\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791372564\"><div data-type=\"problem\" id=\"fs-id1171791372566\"><p id=\"fs-id1171791613667\">\\({\\left(8{p}^{3}-3\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791656030\"><p id=\"fs-id1171791656032\">\\(64{p}^{6}-48{p}^{3}+9\\)<\/p><\/div><\/div><p id=\"fs-id1167829719256\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1167829719260\"><div data-type=\"problem\" id=\"fs-id1167829719263\"><p id=\"fs-id1167829719265\">\\(\\left(5k+6\\right)\\left(5k-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836624209\"><div data-type=\"problem\" id=\"fs-id1167836624211\"><p id=\"fs-id1167836624213\">\\(\\left(8j+4\\right)\\left(8j-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171781866227\"><p id=\"fs-id1171781866230\">\\(64{j}^{2}-16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836595091\"><div data-type=\"problem\" id=\"fs-id1167836595093\"><p id=\"fs-id1167836595095\">\\(\\left(11k+4\\right)\\left(11k-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833310901\"><div data-type=\"problem\" id=\"fs-id1167833310903\"><p id=\"fs-id1167833310905\">\\(\\left(9c+5\\right)\\left(9c-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791622738\"><p id=\"fs-id1171791622740\">\\(81{c}^{2}-25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833227141\"><div data-type=\"problem\" id=\"fs-id1167833227143\"><p id=\"fs-id1167833227145\">\\(\\left(9c-2d\\right)\\left(9c+2d\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836600150\"><div data-type=\"problem\" id=\"fs-id1167836600152\"><p id=\"fs-id1167836600154\">\\(\\left(7w+10x\\right)\\left(7w-10x\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791394525\"><p id=\"fs-id1171791394527\">\\(49{w}^{2}-100{x}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082358\"><div data-type=\"problem\" id=\"fs-id1167833082360\"><p id=\"fs-id1167833082363\">\\(\\left(m+\\frac{2}{3}n\\right)\\left(m-\\frac{2}{3}n\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824720904\"><div data-type=\"problem\" id=\"fs-id1167824720906\"><p id=\"fs-id1167824720908\">\\(\\left(p+\\frac{4}{5}q\\right)\\left(p-\\frac{4}{5}q\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171791640999\"><p id=\"fs-id1171791641001\">\\({p}^{2}-\\frac{16}{25}{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829620179\"><div data-type=\"problem\" id=\"fs-id1167829620181\"><p id=\"fs-id1167829620183\">\\(\\left(ab-4\\right)\\left(ab+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791687588\"><div data-type=\"problem\" id=\"fs-id1171791687590\"><p id=\"fs-id1171791396621\">\\(\\left(xy-9\\right)\\left(xy+9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171784001828\"><p id=\"fs-id1171784001830\">\\({x}^{2}{y}^{2}-81\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829878674\"><div data-type=\"problem\" id=\"fs-id1167829878677\"><p id=\"fs-id1167829878679\">\\(\\left(12{p}^{3}-11{q}^{2}\\right)\\left(12{p}^{3}+11{q}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1171791544836\"><div data-type=\"problem\" id=\"fs-id1171791544839\"><p id=\"fs-id1171791544841\">\\(\\left(15{m}^{2}-8{n}^{4}\\right)\\left(15{m}^{2}+8{n}^{4}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171784084359\"><p id=\"fs-id1171784084361\">\\(225{m}^{4}-64{n}^{8}\\)<\/p><\/div><\/div><p id=\"fs-id1167836560778\">In the following exercises, find each product.<\/p><div data-type=\"exercise\" id=\"fs-id1167836409046\"><div data-type=\"problem\" id=\"fs-id1167836409048\"><p id=\"fs-id1167836409050\">\\(\\left(p-3\\right)\\left(p+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409100\"><div data-type=\"problem\" id=\"fs-id1167836409102\"><p id=\"fs-id1167836409104\">\\({\\left(t-9\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832971208\"><p id=\"fs-id1167832971210\">\\({t}^{2}-18t+81\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832971232\"><div data-type=\"problem\" id=\"fs-id1167832971234\"><p id=\"fs-id1167832971236\">\\({\\left(m+n\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836480437\"><div data-type=\"problem\" id=\"fs-id1167836480439\"><p id=\"fs-id1167836480441\">\\(\\left(2x+y\\right)\\left(x-2y\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836480480\"><p id=\"fs-id1167836480482\">\\(2{x}^{2}-3xy-2{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826212507\"><div data-type=\"problem\" id=\"fs-id1167826212509\"><p id=\"fs-id1167826212512\">\\({\\left(2r+12\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824584387\"><div data-type=\"problem\" id=\"fs-id1167824584389\"><p id=\"fs-id1167824584391\">\\(\\left(3p+8\\right)\\left(3p-8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824584429\"><p id=\"fs-id1167824584432\">\\(9{p}^{2}-64\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824764481\"><div data-type=\"problem\" id=\"fs-id1167824764483\"><p id=\"fs-id1167824764486\">\\(\\left(7a+b\\right)\\left(a-7b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833129840\"><div data-type=\"problem\" id=\"fs-id1167833129842\"><p id=\"fs-id1167833129844\">\\({\\left(k-6\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833129867\"><p id=\"fs-id1167833129869\">\\({k}^{2}-12k+36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833129891\"><div data-type=\"problem\" id=\"fs-id1167833129893\"><p id=\"fs-id1167833129896\">\\({\\left({a}^{5}-7b\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824735076\"><div data-type=\"problem\" id=\"fs-id1167824735078\"><p id=\"fs-id1167824735080\">\\(\\left({x}^{2}+8y\\right)\\left(8x-{y}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829832169\"><p id=\"fs-id1167829832171\">\\(8{x}^{3}-{x}^{2}{y}^{2}+64xy-8{y}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833061674\"><div data-type=\"problem\" id=\"fs-id1167833061676\"><p id=\"fs-id1167833061679\">\\(\\left({r}^{6}+{s}^{6}\\right)\\left({r}^{6}-{s}^{6}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829784891\"><div data-type=\"problem\" id=\"fs-id1167829784893\"><p id=\"fs-id1167829784895\">\\({\\left({y}^{4}+2z\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829784925\"><p id=\"fs-id1167829784927\">\\({y}^{8}+4{y}^{4}z+4{z}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824715051\"><div data-type=\"problem\" id=\"fs-id1167824715054\"><p id=\"fs-id1167824715056\">\\(\\left({x}^{5}+{y}^{5}\\right)\\left({x}^{5}-{y}^{5}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836605561\"><div data-type=\"problem\" id=\"fs-id1167836605563\"><p id=\"fs-id1167836605565\">\\({\\left({m}^{3}-8n\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836605595\"><p id=\"fs-id1167836605597\">\\({m}^{6}-16{m}^{3}n+64{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829829968\"><div data-type=\"problem\" id=\"fs-id1167829829970\"><p id=\"fs-id1167829829972\">\\({\\left(9p+8q\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826188159\"><div data-type=\"problem\" id=\"fs-id1167826188161\"><p id=\"fs-id1167826188163\">\\(\\left({r}^{2}-{s}^{3}\\right)\\left({r}^{3}+{s}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826188211\"><p id=\"fs-id1167826188213\">\\({r}^{5}+{r}^{2}{s}^{2}-{r}^{3}{s}^{3}-{s}^{5}\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829715667\"><h4 data-type=\"title\">Mixed Practice<\/h4><div data-type=\"exercise\" id=\"fs-id1167829715673\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829715675\"><p id=\"fs-id1167829715677\">\\(\\left(10y-6\\right)+\\left(4y-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829968226\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829968228\"><p id=\"fs-id1167829968231\">\\(\\left(15p-4\\right)+\\left(3p-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829968268\"><p id=\"fs-id1167829968270\">\\(18p-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833009478\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833009480\"><p id=\"fs-id1167833009482\">\\(\\left({x}^{2}-4x-34\\right)-\\left({x}^{2}+7x-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830123628\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830123630\"><p id=\"fs-id1167830123632\">\\(\\left({j}^{2}-8j-27\\right)-\\left({j}^{2}+2j-12\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830123685\"><p id=\"fs-id1167830123687\">\\(-10j-15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833349987\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833349989\"><p id=\"fs-id1167833349991\">\\(\\left(\\frac{1}{5}{f}^{8}\\right)\\left(20{f}^{3}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833350047\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833350049\"><p id=\"fs-id1167829924534\">\\(\\left(\\frac{1}{4}{d}^{5}\\right)\\left(36{d}^{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829924573\"><p id=\"fs-id1167829924575\">\\(9{d}^{7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829924589\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829924591\"><p id=\"fs-id1167829924593\">\\(\\left(4{a}^{3}b\\right)\\left(9{a}^{2}{b}^{6}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836666692\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836666694\"><p id=\"fs-id1167836666696\">\\(\\left(6{m}^{4}{n}^{3}\\right)\\left(7m{n}^{5}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833085435\"><p id=\"fs-id1167833085437\">\\(72{m}^{5}{n}^{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833085457\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833085459\"><p id=\"fs-id1167833085461\">\\(-5m\\left({m}^{2}+3m-18\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832936619\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832936621\"><p id=\"fs-id1167832936623\">\\(5{q}^{3}\\left({q}^{2}-2q+6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826128373\"><p id=\"fs-id1167826128376\">\\(5{q}^{5}-10{q}^{4}+30{q}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826128409\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826128411\"><p id=\"fs-id1167826128413\">\\(\\left(s-7\\right)\\left(s+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744835\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829744837\"><p id=\"fs-id1167829744839\">\\(\\left({y}^{2}-2y\\right)\\left(y+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829849587\"><p id=\"fs-id1167829849589\">\\({y}^{3}-{y}^{2}-2y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829849614\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829849616\"><p id=\"fs-id1167829849619\">\\(\\left(5x-y\\right)\\left(x-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836601627\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836601630\"><p id=\"fs-id1167836601632\">\\(\\left(6k-1\\right)\\left({k}^{2}+2k-4\\right)\\)<\/p><\/div><div data-type=\"solution\"><p>\\(6{k}^{3}-11{k}^{2}-26k+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755828\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829755830\"><p id=\"fs-id1167829755832\">\\(\\left(3x-11y\\right)\\left(3x-11y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829830755\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829830757\"><p id=\"fs-id1167829830759\">\\(\\left(11-b\\right)\\left(11+b\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825702931\"><p id=\"fs-id1167825702933\">\\(121-{b}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825702949\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167825702951\"><p id=\"fs-id1167825702953\">\\(\\left(rs-\\frac{2}{7}\\right)\\left(rs+\\frac{2}{7}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836440801\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836440803\"><p id=\"fs-id1167836440805\">\\(\\left(2{x}^{2}-3{y}^{4}\\right)\\left(2{x}^{2}+3{y}^{4}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836536075\"><p id=\"fs-id1167836536078\">\\(4{x}^{4}-9{y}^{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836536101\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836536103\"><p id=\"fs-id1167836536105\">\\({\\left(m-15\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706906\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836706908\"><p id=\"fs-id1167836706910\">\\({\\left(3d+1\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836706939\">\\(9{d}^{2}+6d+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829692074\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829692077\"><p id=\"fs-id1167829692079\">\\({\\left(4a+10\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829692130\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829692132\"><p id=\"fs-id1167833361629\">\\({\\left(3z+\\frac{1}{5}\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833361657\"><p id=\"fs-id1167833361659\">\\(9{z}^{2}-\\frac{6}{5}z+\\frac{1}{25}\\)<\/p><\/div><\/div><p id=\"fs-id1167829594189\"><strong data-effect=\"bold\">Multiply Polynomial Functions<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167833361698\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833361700\"><p id=\"fs-id1167833361702\">For functions \\(f\\left(x\\right)=x+2\\) and \\(g\\left(x\\right)=3{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><div data-type=\"exercise\" id=\"fs-id1167833137952\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833137954\"><p id=\"fs-id1167833137956\">For functions \\(f\\left(x\\right)=x-1\\) and \\(g\\left(x\\right)=4{x}^{2}+3x-5,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836598716\"><p id=\"fs-id1167836598718\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(f\u00b7g\\right)\\left(x\\right)=4{x}^{3}-{x}^{2}-8x+5\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(f\u00b7g\\right)\\left(-2\\right)=-15\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833227311\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833227314\"><p id=\"fs-id1167833227316\">For functions \\(f\\left(x\\right)=2x-7\\) and \\(g\\left(x\\right)=2x+7,\\) find <span class=\"token\">\u24d0<\/span>\\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836553962\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836553964\"><p id=\"fs-id1167836553966\">For functions \\(f\\left(x\\right)=7x-8\\) and \\(g\\left(x\\right)=7x+8,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(f\u00b7g\\right)\\left(-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824578474\"><p id=\"fs-id1167824578475\"><span class=\"token\">\u24d0<\/span>\\(\\left(f\u00b7g\\right)\\left(x\\right)=49{x}^{2}-64\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(f\u00b7g\\right)\\left(-2\\right)=187\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829747512\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829747514\"><p id=\"fs-id1167829747517\">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-id1167829767053\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829767055\"><p id=\"fs-id1167829767058\">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-id1167829594113\"><p id=\"fs-id1167836508796\"><\/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><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167824648978\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167824648986\"><div data-type=\"problem\" id=\"fs-id1167824648988\"><p id=\"fs-id1167824648990\">Which method do you prefer to use when multiplying two binomials: the Distributive Property or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the Distributive Property or the Vertical Method? Why?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824649004\"><div data-type=\"problem\" id=\"fs-id1167824649006\"><p id=\"fs-id1167824649008\">Multiply the following:<\/p><p>\\(\\begin{array}{c}\\left(x+2\\right)\\left(x-2\\right)\\hfill \\\\ \\left(y+7\\right)\\left(y-7\\right)\\hfill \\\\ \\left(w+5\\right)\\left(w-5\\right)\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167829893380\">Explain the pattern that you see in your answers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829893384\"><p id=\"fs-id1167829893386\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829893392\"><div data-type=\"problem\" id=\"fs-id1167829893394\"><p id=\"fs-id1167829893396\">Multiply the following:<\/p><p id=\"fs-id1167829893399\">\\(\\begin{array}{c}\\left(p+3\\right)\\left(p+3\\right)\\hfill \\\\ \\left(q+6\\right)\\left(q+6\\right)\\hfill \\\\ \\left(r+1\\right)\\left(r+1\\right)\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167829811854\">Explain the pattern that you see in your answers.<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833109686\"><p id=\"fs-id1167833109688\">Why does \\({\\left(a+b\\right)}^{2}\\) result in a trinomial, but \\(\\left(a-b\\right)\\left(a+b\\right)\\) result in a binomial?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833109746\"><p id=\"fs-id1167833109748\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833109754\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167826205098\"><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-id1167826205106\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><p id=\"fs-id1167826205117\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167826205131\"><dt>conjugate pair<\/dt><dd id=\"fs-id1167826205135\">A conjugate pair is two binomials of the form \\(\\left(a-b\\right),\\left(a+b\\right).