{"id":3047,"date":"2018-12-11T13:50:54","date_gmt":"2018-12-11T18:50:54","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/factor-trinomials\/"},"modified":"2018-12-11T13:50:54","modified_gmt":"2018-12-11T18:50:54","slug":"factor-trinomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/factor-trinomials\/","title":{"raw":"Factor Trinomials","rendered":"Factor Trinomials"},"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>Factor trinomials of the form \\({x}^{2}+bx+c\\)<\/li><li>Factor trinomials of the form \\(a{x}^{2}+bx+c\\) using trial and error<\/li><li>Factor trinomials of the form \\(a{x}^{2}+bx+c\\) using the \u2018ac\u2019 method<\/li><li>Factor using substitution<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167833052618\" class=\"be-prepared\"><p id=\"fs-id1167836363691\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167833047839\" type=\"1\"><li>Find all the factors of 72.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167829937177\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Find the product: \\(\\left(3y+4\\right)\\left(2y+5\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836544266\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(-9\\left(6\\right);\\) \\(-9\\left(-6\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536325\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836601351\"><h3 data-type=\"title\">Factor Trinomials of the Form \\({x}^{2}+bx+c\\)<\/h3><p id=\"fs-id1167824578628\">You have already learned how to multiply binomials using <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>. Now you\u2019ll need to \u201cundo\u201d this multiplication. To factor the trinomial means to start with the product, and end with the factors.<\/p><span data-type=\"media\" id=\"fs-id1167836508876\" data-alt=\"Figure shows the equation open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 5 x plus 6. The left side of the equation is labeled factors and the right is labeled product. An arrow pointing right is labeled multiply. An arrow pointing left is labeled factor.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the equation open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 5 x plus 6. The left side of the equation is labeled factors and the right is labeled product. An arrow pointing right is labeled multiply. An arrow pointing left is labeled factor.\"><\/span><p id=\"fs-id1167836545205\">To figure out how we would factor a <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> of the form \\({x}^{2}+bx+c,\\) such as \\({x}^{2}+5x+6\\) and factor it to \\(\\left(x+2\\right)\\left(x+3\\right),\\) let\u2019s start with two general binomials of the form \\(\\left(x+m\\right)\\) and \\(\\left(x+n\\right).\\)<\/p><table id=\"fs-id1167836549815\" class=\"unnumbered unstyled\" summary=\"We have open parentheses x plus m close parentheses open parentheses x plus n close parentheses. Foil to find the product x squared plus mx plus nx plus mn. Factor the GCF from the middle terms x squared plus open parentheses m plus n close parentheses x plus mn. Now our trinomial is of the form x squared plus bx plus c, where b is m plus n and c is mn\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836444935\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Foil to find the product.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836366433\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the GCF from the middle 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_06_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Our trinomial is of the form \\({x}^{2}+bx+c.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056814\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836625705\">This tells us that to factor a trinomial of the form \\({x}^{2}+bx+c,\\) we need two factors \\(\\left(x+m\\right)\\) and \\(\\left(x+n\\right)\\) where the two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> multiply to <em data-effect=\"italics\">c<\/em> and add to <em data-effect=\"italics\">b<\/em>.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor a Trinomial of the form \\({x}^{2}+bx+c\\)<\/div><div data-type=\"exercise\" id=\"fs-id1167833386897\"><div data-type=\"problem\" id=\"fs-id1167836525654\"><p>Factor: \\({x}^{2}+11x+24.\\)<\/p><\/div><div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167829714211\" data-alt=\"Step 1 is to write the factors of x squared plus 11x plus 24 as two binomials with first terms x. Write two sets of parentheses and put x as the first term.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the factors of x squared plus 11x plus 24 as two binomials with first terms x. Write two sets of parentheses and put x as the first term.\"><\/span><span data-type=\"media\" id=\"fs-id1167829676303\" data-alt=\"Step 2 is to find two numbers m and n that multiply to c, m times n is c and add to b, m plus n is b. So, find two numbers that multiply to 24 and add to 11. Factors of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6. Sum of factors: 1 plus 24 is 25, 2 plus 12 is 14, 3 plus 8 is 11 and 4 plus 6 is 10.