\\) The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/dd><\/dl><\/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>Multiply monomials<\/li>\n<li>Multiply a polynomial by a monomial<\/li>\n<li>Multiply a binomial by a binomial<\/li>\n<li>Multiply a polynomial by a polynomial<\/li>\n<li>Multiply special products<\/li>\n<li>Multiply polynomial functions<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836299778\" class=\"be-prepared\">\n<p id=\"fs-id1167829624120\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167833196772\" type=\"1\">\n<li>Distribute: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c46df630b9676f218b0e046bfec9ec18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829789060\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74deaa134f7b1f68fcc620bbda92abe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#57;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" 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-6bcd30149d10993d9904c9b42353f85f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" 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-67129ed26a111b5ab76f1d6e6fda0d7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#57;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e202bfb65e9e8db798b892727296e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/> for <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;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833303374\">\n<h3 data-type=\"title\">Multiply Monomials<\/h3>\n<p id=\"fs-id1167833021539\">We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829788772\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836525743\">\n<div data-type=\"problem\" id=\"fs-id1167833387010\">\n<p id=\"fs-id1167833022571\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c060cb4be436cd4e518c471f93d9b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cf505abcfcf619a7f7a7a1ce7ea684b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"123\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836520121\">\n<p id=\"fs-id1167836300221\"><span class=\"token\">\u24d0<\/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-4171eddbdf61bf1250e2f43e10ffa5c6_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#67;&#111;&#109;&#109;&#117;&#116;&#97;&#116;&#105;&#118;&#101;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#114;&#101;&#97;&#114;&#114;&#97;&#110;&#103;&#101;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#51;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&middot;&#123;&#120;&#125;&#94;&#123;&#51;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#50;&#123;&#120;&#125;&#94;&#123;&#53;&#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=\"62\" width=\"617\" style=\"vertical-align: -25px;\" \/><\/p>\n<p id=\"fs-id1167836689313\"><span class=\"token\">\u24d1<\/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-91ee3a91be9586d637dedb9ec3719df1_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#67;&#111;&#109;&#109;&#117;&#116;&#97;&#116;&#105;&#118;&#101;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#114;&#101;&#97;&#114;&#114;&#97;&#110;&#103;&#101;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&middot;&#49;&#50;&middot;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&middot;&#120;&middot;&#121;&middot;&#123;&#121;&#125;&#94;&#123;&#50;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#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=\"64\" width=\"633\" style=\"vertical-align: -26px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836353225\">\n<div data-type=\"problem\" id=\"fs-id1167836516000\">\n<p id=\"fs-id1167833056558\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c800c0cfed378583bcf9207a1a283db1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#121;&#125;&#94;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48bc29699dba587ffe3f641711d9b1c8_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;&#53;&#125;&#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;&#49;&#53;&#97;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"126\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836616379\">\n<p id=\"fs-id1167829709229\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e71a34a30fbbdb19fb0a8dc2f3f61311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#53;&#123;&#121;&#125;&#94;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" 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-358930741e27e8087988b4bf38e5722a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#123;&#98;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833050779\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829905091\">\n<p id=\"fs-id1167833031437\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9be9eb3b31b24f523c88c5b11c3c56a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69ee6ad703ab1aa97633714ba8bc84d2_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;&#114;&#125;&#94;&#123;&#53;&#125;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#114;&#125;&#94;&#123;&#54;&#125;&#123;&#115;&#125;&#94;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"126\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829713143\">\n<p id=\"fs-id1167836538207\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c710e8a163d832703c50c9f794a7d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;&#123;&#98;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"33\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbf0f230c3b7131ac0fd392e11302821_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#114;&#125;&#94;&#123;&#49;&#49;&#125;&#123;&#115;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836502215\">\n<h3 data-type=\"title\">Multiply a Polynomial by a Monomial<\/h3>\n<p>Multiplying a polynomial by a monomial is really just applying the Distributive Property.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829810683\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836699941\">\n<div data-type=\"problem\" id=\"fs-id1167836448130\">\n<p id=\"fs-id1167836665188\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b3bc89d642dbbaa6dd46670ba11f352_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"144\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-210ab06be826268ea57a626a6ebcaaa5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#121;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836293422\">\n<p id=\"fs-id1167836576202\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836568423\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of negative 2 y with the polynomial 4 y squared plus 3 y minus 5 in parentheses. Three arrows are drawn from the negative 2y pointing to each term in the polynomial in parentheses indicating the three multiplications. The next line shows the result when the negative 2 y is distributed: negative 2 y times 4 y squared plus negative 2 y times 3 y minus negative 2 y times 5. The simplified form is then negative 8 y to power of 3 minus 6 y to the power of 2 plus 10 y.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167829807325\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41964c4995689dcf95c6c54d9bdef249_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;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836515792\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167836543231\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836663108\"><span class=\"token\">\u24d1<\/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-365045dbb60d7a8d00359a0d25b91928_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#121;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#114;&#105;&#98;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&middot;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#120;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#123;&#121;&#125;&#94;&#123;&#50;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#121;&#45;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#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=\"63\" width=\"442\" style=\"vertical-align: -26px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836737859\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829720987\">\n<div data-type=\"problem\" id=\"fs-id1167836648572\">\n<p id=\"fs-id1167836512150\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3940a37acf53b7a505c0ff369398398c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"144\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b50dfa243fe386c52da71bb45da4ec6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#121;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#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-id1167836791173\">\n<p id=\"fs-id1167836613889\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28e71c999a73382022938b4c508bd6c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/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-799af038bc62240394973f0f8c0a6db5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"199\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833350613\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829810693\">\n<div data-type=\"problem\" id=\"fs-id1167829689196\">\n<p id=\"fs-id1167829737462\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0d015e6c6accfcd82e3aa0440af90eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"142\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f580c229578d24fb4a564e907f66ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#98;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"192\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829968361\">\n<p id=\"fs-id1167836730774\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb2a2e39d2626ddcb946badd077582a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/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-33d9d36d025cd97d064252825e3b644b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#98;&#43;&#49;&#50;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"198\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Multiply a Binomial by a Binomial<\/h3>\n<p id=\"fs-id1167833349690\">Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836495233\">\n<div data-type=\"problem\" id=\"fs-id1167836521435\">\n<p id=\"fs-id1167836309885\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-299c598e3c643e70ede2cf5c1e2cbcf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" 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-a14b2fd4dd84f1e8ac2fb219b57cef91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836352593\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829807702\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of y plus 5 in parentheses with y plus 8 in parentheses. y plus 8 in parentheses is colored red with two red arrows drawn from y plus 8 in parentheses pointing to the y and 5 in the y plus 5 factor. The next line shows the result when the y plus 8 is distributed: y times the quantity y plus 8 in parentheses plus 5 times the quantity y plus 8 in parentheses. Then we distribute again, distributing the y to the y plus 8 and the 5 to the y plus 8. The result is y squared plus 8 y plus 5 y plus 40. Combining like terms results in the simplified form y to power of 2 plus 13 y plus 40.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/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_03_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d06c68c7b2952f35fa444615a27d4c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836511238\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute again.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829596696\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811237\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836684819\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833274109\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of 4 y plus 3 in parentheses with 2 y minus 5 in parentheses. After distributing the quantity 2 y minus 5 in parentheses the result is 4 y times the quantity 2 y minus 5 in parentheses plus 3 times the quantity 2 y minus 5 in parentheses. Then we distribute the 4 y to the 2 y minus 5 and the 3 to the 2 y minus 5. The result is 8 y squared minus 20 y plus 6 y minus 15. Combining like terms results in the simplified form 8 y to power of 2 minus 14 y minus 15.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/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_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829879544\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute again.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836484890\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513988\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836409575\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836286748\">\n<div data-type=\"problem\" id=\"fs-id1167836326706\">\n<p id=\"fs-id1167833366001\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d479633810528f99882bd0ec45c5be00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#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;\" \/> <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04bcb0ae27a462ead6287c8cb9ddee09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#99;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#99;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836364079\">\n<p id=\"fs-id1167833310407\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc88d688fc794624259bd6d958273bd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#120;&#43;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/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-539652452c1567d7a65aaf2dcff50f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#99;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836526818\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836549431\">\n<div data-type=\"problem\" id=\"fs-id1167836623888\">\n<p id=\"fs-id1167829594717\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db3929afa5341e68ce1349cffcc342d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-4579317d812474580f292d06e93bcf8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836714500\">\n<p id=\"fs-id1167836685227\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d339dce86374b6734bb3e5d128ea679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#49;&#120;&#43;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/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-53d05ceef6d19db42f5fcbe6aff8d52e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#121;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836327796\">If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the <em data-effect=\"italics\">first<\/em> terms in each binomial. The second and third terms are the product of multiplying the two <em data-effect=\"italics\">outer<\/em> terms and then the two <em data-effect=\"italics\">inner<\/em> terms. And the last term results from multiplying the two <em data-effect=\"italics\">last<\/em> terms,<\/p>\n<p id=\"fs-id1167836597858\">We abbreviate \u201cFirst, Outer, Inner, Last\u201d as FOIL. The letters stand for \u2018First, Outer, Inner, Last\u2019. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.<\/p>\n<p>Let\u2019s multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f7d7d4da128ed48e4b1bf11475168e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> using both methods.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829739282\" data-alt=\"The figure shows how four terms in the product of two binomials can be remembered according to the mnemonic acronym FOIL. The example is the quantity x plus 3 in parentheses times the quantity x plus 7 in parentheses. The expression is expanded as in the previous examples by using the distributive property twice. After distributing the quantity x plus 7 in parentheses the result is x times the quantity x plus 7 in parentheses plus 3 times the quantity x plus 7 in parentheses. Then the x is distributed the x plus 7 and the 3 is distributed to the x plus 7 to get x squared plus 7 x plus 3 x plus 21. The letter F is written under the term x squared since it was the product of the first terms in the binomials. The letter O is written under the 7 x term sine it was the product of the outer terms in the binomials. The letter I is written under the 3 x term since it was the product of the inner terms in the binomials. The letter L is written under the 21 since it was the product of the last terms in the binomial. The original expression is shown again with four arrows connecting the first, outer, inner, and last terms in the binomials showing how the four terms can be determined directly from the factored form.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how four terms in the product of two binomials can be remembered according to the mnemonic acronym FOIL. The example is the quantity x plus 3 in parentheses times the quantity x plus 7 in parentheses. The expression is expanded as in the previous examples by using the distributive property twice. After distributing the quantity x plus 7 in parentheses the result is x times the quantity x plus 7 in parentheses plus 3 times the quantity x plus 7 in parentheses. Then the x is distributed the x plus 7 and the 3 is distributed to the x plus 7 to get x squared plus 7 x plus 3 x plus 21. The letter F is written under the term x squared since it was the product of the first terms in the binomials. The letter O is written under the 7 x term sine it was the product of the outer terms in the binomials. The letter I is written under the 3 x term since it was the product of the inner terms in the binomials. The letter L is written under the 21 since it was the product of the last terms in the binomial. The original expression is shown again with four arrows connecting the first, outer, inner, and last terms in the binomials showing how the four terms can be determined directly from the factored form.\" \/><\/span><\/p>\n<p id=\"fs-id1167836376675\">We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!<\/p>\n<div data-type=\"note\" id=\"fs-id1167833019452\" class=\"howto\">\n<div data-type=\"title\">Use the FOIL method to multiply two binomials.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836299260\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\" \/><\/span><\/div>\n<p id=\"fs-id1167836569080\">When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.<\/p>\n<p id=\"fs-id1167829741782\">Now we will do an example where we use the FOIL pattern to multiply two binomials.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836626974\">\n<p id=\"fs-id1167836731853\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97c5d9a5f09b64515544eb020bb0e2a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" 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-a0f861a34de4303f7d5571b15b0209ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<ol id=\"fs-id1171791694878\" type=\"1\" class=\"circled\">\n<li><span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836536657\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity y minus 7 in parentheses times the quantity y plus 4 in parentheses. Step 1. Multiply the First terms. The terms y and y are colored red with an arrow connecting them. The result is y squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The terms y and 4 are colored red with an arrow connecting them. The result is 4 y and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The terms negative 7 and y are colored red with an arrow connecting them. The result is negative 7 y squared and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The terms negative 7 and 4 are colored red with an arrow connecting them. The result is negative 28 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is y squared minus 3 y minus 28.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity y minus 7 in parentheses times the quantity y plus 4 in parentheses. Step 1. Multiply the First terms. The terms y and y are colored red with an arrow connecting them. The result is y squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The terms y and 4 are colored red with an arrow connecting them. The result is 4 y and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The terms negative 7 and y are colored red with an arrow connecting them. The result is negative 7 y squared and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The terms negative 7 and 4 are colored red with an arrow connecting them. The result is negative 28 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is y squared minus 3 y minus 28.\" \/><\/span><\/li>\n<li>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 4 x plus 3 in parentheses times the quantity 2 x minus 5 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms 4 x and 2 x. The product of the first terms is 8 x squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms 4 x and negative 5. The result is negative 20 x and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms 3 and 2 x. The result is 6 x and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms 3 and negative 5. The result is negative 15 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 8 y squared minus 14 x minus 15.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 4 x plus 3 in parentheses times the quantity 2 x minus 5 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms 4 x and 2 x. The product of the first terms is 8 x squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms 4 x and negative 5. The result is negative 20 x and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms 3 and 2 x. The result is 6 x and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms 3 and negative 5. The result is negative 15 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 8 y squared minus 14 x minus 15.\" \/><\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832999172\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829749818\">\n<div data-type=\"problem\" id=\"fs-id1167836548589\">\n<p id=\"fs-id1167836432034\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cfd4d1332bcb8056bacf47e7f1ff16e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#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;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cedb852e72cb8c8f409edf34b11760d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836309368\">\n<p id=\"fs-id1167836513138\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd5e0c8bc26b94ab11fec8330adc16b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"97\" style=\"vertical-align: 0px;\" \/><\/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-151c8dbd4ccd89f049440035ae00f6af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#57;&#120;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836510862\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836552129\">\n<p>Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75452931ebe4ae363a4800d70fb6c553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" 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-8785de810f149746ad34e702ae7f347a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836327641\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ae3c9d017b37128176df302f1f6ad33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#98;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/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-434572cd6d6c3f979d57ca99dee3c9d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#121;&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836356398\">The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836408987\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833060099\">\n<div data-type=\"problem\" id=\"fs-id1167829609240\">\n<p id=\"fs-id1167836597450\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ad1a7f575645e41ca08db7335c2f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a25745958ad43924f1a37a8de63a73c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#113;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#112;&#113;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836550155\">\n<p id=\"fs-id1167829984332\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836608558\" class=\"unnumbered unstyled\" summary=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity n squared plus 4 in parentheses times the quantity n minus 1 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms. The product of the first terms is n to the power of 3 and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The result is negative n squared and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The result is 4 n and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The result is negative 4 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is n to the power of 3 minus n squared plus 4 n minus 4.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547182\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/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_03_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Multiply the <em data-effect=\"italics\">First<\/em> terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836442295\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Multiply the <em data-effect=\"italics\">Outer<\/em> terms.<\/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_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Multiply the <em data-effect=\"italics\">Inner<\/em> terms.<\/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_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Multiply the <em data-effect=\"italics\">Last<\/em> terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507744\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Combine like terms\u2014there are none.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836650129\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829696676\" class=\"unnumbered unstyled\" summary=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity 3 p q plus 5 in parentheses times the quantity 6 p q minus 11 in parentheses. The expression is show with four red arrows connecting the First. Outer, Inner, and Last terms. Step 1. Multiply the First terms. The product of the first terms is 18 p squared q squared and is shown above the letter F in the word FOIL. Step 2. Multiply the Outer terms. The result is negative 33 p q and is shown above the letter O in the word FOIL. Step 3. Multiply the Inner terms. The result is 30 p q and is shown above the letter I in the word FOIL. Step 4. Multiply the Last terms. The result is negative 55 and is shown above the letter L in the word FOIL. Step 5. Combine like terms. The simplified result is 18 p squared q squared minus 3 p q minus 55.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792364493\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836501741\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Multiply the <em data-effect=\"italics\">First<\/em> terms.\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836754795\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Multiply the <em data-effect=\"italics\">Outer<\/em> terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836429820\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Multiply the <em data-effect=\"italics\">Inner<\/em> terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550146\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Multiply the <em data-effect=\"italics\">Last<\/em> terms.<\/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_03_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Combine like terms.<\/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_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829879584\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836309297\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833008167\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d65c3368acde3eede3ab9ec626a2abc_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;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"119\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0faf3b18bf7292a4092532fe054ecad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#98;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#98;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836570171\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d872753c83986fc947f47ff9227c0e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/><\/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-5e5362e7541fa2e20cd3e9f8bb9df5bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#97;&#98;&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833046878\">\n<p id=\"fs-id1167836688632\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10f90efe303c50e24213c12c8b39343a_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;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12176dc8ab08d5c086446fccebd30991_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829579784\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db1a6e114396760c9ba5bd769675f905_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#121;&#45;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/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-e69d66d8c7dd27ebcb03870f86cbd591_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#121;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171791276732\">The FOIL method is usually the quickest method for multiplying two binomials, but it <em data-effect=\"italics\">only<\/em> works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1171789718031\" data-alt=\"This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says \u201cYou start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products.\u201d\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says \u201cYou start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products.\u201d\" \/><\/span><\/p>\n<p id=\"fs-id1171791459374\">Now we\u2019ll apply this same method to multiply two binomials.<\/p>\n<div data-type=\"example\" id=\"fs-id1171791417364\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1171791398514\">\n<div data-type=\"problem\" id=\"fs-id1171791446238\">\n<p id=\"fs-id1171784054248\">Multiply using the Vertical Method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40b064239f35a98dc99b536e04dead80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171789922301\">\n<p id=\"fs-id1171787442859\">It does not matter which binomial goes on the top.<\/p>\n<p id=\"fs-id1171789582920\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f0366965a5dd4f0fa4fd49e4d47a31a_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;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#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;&#51;&#121;&#45;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#121;&#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;&#45;&#54;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#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;&#51;&#121;&#45;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#121;&#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;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#121;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#57;&#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;&#50;&#121;&#45;&#54;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#56;&#121;&#43;&#54;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#121;&#43;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#116;&#105;&#97;&#108;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#116;&#105;&#97;&#108;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#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=\"158\" width=\"552\" style=\"vertical-align: -75px;\" \/><\/p>\n<p id=\"fs-id1171791320793\">Notice the partial products are the same as the terms in the FOIL method.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1171791584225\" data-alt=\"This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1171791637725\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1171791450879\">\n<div data-type=\"problem\" id=\"fs-id1171789766013\">\n<p id=\"fs-id1171791631148\">Multiply using the Vertical Method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4c782548960b43bf986d3f0a3f3e8fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#109;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171784055217\">\n<p id=\"fs-id1171784121997\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b26c82838ca9b8bcdb241442307bfd0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#49;&#109;&#43;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1171789589582\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1171789574277\">\n<div data-type=\"problem\" id=\"fs-id1171782047419\">\n<p id=\"fs-id1171791419018\">Multiply using the Vertical Method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5a78257df69a98105bb7725bd2fd2a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#98;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791459738\">\n<p id=\"fs-id1171791319180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a14236a9c835689848671c3f291c56c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#51;&#98;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171791574620\">We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.<\/p>\n<div data-type=\"note\" id=\"fs-id1171782147854\">\n<div data-type=\"title\">Multiplying Two Binomials<\/div>\n<p id=\"fs-id1171784140960\">To multiply binomials, use the:<\/p>\n<ul id=\"fs-id1171791449840\" data-bullet-style=\"bullet\">\n<li>Distributive Property<\/li>\n<li>FOIL Method<\/li>\n<li>Vertical Method<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Multiply a Polynomial by a Polynomial<\/h3>\n<p id=\"fs-id1167836571662\">We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we\u2019re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836660220\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829738325\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836528267\">Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-436d4d9e2672e236ad4b9d2d639b5744_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#98;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -7px;\" \/> using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829597387\">\n<p id=\"fs-id1167833349853\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167824781546\" class=\"unnumbered unstyled\" summary=\"This figure shows how to distribute the multiplication of b plus 3 in parentheses with 2 b squared minus 5 b plus 8 in parentheses. After distributing the trinomial the result is b times the quantity 2 b squared minus 5 b plus 8 in parentheses plus 3 times the quantity 2 b squared minus 5 b plus 8 in parentheses 2 y minus 5 in parentheses. Then we distribute the b to the trinomial to get 2 b to the power of 3 minus 5 b squared plus 8 b and distribute the 3 to the trinomial to get 6 b squared minus 15 b plus 24. Combining like terms results in the simplified form 2 b to power of 3 plus b squared minus 7 b plus 24.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829711887\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829690938\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550571\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.\u2003\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833022850\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833346816\"><span class=\"token\">\u24d1<\/span> It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833377138\" class=\"unnumbered unstyled\" summary=\"This figure shows how to multiply the polynomials with the vertical method. The polynomial 2 b squared minus 5 b plus 8 is written directly over the polynomial b plus 3. The 8 is directly over the 3 and the negative 5 b is directly over the b. A horizontal line is drawn below the b plus 3. The result of multiplying 3 with the quantity 2 b squared minus 5 b plus 8 is written below the horizontal line. The result is get 6 b squared minus 15 b plus 24 with the 24 under the 3 and 8. The result of multiplying the b with the quantity 2 b squared minus 5 b plus 8 is written below the last calculation but shifted one term to the left. The result is 2 b to the power of 3 minus 5 b squared plus 8 b with the 8 b under the negative 15 b from the first multiplication. A second horizontal line is draw below the last result. The two multiplications are then added column by column. 24 is brought down since nothing is below it. Negative 15 b is added to 8 b to get negative 7 b. 6 b squared is added to negative 5 b squared to get b squared. 2 b to the power of 3 is brought down since nothing is above it. The final result is 2 b to power of 3 plus b squared minus 7 b plus 24.\" data-label=\"\">\n<tbody>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74606cbc614c4cc984b1e5e4d3b199f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#98;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -7px;\" \/> by 3.<\/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-74606cbc614c4cc984b1e5e4d3b199f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#98;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -7px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/>.<\/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_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2003\u2003<span data-type=\"media\" id=\"fs-id1167836309930\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>\u2003\u2003<span data-type=\"media\" id=\"fs-id1167836627519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836608523\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829744225\">\n<div data-type=\"problem\" id=\"fs-id1167836558379\">\n<p id=\"fs-id1167836730622\">Multiply<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bf607d3b9ccb480711f3609815e69cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -7px;\" \/> using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> the Vertical Method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836520603\">\n<p id=\"fs-id1167833008070\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01ad127b8e9f763be1f817ef69e2f1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#121;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/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-01ad127b8e9f763be1f817ef69e2f1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#121;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829790877\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833142588\">Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f8752048bd1ceca43b9f7691dd78d06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/> using <span class=\"token\">\u24d0<\/span> the Distributive Property and <span class=\"token\">\u24d1<\/span> The Vertical Method.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829880204\">\n<p id=\"fs-id1167824701386\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0276452f1a221ce4bf09fa5af933738f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/><\/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-01ad127b8e9f763be1f817ef69e2f1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#121;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836549264\">We have now seen two methods you can use to multiply a polynomial by a polynomial. After you practice each method, you\u2019ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836305065\">\n<div data-type=\"title\">Multiplying a Polynomial by a Polynomial<\/div>\n<p id=\"fs-id1167829594694\">To multiply a trinomial by a binomial, use the:<\/p>\n<ul id=\"fs-id1167829807211\" data-bullet-style=\"bullet\">\n<li>Distributive Property<\/li>\n<li>Vertical Method<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830123961\">\n<h3 data-type=\"title\">Multiply Special Products<\/h3>\n<p id=\"fs-id1167833196636\">Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let\u2019s start by looking at three examples and look for a pattern.<\/p>\n<p id=\"fs-id1167829786669\">Look at these results. Do you see any patterns?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833240126\" data-alt=\"The figure shows three examples of squaring a binomial. In the first example x plus 9 is squared to get x plus 9 times x plus 9 which is x squared plus 9 x plus 9 x plus 81 which simplifies to x squared plus 18 x plus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 7 is squared to get y minus y times y minus 7 which is y squared minus 7 y minus 7 y plus 49 which simplifies to y squared minus 14 y plus 49. Colors show that y squared comes from the square of the y in the original binomial and 49 comes from the square of the negative 7 in the original binomial. In the third example 2 x plus 3 is squared to get 2 x plus 3 times 2 x plus 3 which is 4 x squared plus 6 x plus 6 x plus 9 which simplifies to 4 x squared plus 12 x plus 9. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 9 comes from the square of the 3 in the original binomial.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three examples of squaring a binomial. In the first example x plus 9 is squared to get x plus 9 times x plus 9 which is x squared plus 9 x plus 9 x plus 81 which simplifies to x squared plus 18 x plus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 7 is squared to get y minus y times y minus 7 which is y squared minus 7 y minus 7 y plus 49 which simplifies to y squared minus 14 y plus 49. Colors show that y squared comes from the square of the y in the original binomial and 49 comes from the square of the negative 7 in the original binomial. In the third example 2 x plus 3 is squared to get 2 x plus 3 times 2 x plus 3 which is 4 x squared plus 6 x plus 6 x plus 9 which simplifies to 4 x squared plus 12 x plus 9. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 9 comes from the square of the 3 in the original binomial.\" \/><\/span><\/p>\n<p id=\"fs-id1167836628650\">What about the number of terms? In each example we squared a binomial and the result was a trinomial.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836698747\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28ff37e22569671969314100ad5a3621_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#95;&#95;&#95;&#43;&#95;&#95;&#95;&#43;&#95;&#95;&#95;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"97\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1167829609160\">Now look at the <em data-effect=\"italics\">first term<\/em> in each result. Where did it come from?<\/p>\n<p id=\"fs-id1167829947837\">The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!<\/p>\n<div data-type=\"equation\" id=\"fs-id1167830093894\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ffd1d7418bfb3bf4e5a73855456649d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#95;&#95;&#95;&#43;&#95;&#95;&#95;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"123\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"fs-id1167829715919\">\u2003\u2003<em data-effect=\"italics\">To get the first term of the product, square the first term.<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Where did the <em data-effect=\"italics\">last term<\/em> come from? Look at the examples and find the pattern.<\/p>\n<p id=\"fs-id1167836325829\">The last term is the product of the last terms, which is the square of the last term.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836547530\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d32332e5aca2ec8d6b69a033fd3e4b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#95;&#95;&#95;&#43;&#95;&#95;&#95;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"109\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"fs-id1167829738246\">\u2003\u2003<em data-effect=\"italics\">To get the last term of the product, square the last term.<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p id=\"fs-id1167833053517\">Finally, look at the <em data-effect=\"italics\">middle term<\/em>. Notice it came from adding the \u201couter\u201d and the \u201cinner\u201d terms\u2014which are both the same! So the middle term is double the product of the two terms of the binomial.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832951176\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53344e0e7643aa4f240c30b89f08b9cb_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;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#95;&#95;&#95;&#43;&#50;&#97;&#98;&#43;&#95;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#95;&#95;&#95;&#45;&#50;&#97;&#98;&#43;&#95;&#95;&#95;&#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=\"476\" style=\"vertical-align: -196px;\" \/><\/div>\n<p id=\"fs-id1167829879462\">\u2003\u2003<em data-effect=\"italics\">To get the middle term of the product, multiply the terms and double their product.<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p id=\"fs-id1167829694908\">Putting it all together:<\/p>\n<div data-type=\"note\" id=\"fs-id1167836524528\">\n<div data-type=\"title\">Binomial Squares Pattern<\/div>\n<p id=\"fs-id1167826188116\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829905170\" data-alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\" \/><\/span><\/p>\n<p id=\"fs-id1171791310441\">To square a binomial, square the first term, square the last term , double their product.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829711970\">\n<p id=\"fs-id1167836729154\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35e24c74083d85a2eea096d93b19410f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" 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-230f9bd27a56a45e0bd766d5226e0b6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836713911\">\n<p id=\"fs-id1167829713617\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833338740\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply x plus 5 squared using the formula a plus b squared equals a squared plus 2 a b plus b squared. Squaring the first term results in x squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times x times 5 which matches up with 2 a b in the formula. The simplified version is x squared plus 10 x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836511568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Square the first term.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829694664\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Square the last term.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829689053\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Double their product.\u2003\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836534247\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836386651\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836481274\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x minus 3 y squared using the formula a minus b squared equals a squared minus 2 a b plus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 3 y squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 2 x times 3 y which matches up with 2 a b in the formula. The simplified version is 4 x squared minus 12 x y plus 9 y squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836293412\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833057054\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829715332\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836519176\">\n<div data-type=\"problem\" id=\"fs-id1167833066308\">\n<p>Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbb904421ab1369a10c2a5bdb9a200dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" 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-800ca32276bceec60495183ac7f2cb22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#99;&#45;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833365593\">\n<p id=\"fs-id1167833018052\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-312fdde785aba692e6b3c496fe49d1ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/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-d01fe019b6b96095576ed18f982cdcb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#99;&#100;&#43;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836667065\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836543225\">\n<div data-type=\"problem\" id=\"fs-id1167833021492\">\n<p id=\"fs-id1167836697177\">Multiply: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2071674ff9bc6314daf2109d13e55f5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#49;&#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;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1382e2373a536a46db3ab5dad037134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836367497\">\n<p id=\"fs-id1167833328786\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dea78244edb73f1bceeea012993105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#50;&#121;&#43;&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/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-be317c1b4ea055bddb3b229703980226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#48;&#120;&#121;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836686187\">We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.<\/p>\n<p id=\"fs-id1167824617566\">A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a <strong data-effect=\"bold\">conjugate pair<\/strong> and is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df1b1ce95678059243781252da3af8a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167833345895\">\n<div data-type=\"title\">Conjugate Pair<\/div>\n<p id=\"fs-id1167836716282\">A <span data-type=\"term\" class=\"no-emphasis\">conjugate pair<\/span> is two binomials of the form<\/p>\n<div data-type=\"equation\" id=\"fs-id1167825835948\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df1b1ce95678059243781252da3af8a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836485185\">The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/p>\n<\/div>\n<p id=\"fs-id1167836408384\">There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let\u2019s look for the pattern by using FOIL to multiply some conjugate pairs.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167824617552\" data-alt=\"The figure shows three examples of multiplying a binomial with its conjugate. In the first example x plus 9 is multiplied with x minus 9 to get x squared minus 9 x plus 9 x minus 81 which simplifies to x squared minus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 8 is multiplied with y plus 8 to get y squared plus 8 y minus 8 y minus 64 which simplifies to y squared minus 64. Colors show that y squared comes from the square of the y in the original binomial and 64 comes from the square of the 8 in the original binomial. In the third example 2 x minus 5 is multiplied with 2 x plus 5 to get 4 x squared plus 10 x minus 10 x minus 25 which simplifies to 4 x squared minus 25. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 25 comes from the square of the 5 in the original binomial.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three examples of multiplying a binomial with its conjugate. In the first example x plus 9 is multiplied with x minus 9 to get x squared minus 9 x plus 9 x minus 81 which simplifies to x squared minus 81. Colors show that x squared comes from the square of the x in the original binomial and 81 comes from the square of the 9 in the original binomial. In the second example y minus 8 is multiplied with y plus 8 to get y squared plus 8 y minus 8 y minus 64 which simplifies to y squared minus 64. Colors show that y squared comes from the square of the y in the original binomial and 64 comes from the square of the 8 in the original binomial. In the third example 2 x minus 5 is multiplied with 2 x plus 5 to get 4 x squared plus 10 x minus 10 x minus 25 which simplifies to 4 x squared minus 25. Colors show that 4 x squared comes from the square of the 2 x in the original binomial and 25 comes from the square of the 5 in the original binomial.\" \/><\/span><\/p>\n<p id=\"fs-id1167829750123\">What do you observe about the products?<\/p>\n<p id=\"fs-id1167825702482\">The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.<\/p>\n<p id=\"fs-id1167829748237\">Each <em data-effect=\"italics\">first term<\/em> is the product of the first terms of the binomials, and since they are identical it is the square of the first term.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167833135469\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c2d08e6073d18ded0a5179dbe1a5189_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#95;&#95;&#95;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836664763\">\u2003\u2003\u2003<em data-effect=\"italics\">To get the first term, square the first term.<\/em><\/p>\n<p id=\"fs-id1167826172060\">The <em data-effect=\"italics\">last term<\/em> came from multiplying the last terms, the square of the last term.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832935990\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e6d02561987b611a25e33468aa7c1a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836606241\">\u2003\u2003\u2003<em data-effect=\"italics\">To get the last term, square the last term<\/em>.<\/p>\n<p id=\"fs-id1167829783760\">Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.<\/p>\n<p id=\"fs-id1167833139755\">The product of conjugates is always of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31eb1e3b99dd578a13ed1862b52a58b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/> This is called a <strong data-effect=\"bold\">difference of squares<\/strong>.<\/p>\n<p id=\"fs-id1167836360699\">This leads to the pattern:<\/p>\n<div data-type=\"note\" id=\"fs-id1167833349943\">\n<div data-type=\"title\">Product of Conjugates Pattern<\/div>\n<p id=\"fs-id1167833349948\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167825918985\" data-alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\" \/><\/span><\/p>\n<p id=\"fs-id1167824578716\">The product is called a difference of squares.<\/p>\n<p id=\"fs-id1167836717037\">To multiply conjugates, square the first term, square the last term, write it as a difference of squares.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836717042\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833020420\">\n<div data-type=\"problem\" id=\"fs-id1167833020422\">\n<p id=\"fs-id1167833020424\">Multiply using the product of conjugates pattern: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87f4e3e0a9f1e7bf43c54efb411e3874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-51d3756889be09fd4a06cb06920a73f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#109;&#45;&#57;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#109;&#43;&#57;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836722468\">\n<p id=\"fs-id1167836722470\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829694181\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x plus 5 times 2 x minus 5 using the formula a plus b times a minus b equals a squared minus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Simplifying the product results in 4 x squared minus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Are the binomials conjugates?<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836525165\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">It is the product of conjugates.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836648845\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Square the first term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77449fb24fbd9e26ca06af6cbe37217c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833051940\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Square the last term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6068d7b853acab337e4ed43538adccf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829739552\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify. The product is a difference of squares.\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833175364\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167830121978\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829689596\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 5 m minus 9 n times 5 m plus 9 n using the formula a minus b times a plus b equals a squared minus b squared. Squaring the first term results in 5 m squared which matches up with the term a squared in the formula. Squaring the last term results in 9 n squared which matches up with the term b squared in the formula. Simplifying the product results in 25 m squared minus 81 n squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824733940\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">This fits the pattern.\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829739570\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the pattern.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167825836088\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833186407\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836429507\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836429510\">\n<div data-type=\"problem\" id=\"fs-id1167836429512\">\n<p id=\"fs-id1167836429514\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67ebb0b3b20a5b16c4411f356f4bebea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-e52a1a325ea088e3a69aa18fea0fa86e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#45;&#55;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#43;&#55;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824733730\">\n<p id=\"fs-id1167824733732\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41800faa6af8a4f35c7b7ec074cbf243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: 0px;\" \/><\/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-8e5b0ca0dff5f9f08fcbc95fb0717430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836791451\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836791454\">\n<div data-type=\"problem\" id=\"fs-id1167836791456\">\n<p id=\"fs-id1167836791458\">Multiply: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7740c780253bcb542804f2420540c0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#55;&#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;\" \/> <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac3041a00a1f2c4f257382004700ca8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829860523\">\n<p id=\"fs-id1167833396771\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d7a3dbd8ebef181b79b07d42f2b1fb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f1db314b96c71687613e1c29debcb4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833202398\">We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836399600\">\n<div data-type=\"title\">Comparing the Special Product Patterns<\/div>\n<table id=\"fs-id1167836399606\" summary=\"Table has two columns. The left column lists binomial squares and shows two equations: a plus b in parentheses square equals a square plus 2 a b plus b squared and a minus b in parentheses squared equals a squared minus 2 ab plus b squared. Below the equations the text states squaring a binomial, product is a trinomial, inner and outer terms with foil are the same, and middle term is double the product of the terms. The right column lists product of conjugates with the equation a minus b times a plus b equals a squared minus b squared, Below the equation the test states multiplying conjugates, product is a binomial, inner and outer terms with foil are opposites, and there is no middle term.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"top\" data-align=\"center\">Binomial Squares<\/th>\n<th data-valign=\"top\" data-align=\"center\">Product of Conjugates<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\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-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" 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-84329933bdaada5a04696a6506bb6d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\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-a05bb1a6b69ab51f7d68b9cad2326fd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Squaring a binomial<\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Multiplying conjugates<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">trinomial<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">binomial.<\/strong><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">the same.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">opposites.<\/strong><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Middle term is <strong data-effect=\"bold\">double the product<\/strong> of the terms<\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003There is <strong data-effect=\"bold\">no<\/strong> middle term.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829748535\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829748537\">\n<div data-type=\"problem\" id=\"fs-id1167829748539\">\n<p id=\"fs-id1167829748541\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1167829748544\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-834f4efc848e7c65b64ca5db80586153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-2ad1e42600709f81c5ce8431d5cee912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#56;&#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;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-001bd5e0af0743a5f349cf18b4cb65c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#109;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" 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-a9cc49bfe6ab3a31a31ed19894cf34d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172057\">\n<p id=\"fs-id1167824733785\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-834f4efc848e7c65b64ca5db80586153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167826172195\">These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.<\/p>\n<table id=\"fs-id1167826172199\" class=\"unnumbered unstyled\" summary=\"The example shows how to multiply 2 x minus 3 times 2 x plus 3 using the formula a minus b times a plus b equals a squared minus b squared. Squaring the first term results in 2 x squared which matches up with the term a squared in the formula. Squaring the last term results in 3 squared which matches up with the term b squared in the formula. Simplifying the product results in 4 x squared minus 9.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836791299\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836791320\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824733756\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167824733770\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c65b9e9c4c1640438744a050eef5928e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#45;&#53;&#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<p id=\"fs-id1167825987072\">We are asked to square a binomial. It fits the binomial squares pattern.<\/p>\n<table id=\"fs-id1167825987075\" class=\"unnumbered unstyled\" summary=\"The example shows how to square 8 x minus 5 using the formula a minus b squared equals a squared minus 2 a b plus b squared. Squaring the first term results in the quantity 8 x in parentheses squared which matches up with the term a squared in the formula. Squaring the last term results in 5 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 8 x times 5 which matches up with 2 a b in the formula. The simplified version is 64 x squared minus 80 x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829738827\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829738848\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836713594\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836713608\"><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-001bd5e0af0743a5f349cf18b4cb65c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#109;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836700258\">Again, we will square a binomial so we use the binomial squares pattern.<\/p>\n<table id=\"fs-id1167836700261\" class=\"unnumbered unstyled\" summary=\"The example shows how to square 6 m plus 7 using the formula a plus b squared equals a squared plus 2 a b plus b squared. Squaring the first term results in the quantity 6 m in parentheses squared which matches up with the term a squared in the formula. Squaring the last term results in 7 squared which matches up with the term b squared in the formula. Doubling the product results in 2 times 6 m times 7 which matches up with 2 a b in the formula. The simplified version is 36 m squared plus 84 m plus 49.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829860644\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the pattern.\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829860665\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833349602\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833349616\"><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d951078281db37e1ee7c5bf0d394408b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167829715204\">This product does not fit the patterns, so we will use FOIL.<\/p>\n<p id=\"fs-id1167829715207\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8081fa04ed6ef6f9fa113ad8985a3092_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#70;&#79;&#73;&#76;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#120;&#45;&#51;&#54;&#120;&#45;&#51;&#48;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#45;&#51;&#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=\"61\" width=\"420\" style=\"vertical-align: -25px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833051916\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833051919\">\n<div data-type=\"problem\" id=\"fs-id1167833051922\">\n<p id=\"fs-id1167833051924\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1167833051927\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71424103589302253a54cbed32b13071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#98;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" 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-464aee29ee8dd3429d1cfc140a12338f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" 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-4c345bbd1319914d9e574f8a03bbc09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#121;&#43;&#49;&#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;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dbcda1e271d5fd52a513ce649e935d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#114;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#114;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836690445\">\n<p id=\"fs-id1167836690447\"><span class=\"token\">\u24d0<\/span> FOIL; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05bff08383e0bf45bb73f85986a173e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#55;&#98;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afebec9210e4d15ba9da87863f392dec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#112;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7993739088453adf9654bac6e7c8b714_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Product of Conjugates; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-445c0c14b1ae0638eaceceb22c8e7b0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825836245\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167825836248\">\n<div data-type=\"problem\" id=\"fs-id1167825836250\">\n<p id=\"fs-id1167825836252\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1167825836256\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc5c7e5121a038e94ca009de9e2e4832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#55;&#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;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8f5ab687361994c8169fc2810b7b16c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-5b8de97089b547b1f50ab4525e58cccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" 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-db5525acfb662b1c112501d558be8f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825987123\">\n<p id=\"fs-id1167825987125\"><span class=\"token\">\u24d0<\/span> Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52cc53827d98c39eb294a7f2e07f7071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#120;&#43;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d1<\/span> Product of Conjugates; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b696c75cf014e5f253b533b9fce13f95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d2<\/span> FOIL; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfad41ec954bdcdb7bdcd16a7877753c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#57;&#120;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d3<\/span> Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a93d37c279b13794c059a35a3e5a719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#110;&#43;&#49;\" 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\" id=\"fs-id1167832936015\">\n<h3 data-type=\"title\">Multiply Polynomial Functions<\/h3>\n<p id=\"fs-id1167832936020\">Just as polynomials can be multiplied, polynomial functions can also be multiplied.<\/p>\n<div data-type=\"note\" id=\"fs-id1167832936023\">\n<div data-type=\"title\">Multiplication of Polynomial Functions<\/div>\n<p id=\"fs-id1167832936029\">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;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167829878976\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f674001d8ff512c0fd188de36ec714a3_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;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829621729\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829621731\">\n<div data-type=\"problem\" id=\"fs-id1167829621733\">\n<p id=\"fs-id1167829621735\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e650278017e06b9362c36cf971a111fe_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;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fb205d691461c1dbf388e768ea59c3f_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;&#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-c17c2a6bcf851bf9f50add5faaa8715b_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172148\">\n<p id=\"fs-id1167826172150\"><span class=\"token\">\u24d0<\/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-11e09efc810fb53a44fe692f0ddd8a95_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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;&#50;&#125;&#45;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#114;&#105;&#98;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"637\" style=\"vertical-align: -48px;\" \/><\/p>\n<p id=\"fs-id1167829739447\"><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-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;\" \/> and now are asked to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c17c2a6bcf851bf9f50add5faaa8715b_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836554557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e48002917dccf0ccbdd67f97aed882e_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#45;&#56;&#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;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#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;&#50;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&middot;&#50;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#45;&#52;&#45;&#50;&#48;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#52;&#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=\"556\" style=\"vertical-align: -37px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829718351\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829718355\">\n<div data-type=\"problem\" id=\"fs-id1167829718357\">\n<p id=\"fs-id1167829718360\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2acc44aa785d9addd2704fdb32cb0f9_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;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93f342a73a3c6cdaecba1a8168b23777_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;&#50;&#120;&#43;&#51;&#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-c17c2a6bcf851bf9f50add5faaa8715b_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829689565\">\n<p id=\"fs-id1167829689567\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee9fa5203295142aad9e27e3fc26054c_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;&#51;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#120;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/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-114e462c984941d1164e5d713fc090c7_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836648781\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836648785\">\n<div data-type=\"problem\" id=\"fs-id1167836648787\">\n<p id=\"fs-id1167836648789\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfddbed7967a580925ee45a62e31a76e_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;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e4e3e6d733da0920c08155fbe0386ab_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;&#56;&#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-c17c2a6bcf851bf9f50add5faaa8715b_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829614450\">\n<p id=\"fs-id1167829614452\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bcb924f4437dde5c02ee045d5b10e2a_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;&#51;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#50;&#120;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"229\" style=\"vertical-align: -4px;\" \/><\/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-80b87d7a5b5364a3cc871dc60e84488c_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825681186\" class=\"media-2\">\n<p id=\"fs-id1167825681190\">Access this online resource for additional instruction and practice with multiplying polynomials.<\/p>\n<ul id=\"fs-id1167825681194\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37Introspecprod\">Introduction to special products of binomials<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167825681208\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167825681215\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to use the FOIL method to multiply two binomials.<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836409131\" data-alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows how to use the FOIL method to multiply two binomials. The example is the quantity a plus b in parentheses times the quantity c plus d in parentheses. The numbers a and c are labeled first and the numbers b and d are labeled last. The numbers b and c are labeled inner and the numbers a and d are labeled outer. A note on the side of the expression tells you to Say it as you multiply! FOIL First Outer Inner Last. The directions are then given in numbered steps. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last Terms. Step 5. Combine like terms when possible.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Multiplying Two Binomials:<\/strong> To multiply binomials, use the:\n<ul id=\"fs-id1167836409155\" data-bullet-style=\"open-circle\">\n<li>Distributive Property<\/li>\n<li>FOIL Method<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Multiplying a Polynomial by a Polynomial:<\/strong> To multiply a trinomial by a binomial, use the:\n<ul id=\"fs-id1167836409175\" data-bullet-style=\"open-circle\">\n<li>Distributive Property<\/li>\n<li>Vertical Method<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Binomial Squares Pattern<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers, <span data-type=\"media\" id=\"fs-id1167836536277\" data-alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of squaring two binomials. The first example is a plus b squared equals a squared plus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a plus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared. The second example is a minus b squared equals a squared minus 2 a b plus b squared. The equation is written out again with each part labeled. The quantity a minus b squared is labeled binomial squared. The terms a squared is labeled first term squared. The term negative 2 a b is labeled 2 times product of terms. The term b squared is labeled last term squared.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Product of Conjugates Pattern<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfaed44949cf9cfbeb3445de33aabd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"25\" style=\"vertical-align: -4px;\" \/> are real numbers<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167836536314\" data-alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the result of multiplying a binomial with its conjugate. The formula is a plus b times a minus b equals a squared minus b squared. The equation is written out again with labels. The product a plus b times a minus b is labeled conjugates. The result a squared minus b squared is labeled difference of squares.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The product is called a difference of squares.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> To multiply conjugates, square the first term, square the last term, write it as a difference of squares.<\/li>\n<li><strong data-effect=\"bold\">Comparing the Special Product Patterns<\/strong>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833346332\" class=\"unnumbered\" summary=\"Table has two columns. The left column lists binomial squares and shows two equations: a plus b in parentheses square equals a square plus 2 a b plus b squared and a minus b in parentheses squared equals a squared minus 2 ab plus b squared. Below the equations the text states squaring a binomial, product is a trinomial, inner and outer terms with foil are the same, and middle term is double the product of the terms. The right column lists product of conjugates with the equation a minus b times a plus b equals a squared minus b squared, Below the equation the test states multiplying conjugates, product is a binomial, inner and outer terms with foil are opposites, and there is no middle term.\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"top\" data-align=\"center\">Binomial Squares<\/th>\n<th data-valign=\"top\" data-align=\"center\">Product of Conjugates<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\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-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" 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-a05bb1a6b69ab51f7d68b9cad2326fd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\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-84329933bdaada5a04696a6506bb6d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Squaring a binomial<\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Multiplying conjugates<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">trinomial<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Product is a <strong data-effect=\"bold\">binomial.<\/strong><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">the same.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Inner and outer terms with FOIL are <strong data-effect=\"bold\">opposites.<\/strong><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003Middle term is <strong data-effect=\"bold\">double the product<\/strong> of the terms<\/td>\n<td data-valign=\"top\" data-align=\"left\">\u2022\u2003\u2003There is <strong data-effect=\"bold\">no<\/strong> middle term.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong data-effect=\"bold\">Multiplication of Polynomial Functions:<\/strong>\n<ul id=\"fs-id1167829828527\" data-bullet-style=\"open-circle\">\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;\" \/>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167829828567\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f674001d8ff512c0fd188de36ec714a3_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;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836788245\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836788250\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836788257\"><strong data-effect=\"bold\">Multiply Monomials<\/strong><\/p>\n<p id=\"fs-id1167836788263\">In the following exercises, multiply the monomials.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829833532\">\n<div data-type=\"problem\" id=\"fs-id1167829833535\">\n<p id=\"fs-id1167829833537\">\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-31c3322fce516081c74616b2075ed0e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#121;&#125;&#94;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -7px;\" \/><\/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-913f04d0871c67e9e653f2430ebf7739_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#55;&#125;&#114;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#114;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"109\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826024750\">\n<div data-type=\"problem\" id=\"fs-id1167833021418\">\n<p id=\"fs-id1167833021421\">\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-6c541d34c713f8ba8d86b3ddee047fbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/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-f61e55ae5486cae7013bfcd41c58f96f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836664315\">\n<p id=\"fs-id1167836664317\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a24bd20e3bd661cd3e50ad2721c4bb9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"35\" 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-1e8cf6cec08c583438752ffb402ce938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836664356\">\n<div data-type=\"problem\" id=\"fs-id1167836664358\">\n<p id=\"fs-id1167836738069\">\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-f860f4cac507668ac835af46c59be7ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#123;&#117;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#117;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -7px;\" \/><\/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-ffd8d7c855a995949b70feb90f54c35b_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;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548299\">\n<div data-type=\"problem\" id=\"fs-id1167829688071\">\n<p id=\"fs-id1167829688073\">\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-ba99dfdab22207ef5bef22bc1497e436_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#123;&#99;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"108\" style=\"vertical-align: -7px;\" \/><\/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-7d425544d024e7b91192beeb3daf13d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440683\">\n<p id=\"fs-id1167836440685\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d35fc17b90b0eef1fc43bb812c635e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#123;&#99;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" 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-2326a8d8f603d6a30f8e1afa98bab8e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826172074\"><strong data-effect=\"bold\">Multiply a Polynomial by a Monomial<\/strong><\/p>\n<p id=\"fs-id1167826172080\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826172083\">\n<div data-type=\"problem\" id=\"fs-id1167826172085\">\n<p id=\"fs-id1167826172087\">\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-9f5987203aa978d4772c9c55689856a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -7px;\" \/><\/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-109adae9dc3c44da88c7b66372b2c1c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#112;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#112;&#113;&#43;&#54;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"162\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829732189\">\n<div data-type=\"problem\" id=\"fs-id1167829732191\">\n<p id=\"fs-id1167829732193\">\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-f7afa2dc5ed9ec4fbb1b911f5d03ef58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#116;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#116;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"145\" style=\"vertical-align: -7px;\" \/><\/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-694708fb6cd7b0b28c3a8b790db7f4b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#115;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#114;&#115;&#43;&#53;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"160\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167826171307\">\n<p id=\"fs-id1167826171310\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-949ba0744ca256426447ada86ceb5d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#123;&#116;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#53;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#48;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" style=\"vertical-align: -2px;\" \/><\/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-e1f22fe42089b893b30c1562ddbe3324_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#115;&#123;&#114;&#125;&#94;&#123;&#53;&#125;&#45;&#50;&#55;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#123;&#114;&#125;&#94;&#123;&#52;&#125;&#43;&#52;&#53;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#123;&#114;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"176\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836400053\">\n<div data-type=\"problem\" id=\"fs-id1167836400055\">\n<p id=\"fs-id1167836400057\">\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-8055b02284ef87719c482c40e8a20b3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"145\" style=\"vertical-align: -7px;\" \/><\/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-b6a3e26b7a2957ae6b32ed633a91a20b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#122;&#45;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"194\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836429551\">\n<div data-type=\"problem\" id=\"fs-id1167836429554\">\n<p id=\"fs-id1167836429556\">\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-84c02901046358eb7df595bf5dcea86c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -7px;\" \/><\/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-8ebcc60c938fc935921ef168363f67d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#121;&#45;&#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;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833232422\">\n<p id=\"fs-id1167833232424\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c8bae577bce6647e31cab2a62c300f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#48;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/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-6edd19e1cdb689d0c4d29c7ace010a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836573095\"><strong data-effect=\"bold\">Multiply a Binomial by a Binomial<\/strong><\/p>\n<p id=\"fs-id1167836573100\">In the following exercises, multiply the binomials using <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-id1167836573104\">\n<div data-type=\"problem\" id=\"fs-id1167836573106\">\n<p id=\"fs-id1167836573108\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36d28eaf0758e07a0e58f43044beb90b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829809026\">\n<div data-type=\"problem\" id=\"fs-id1167829809029\">\n<p id=\"fs-id1167833020625\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-986c796b3eefa77cb6389970606e66e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#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 data-type=\"solution\" id=\"fs-id1167829809070\">\n<p id=\"fs-id1167829809072\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dbc28b3a3bfb580af571c234c5b92d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#43;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833020623\">\n<p id=\"fs-id1167829809031\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-126d7966601163538f282f205c0e8fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826172576\">\n<div data-type=\"problem\" id=\"fs-id1167826172578\">\n<p id=\"fs-id1167826172580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-947e1e6de1ce7c959f2a8a36f222b02a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#113;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#113;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172619\">\n<p id=\"fs-id1167826172621\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8fa8416ddc8daf7957197e37bf95986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#52;&#113;&#45;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829614536\">In the following exercises, multiply the binomials. Use any method.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829614542\">\n<p id=\"fs-id1167829614544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85b95c6e4332c9830469d033f971202f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167833224114\">\n<div data-type=\"problem\" id=\"fs-id1167833224116\">\n<p id=\"fs-id1167833224118\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7f1545f5631ee9289ee050407e80912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#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 data-type=\"solution\" id=\"fs-id1167832980635\">\n<p id=\"fs-id1167832980637\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fd59f553ce82c9f9034e63691383e73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#121;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832980592\">\n<div data-type=\"problem\" id=\"fs-id1167832980594\">\n<p id=\"fs-id1167832980596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7a142124ac672355434433fa0552623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#116;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#116;&#43;&#49;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167836615837\">\n<div data-type=\"problem\" id=\"fs-id1167836615839\">\n<p id=\"fs-id1167836615842\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9fe37138bf6c71f8f2d7a2aae65ee8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#112;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#49;&#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-id1167836615878\">\n<p id=\"fs-id1167836615880\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7289757c9a50676abab6c9c5c8ea3dc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#112;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836498060\">\n<div data-type=\"problem\" id=\"fs-id1167836498062\">\n<p id=\"fs-id1167836498064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c07f4e7de24573a21b1699a8b02fd039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833007565\">\n<div data-type=\"problem\" id=\"fs-id1167833007567\">\n<p id=\"fs-id1167833007569\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9c12837c4efbae147884e4612278b20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#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-id1167836536203\">\n<p id=\"fs-id1167836536205\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac31c98983f98e6e5ee556faa625262c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#109;&#45;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836536162\">\n<div data-type=\"problem\" id=\"fs-id1167836536164\">\n<p id=\"fs-id1167836536166\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f06747517fecebc3a91cf2e01e3e00b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#51;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167833327993\">\n<div data-type=\"problem\" id=\"fs-id1167833327995\">\n<p id=\"fs-id1167833327997\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f7a8571bf5625348e1025c6286364f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#114;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#114;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833328036\">\n<p id=\"fs-id1167833328038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-186d2d50e21eab4411a1db8505f840d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#53;&#114;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"112\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829849451\">\n<div data-type=\"problem\" id=\"fs-id1167829849453\">\n<p id=\"fs-id1167829849455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6239779c34d3d4ab417908866c8d9910_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"119\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833407519\">\n<div data-type=\"problem\" id=\"fs-id1167833407521\">\n<p id=\"fs-id1167833407523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5faf4187fe31101b8486011a76abcd61_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;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833407485\">\n<p id=\"fs-id1167833407487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4285c5ea9f5bf2e0c5b77d33b0483392_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829713072\">\n<div data-type=\"problem\" id=\"fs-id1167829713074\">\n<p id=\"fs-id1167829713076\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6343adbb6da15e9eff46ab3b46bf182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#97;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833274042\">\n<div data-type=\"problem\" id=\"fs-id1167833274044\">\n<p id=\"fs-id1167833274046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f865fecdb628c7517454efe1e3524048_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#121;&#43;&#50;&#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 data-type=\"solution\" id=\"fs-id1167836714000\">\n<p id=\"fs-id1167836714002\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fed04928d20955cf89443d42546eceb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#120;&#121;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836714036\">\n<p id=\"fs-id1167836714038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-312f439d91d7af43d2bbc87ebb50002b_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;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"128\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833202435\">\n<div data-type=\"problem\" id=\"fs-id1167833202437\">\n<p id=\"fs-id1167833202439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1ed1e2cb5a29b06de0cda4b933d0718_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;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"127\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836596854\">\n<p id=\"fs-id1167836596856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e8ff66017cc972a37315646e5f7170a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836649660\">\n<div data-type=\"problem\" id=\"fs-id1167836649662\">\n<p id=\"fs-id1167836649665\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7aec0d20c1de5ad3a46b4763cbe947e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#112;&#113;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#113;&#45;&#53;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167829738940\">\n<div data-type=\"problem\" id=\"fs-id1167829738942\">\n<p id=\"fs-id1167829738944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c9d15f071ac47a69613c4a95e61fecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#114;&#115;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#114;&#115;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829738987\">\n<p id=\"fs-id1167829738990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83a33a4a96f9a60e87e63ede87a6bb2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#51;&#114;&#115;&#43;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833316662\"><strong data-effect=\"bold\">Multiply a Polynomial by a Polynomial<\/strong><\/p>\n<p id=\"fs-id1167833316668\">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-id1167833316682\">\n<div data-type=\"problem\" id=\"fs-id1167833316684\">\n<p id=\"fs-id1167833316686\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb3a182e65fc5d773ba06afeac599b77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"160\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829908830\">\n<div data-type=\"problem\" id=\"fs-id1167829908832\">\n<p id=\"fs-id1167829908834\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1490aa45e3fca592d3607e2cd1f399c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#117;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"160\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833346192\">\n<p id=\"fs-id1167833346194\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd95a54e934ecb4edcd1312e0ac48985_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#43;&#55;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#117;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171782121318\">\n<div data-type=\"problem\" id=\"fs-id1171782121321\">\n<p id=\"fs-id1171791619225\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2ecb8f7f99360a74d4a12a020f00749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791696924\">\n<div data-type=\"problem\" id=\"fs-id1171789580637\">\n<p id=\"fs-id1171789580639\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c1292301fb7871560b3dec0dfceb2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#45;&#53;&#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-id1171782121850\">\n<p id=\"fs-id1171782121852\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d67961d6ae109f3d4343e94c4f6174a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#49;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#97;&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833369849\">\n<div data-type=\"problem\" id=\"fs-id1167833369851\">\n<p id=\"fs-id1167833369853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee0861175a1124c16215324a1c46e09e_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;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#45;&#55;&#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-id1167833310592\">\n<div data-type=\"problem\" id=\"fs-id1167833310594\">\n<p id=\"fs-id1167833310596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-774703423dc6a1a591986b437dc4c347_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"224\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829748686\">\n<p id=\"fs-id1167829748688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b6be2dc724094009aed0dd68b39cf93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#51;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#53;&#97;&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"227\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829748732\"><strong data-effect=\"bold\">Multiply Special Products<\/strong><\/p>\n<p id=\"fs-id1171789696083\">In the following exercises, multiply. Use either method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1171789696086\">\n<div data-type=\"problem\" id=\"fs-id1171791619235\">\n<p id=\"fs-id1171791619238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0c8d13ae5d5a2424224dbe5fe479c16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#119;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"178\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791410355\">\n<div data-type=\"problem\" id=\"fs-id1171791410357\">\n<p id=\"fs-id1171791410359\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4fd098bb81ab43dc0a15356853ba39b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#112;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"156\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791605167\">\n<p id=\"fs-id1171791605169\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61a7f77efa1de80ba31dda1ba862c73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#48;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#51;&#112;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791603827\">\n<div data-type=\"problem\" id=\"fs-id1171791603829\">\n<p id=\"fs-id1171791603832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f29f5de48393c506d8874e48f353cb9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#113;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#113;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"164\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791508624\">\n<div data-type=\"problem\" id=\"fs-id1171791508626\">\n<p id=\"fs-id1171791508628\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1e218ec536ce6e871fd2644a1fc5d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#114;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#114;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"164\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791466488\">\n<p id=\"fs-id1171791466490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d54e7a2d6ff559124f4db5c1cfdeecc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#49;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#49;&#114;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"159\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829748738\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836694041\">\n<div data-type=\"problem\" id=\"fs-id1167836694043\">\n<p id=\"fs-id1167836694046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08c387ecac46d590b2780872b2b691c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836556957\">\n<div data-type=\"problem\" id=\"fs-id1167836556960\">\n<p id=\"fs-id1167836556962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-880c4af710d86f2f7ad9b31b4e56c7fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836556925\">\n<p id=\"fs-id1167836556928\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b51add5c0247816bd7d8b1359801b111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#113;&#43;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836694094\">\n<div data-type=\"problem\" id=\"fs-id1167836694096\">\n<p id=\"fs-id1167836694099\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-611965b85a4c9619d830002fc6f0ff44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#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-id1167833046991\">\n<div data-type=\"problem\" id=\"fs-id1167833046994\">\n<p id=\"fs-id1167833046996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b58402b47b15379a3bfd678f77fd1417_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#51;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833047022\">\n<p id=\"fs-id1167833047024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a45104700e1a3f1b961d32b74279349b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#122;&#43;&#57;&#123;&#122;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833053637\">\n<div data-type=\"problem\" id=\"fs-id1167833053639\">\n<p id=\"fs-id1167833053642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f140632a81db0939fb13b19f01887f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024831\">\n<div data-type=\"problem\" id=\"fs-id1167833024833\">\n<p id=\"fs-id1167833024836\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af80f3bfd9e36cde6d0841e2eef0e088_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;&#50;&#125;&#123;&#51;&#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-id1167829720596\">\n<p id=\"fs-id1167829720598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b108173faeae1b9d3f989013f76917b4_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;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720559\">\n<div data-type=\"problem\" id=\"fs-id1167829720561\">\n<p id=\"fs-id1167829720563\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30499140c71aeae44c5d7a1a3e38ca04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#55;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651524\">\n<div data-type=\"problem\" id=\"fs-id1167829651526\">\n<p id=\"fs-id1167829651528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-731f6e1d2e90ac5878c2a93884a181d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829651561\">\n<p id=\"fs-id1167829651563\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-381585b64cd2f44a579ee55fd5affbbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#54;&#125;&#120;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"148\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829620767\">\n<div data-type=\"problem\" id=\"fs-id1167829620769\">\n<p id=\"fs-id1167829620771\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5248ff65b20af3867e896302cad8c7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791328727\">\n<div data-type=\"problem\" id=\"fs-id1171791328729\">\n<p id=\"fs-id1171791591402\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9eb2376df7570f496de3f3fbd929ea35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685510\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e6f8cc2da21a4dfbe359e1cafb3cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#43;&#57;&#48;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824767168\">\n<div data-type=\"problem\" id=\"fs-id1167824767170\">\n<p id=\"fs-id1167824767172\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a273a7442457c5093c8a0f4324640547_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791372564\">\n<div data-type=\"problem\" id=\"fs-id1171791372566\">\n<p id=\"fs-id1171791613667\"><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 data-type=\"solution\" id=\"fs-id1171791656030\">\n<p id=\"fs-id1171791656032\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c144bb025db29c3959ce13f01d519ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#112;&#125;&#94;&#123;&#54;&#125;&#45;&#52;&#56;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829719256\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829719260\">\n<div data-type=\"problem\" id=\"fs-id1167829719263\">\n<p id=\"fs-id1167829719265\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ecc824e2fc55baf13e4e92d0ae2e0ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#107;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836624209\">\n<div data-type=\"problem\" id=\"fs-id1167836624211\">\n<p id=\"fs-id1167836624213\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5341b76eaa9962e35039c5c5d09e728c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#106;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#106;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171781866227\">\n<p id=\"fs-id1171781866230\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31bd689ede5e47b3e009f537f15c2340_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#106;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836595091\">\n<div data-type=\"problem\" id=\"fs-id1167836595093\">\n<p id=\"fs-id1167836595095\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1995ae9214917453a83726cdb80b069f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#107;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#107;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833310901\">\n<div data-type=\"problem\" id=\"fs-id1167833310903\">\n<p id=\"fs-id1167833310905\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02528cb5cc2d9083233d00e1620009e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#53;&#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 data-type=\"solution\" id=\"fs-id1171791622738\">\n<p id=\"fs-id1171791622740\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99ca19c4f8177e6f77d1076bba023655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833227141\">\n<div data-type=\"problem\" id=\"fs-id1167833227143\">\n<p id=\"fs-id1167833227145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286195b24cdae56772b032fd099def5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#43;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836600150\">\n<div data-type=\"problem\" id=\"fs-id1167836600152\">\n<p id=\"fs-id1167836600154\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb4d4dfecf381812f67bbf08871e15bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#119;&#43;&#49;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#119;&#45;&#49;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171791394525\">\n<p id=\"fs-id1171791394527\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2fe29ffdd9b27920592af9cca58d1147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833082358\">\n<div data-type=\"problem\" id=\"fs-id1167833082360\">\n<p id=\"fs-id1167833082363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c60080ce5aab0885d6b1ae15ad902ef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"149\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824720904\">\n<div data-type=\"problem\" id=\"fs-id1167824720906\">\n<p id=\"fs-id1167824720908\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ef336c1f653e10f0064bd7ede16dcee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#113;&#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 data-type=\"solution\" id=\"fs-id1171791640999\">\n<p id=\"fs-id1171791641001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-533b0af717cba25ee8b86cf46175f05c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#50;&#53;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829620179\">\n<div data-type=\"problem\" id=\"fs-id1167829620181\">\n<p id=\"fs-id1167829620183\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e8c1717020ba8d3088954481f1fccf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1171791687588\">\n<div data-type=\"problem\" id=\"fs-id1171791687590\">\n<p id=\"fs-id1171791396621\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f44fc40d410c580abaafbb7dfba17010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#43;&#57;&#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-id1171784001828\">\n<p id=\"fs-id1171784001830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-495cdeb1c778fc1fc10bd1d1506a6f43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;\" 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-id1167829878674\">\n<div data-type=\"problem\" id=\"fs-id1167829878677\">\n<p id=\"fs-id1167829878679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73bca837fc18bac3c2a9b95511a702b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#49;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#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-id1171791544836\">\n<div data-type=\"problem\" id=\"fs-id1171791544839\">\n<p id=\"fs-id1171791544841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48da09faf6b4136e682cfadefeb5e692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#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 data-type=\"solution\" id=\"fs-id1171784084359\">\n<p id=\"fs-id1171784084361\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2521f4f31d63b41b865ac24329ed10be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#53;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#54;&#52;&#123;&#110;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836560778\">In the following exercises, find each product.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836409046\">\n<div data-type=\"problem\" id=\"fs-id1167836409048\">\n<p id=\"fs-id1167836409050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cedef0c8aaf6a2be62c8da925ebe7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#51;&#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-id1167836409100\">\n<div data-type=\"problem\" id=\"fs-id1167836409102\">\n<p id=\"fs-id1167836409104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b16ad6f040394f8967bc6b5d9bc8888b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832971208\">\n<p id=\"fs-id1167832971210\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f37f698ba178e4054f31a01253b4ad9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#116;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"99\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832971232\">\n<div data-type=\"problem\" id=\"fs-id1167832971234\">\n<p id=\"fs-id1167832971236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b18968046794dcfefc0d58d65558b601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836480437\">\n<div data-type=\"problem\" id=\"fs-id1167836480439\">\n<p id=\"fs-id1167836480441\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c804d0967eedc3199dcbad4b8bd5184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#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-id1167836480480\">\n<p id=\"fs-id1167836480482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-097b98f45afd404732bb678ef21f8543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#121;&#45;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826212507\">\n<div data-type=\"problem\" id=\"fs-id1167826212509\">\n<p id=\"fs-id1167826212512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-010754e05eeaf2e133304ec096739027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#114;&#43;&#49;&#50;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167824584387\">\n<div data-type=\"problem\" id=\"fs-id1167824584389\">\n<p id=\"fs-id1167824584391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1af6419091d72a823bb4c200caaa9879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824584429\">\n<p id=\"fs-id1167824584432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e441c3c9b25263613ebc65dd9957d36b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;\" 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-id1167824764481\">\n<div data-type=\"problem\" id=\"fs-id1167824764483\">\n<p id=\"fs-id1167824764486\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26810c6033f2bbd628b69cd33d4d3e18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#55;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833129840\">\n<div data-type=\"problem\" id=\"fs-id1167833129842\">\n<p id=\"fs-id1167833129844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1becfc3c29342c37cd6f9ed7963b469c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833129867\">\n<p id=\"fs-id1167833129869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0b519e3726346b19593c30d9ab80b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#107;&#43;&#51;&#54;\" 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-id1167833129891\">\n<div data-type=\"problem\" id=\"fs-id1167833129893\">\n<p id=\"fs-id1167833129896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39dcc31c0bc5a1c89073d03b5f342916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#45;&#55;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"76\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824735076\">\n<div data-type=\"problem\" id=\"fs-id1167824735078\">\n<p id=\"fs-id1167824735080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7530eb3031f97d7ba7bfba6221fa16a8_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;&#56;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829832169\">\n<p id=\"fs-id1167829832171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76c6d5fd86b777dcdd932771d326b681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#120;&#121;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833061674\">\n<div data-type=\"problem\" id=\"fs-id1167833061676\">\n<p id=\"fs-id1167833061679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62551eb8545ee4c0d0bede50f4a785e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#54;&#125;&#43;&#123;&#115;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#54;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#54;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167829784891\">\n<div data-type=\"problem\" id=\"fs-id1167829784893\">\n<p id=\"fs-id1167829784895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e7c317815c665702604dc37007c5304_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;&#43;&#50;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829784925\">\n<p id=\"fs-id1167829784927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-538ed5ef0f89d5eb96e406cbe1383d4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#56;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#122;&#43;&#52;&#123;&#122;&#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-id1167824715051\">\n<div data-type=\"problem\" id=\"fs-id1167824715054\">\n<p id=\"fs-id1167824715056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ee1303216ae8627c21847dde21969cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836605561\">\n<div data-type=\"problem\" id=\"fs-id1167836605563\">\n<p id=\"fs-id1167836605565\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b4598a3aef0eb2119678554f5131afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"86\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836605595\">\n<p id=\"fs-id1167836605597\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe5f1f9de54421e4d77156ed9c95eed9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#54;&#125;&#45;&#49;&#54;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#110;&#43;&#54;&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829829968\">\n<div data-type=\"problem\" id=\"fs-id1167829829970\">\n<p id=\"fs-id1167829829972\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7be131f431b23bc0aa25e63bb1cb78e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#112;&#43;&#56;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167826188159\">\n<div data-type=\"problem\" id=\"fs-id1167826188161\">\n<p id=\"fs-id1167826188163\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef31476c6bb2df7b20987d0469d503eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#115;&#125;&#94;&#123;&#50;&#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-id1167826188211\">\n<p id=\"fs-id1167826188213\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-865eaa5f7b543726e9aacb39f0f0294d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#53;&#125;&#43;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"161\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829715667\">\n<h4 data-type=\"title\">Mixed Practice<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829715673\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829715675\">\n<p id=\"fs-id1167829715677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae2d3280cc38c8c55fd42ae0f0b0d258_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829968226\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829968228\">\n<p id=\"fs-id1167829968231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2622c9654b2967531d1e1f93fed06b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829968268\">\n<p id=\"fs-id1167829968270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9af0a33c499cf4ad32414e53ea73f357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#112;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833009478\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833009480\">\n<p id=\"fs-id1167833009482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dee68fa2377955fe70f0cfd49ec4654f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#51;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"237\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830123628\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830123630\">\n<p id=\"fs-id1167830123632\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9dc6e43c2c24b6336876dd6846d83ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#106;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#106;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#106;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#106;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"239\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830123685\">\n<p id=\"fs-id1167830123687\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6805bd14c3fb06b142e04296c24ca162_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#106;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833349987\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833349989\">\n<p id=\"fs-id1167833349991\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6ada03819aa030140d098bdd68c85c8_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;&#53;&#125;&#123;&#102;&#125;&#94;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#123;&#102;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"96\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833350047\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833350049\">\n<p id=\"fs-id1167829924534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05482cf6006ff57ad7b0866ac33712ed_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;&#52;&#125;&#123;&#100;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#54;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829924573\">\n<p id=\"fs-id1167829924575\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-defe5acc935bda74ca9cb1ec8a5d35c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#100;&#125;&#94;&#123;&#55;&#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-id1167829924589\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829924591\">\n<p id=\"fs-id1167829924593\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e61e2169b7491a4f440977b5c451db2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836666692\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836666694\">\n<p id=\"fs-id1167836666696\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a550ef75f8e8be7e6586e6e697ea17ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#109;&#123;&#110;&#125;&#94;&#123;&#53;&#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-id1167833085435\">\n<p id=\"fs-id1167833085437\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1da89ab45d2671c5d53c11f9790ece9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833085457\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833085459\">\n<p id=\"fs-id1167833085461\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84c02901046358eb7df595bf5dcea86c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#45;&#49;&#56;&#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\" id=\"fs-id1167832936619\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832936621\">\n<p id=\"fs-id1167832936623\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e411416ddbbdb9c006e7ee2b31a8a22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#113;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"128\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826128373\">\n<p id=\"fs-id1167826128376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c4965907ebcbb6e5aa579f8fe863581_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#45;&#49;&#48;&#123;&#113;&#125;&#94;&#123;&#52;&#125;&#43;&#51;&#48;&#123;&#113;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826128409\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826128411\">\n<p id=\"fs-id1167826128413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b16c5b87daff62e985b5caf0673cc96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744835\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829744837\">\n<p id=\"fs-id1167829744839\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b879c7f84c5586768d1654e47044987_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;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"127\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829849587\">\n<p id=\"fs-id1167829849589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2944ba3f1fc314265e242b31bed7f283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829849614\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829849616\">\n<p id=\"fs-id1167829849619\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3782f7d837cbdf7bc2956061928ffcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#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-id1167836601627\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836601630\">\n<p id=\"fs-id1167836601632\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c48376373886f3068f58b1c8f15c2fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#107;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#107;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"168\" 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-9c833e6f04c108d33ed57bb49619f22c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#107;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#54;&#107;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"163\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755828\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829755830\">\n<p id=\"fs-id1167829755832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54451b60921d800bfcbd216cf12d8372_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#49;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#49;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829830755\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829830757\">\n<p id=\"fs-id1167829830759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc10880b79bc7b849abfbf9eae6848ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#43;&#98;&#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 data-type=\"solution\" id=\"fs-id1167825702931\">\n<p id=\"fs-id1167825702933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7838ebc82c9d712bb58a801547ef312e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825702949\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825702951\">\n<p id=\"fs-id1167825702953\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b57e1a7003f2585db415855c48ac2dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#115;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#115;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"130\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836440801\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836440803\">\n<p id=\"fs-id1167836440805\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae037cde7f0aa7d7215f23bbaaee2d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"180\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836536075\">\n<p id=\"fs-id1167836536078\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99c37ebc5c69090cfcaaa9ce2743cfbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#56;&#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-id1167836536101\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836536103\">\n<p id=\"fs-id1167836536105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2f991b138e56244214379274ba55676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706906\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836706908\">\n<p id=\"fs-id1167836706910\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdfa77ff39a2965fe6cf996f9d2f5264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836706939\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8064c5055ea2516e090c569acfbe8f93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#100;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829692074\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829692077\">\n<p id=\"fs-id1167829692079\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12c9e3330bc8830c08d965920fe04eff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#43;&#49;&#48;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167829692130\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829692132\">\n<p id=\"fs-id1167833361629\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1cd62a04783c35ee74d0d835b2f7cfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#122;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833361657\">\n<p id=\"fs-id1167833361659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0b1cd97c58bf2b3ae5a1ab5e981684f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#122;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829594189\"><strong data-effect=\"bold\">Multiply Polynomial Functions<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833361698\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833361700\">\n<p id=\"fs-id1167833361702\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e650278017e06b9362c36cf971a111fe_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;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b17bc77455a50213c36a2453fcdc07e4_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;&#50;&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" 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-id1167833137952\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833137954\">\n<p id=\"fs-id1167833137956\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a0d850b3fbc28081967637c96067d3a_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;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d57345ec33152e685164fb63f877930_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;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" 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-1a912e2b212dbd98f209d7c25059bc2d_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;&#50;&#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 data-type=\"solution\" id=\"fs-id1167836598716\">\n<p id=\"fs-id1167836598718\">\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-c8ae3ae4af0d86450763dac8ee72c295_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;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -4px;\" \/><\/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-bd808a0e22562a6a6b7263aa993434e5_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833227311\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833227314\">\n<p id=\"fs-id1167833227316\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-756350376238bfbab56a2e3a28c94ae0_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;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02008a37821a4af0acef93a1dfebda6d_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;&#120;&#43;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" 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-24812f695e2b58cc8c320462e1e31a87_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;&#51;&#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-id1167836553962\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836553964\">\n<p id=\"fs-id1167836553966\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3c416ecbd5558af400b250d1110aaf9_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;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a08416d62f2f2d96e35eaefca212d8e_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;&#55;&#120;&#43;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" 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-1a912e2b212dbd98f209d7c25059bc2d_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;&#50;&#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 data-type=\"solution\" id=\"fs-id1167824578474\">\n<p id=\"fs-id1167824578475\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf3c36755ddf326ad231fc51a07b899a_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;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/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-ba8b0e3593871c4edbb8b206697beeea_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#56;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747512\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829747514\">\n<p id=\"fs-id1167829747517\">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-id1167829767053\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829767055\">\n<p id=\"fs-id1167829767058\">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-id1167829594113\">\n<p id=\"fs-id1167836508796\">\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-a940b1631cf603937924484af3ac02f0_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167824648978\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167824648986\">\n<div data-type=\"problem\" id=\"fs-id1167824648988\">\n<p id=\"fs-id1167824648990\">Which method do you prefer to use when multiplying two binomials: the Distributive Property or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the Distributive Property or the Vertical Method? Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824649004\">\n<div data-type=\"problem\" id=\"fs-id1167824649006\">\n<p id=\"fs-id1167824649008\">Multiply the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-725c0f50de5beded1aabfa66033ee2fe_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;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"116\" style=\"vertical-align: -26px;\" \/><\/p>\n<p id=\"fs-id1167829893380\">Explain the pattern that you see in your answers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829893384\">\n<p id=\"fs-id1167829893386\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829893392\">\n<div data-type=\"problem\" id=\"fs-id1167829893394\">\n<p id=\"fs-id1167829893396\">Multiply the following:<\/p>\n<p id=\"fs-id1167829893399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67c08df750cd03390e3ca0fd67c477a1_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;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"108\" style=\"vertical-align: -26px;\" \/><\/p>\n<p id=\"fs-id1167829811854\">Explain the pattern that you see in your answers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833109686\">\n<p id=\"fs-id1167833109688\">Why does <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcf4905322cd42d5aefebd64aa6a523b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"58\" style=\"vertical-align: -4px;\" \/> result in a trinomial, but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b62cb51f5db877a2930ac0090238692d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/> result in a binomial?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833109746\">\n<p id=\"fs-id1167833109748\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833109754\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167826205098\"><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-id1167826205106\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_05_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1167826205117\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167826205131\">\n<dt>conjugate pair<\/dt>\n<dd id=\"fs-id1167826205135\">A conjugate pair is two binomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df1b1ce95678059243781252da3af8a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/> The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2920","chapter","type-chapter","status-publish","hentry"],"part":2727,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2920","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":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2920\/revisions"}],"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\/2920\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2920"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2920"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2920"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}