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find two numbers m and n that multiply to c, m times n is c and add to b, m plus n is b. So, find two numbers that multiply to 24 and add to 11. Factors of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6. Sum of factors: 1 plus 24 is 25, 2 plus 12 is 14, 3 plus 8 is 11 and 4 plus 6 is 10.\"><\/span><span data-type=\"media\" data-alt=\"Step 3 is to use m and n, in this case, 3 and 8, as the last terms of the binomials. So we get open parentheses x plus 3 close parentheses open parentheses x plus 8 close parentheses\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use m and n, in this case, 3 and 8, as the last terms of the binomials. So we get open parentheses x plus 3 close parentheses open parentheses x plus 8 close parentheses\"><\/span><span data-type=\"media\" data-alt=\"Step 4 is to check by multiplying the factors to get the original polynomial.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check by multiplying the factors to get the original polynomial.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836377040\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836538765\"><div data-type=\"problem\" id=\"fs-id1167836543830\"><p id=\"fs-id1167836449827\">Factor: \\({q}^{2}+10q+24.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512368\"><p>\\(\\left(q+4\\right)\\left(q+6\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836609921\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836547465\"><p>Factor: \\({t}^{2}+14t+24.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836282499\"><p id=\"fs-id1167836360137\">\\(\\left(t+2\\right)\\left(t+12\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836545745\">Let\u2019s summarize the steps we used to find the factors.<\/p><div data-type=\"note\" id=\"fs-id1167836732813\" class=\"howto\"><div data-type=\"title\">Factor trinomials of the form \\({x}^{2}+bx+c.\\)<\/div><ol type=\"1\" class=\"stepwise\"><li>Write the factors as two binomials with first terms <em data-effect=\"italics\">x<\/em>. \\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}{x}^{2}+bx+c\\hfill \\\\ \\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\hfill \\end{array}\\)<\/li><li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that <ul id=\"fs-id1167825908932\" data-bullet-style=\"bullet\"><li>multiply to \\(c,m\u00b7n=c\\)<\/li><li>add to \\(b,m+n=b\\)<\/li><\/ul><\/li><li>Use <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> as the last terms of the factors. \\(\\phantom{\\rule{7em}{0ex}}\\left(x+m\\right)\\left(x+n\\right)\\)<\/li><li>Check by multiplying the factors.<\/li><\/ol><\/div><p id=\"fs-id1167836419009\">In the first example, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let\u2019s look first at trinomials with only the middle term negative.<\/p><p>How do you get a <em data-effect=\"italics\">positive product<\/em> and a <em data-effect=\"italics\">negative sum<\/em>? We use two negative numbers.<\/p><div data-type=\"example\" id=\"fs-id1167829716451\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836524746\"><div data-type=\"problem\" id=\"fs-id1167832945802\"><p>Factor: \\({y}^{2}-11y+28.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836516109\"><p id=\"fs-id1167836544225\">Again, with the positive last term, 28, and the negative middle term, \\(-11y,\\) we need two negative factors. Find two numbers that multiply 28 and add to \\(-11.\\)<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}{y}^{2}-11y+28\\hfill \\\\ \\text{Write the factors as two binomials with first terms}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(y\\phantom{\\rule{1.2em}{0ex}}\\right)\\left(y\\phantom{\\rule{1.2em}{0ex}}\\right)\\hfill \\\\ \\text{Find two numbers that: multiply to 28 and add to}\\phantom{\\rule{0.2em}{0ex}}-11.\\hfill &amp; &amp; &amp; \\end{array}\\)<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167830123906\" class=\"unnumbered\" summary=\"This table has two columns showing factors of 28 and sum of the factors. Factors are: minus 1 and minus 28, whose sum is minus 29, minus 2 and minus 14 whose sum is minus 16, minus 4 and minus 7 whose sum is minus 11.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Factors of \\(28\\)<\/th><th data-valign=\"top\" data-align=\"center\">Sum of factors<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\text{\u2212}1,-28\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\text{\u2212}2,-14\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\text{\u2212}4,-7\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\text{\u2212}1+\\left(\\text{\u2212}28\\right)=-29\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\text{\u2212}2+\\left(\\text{\u2212}14\\right)=-16\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.5em}{0ex}}\\text{\u2212}4+\\left(\\text{\u2212}7\\right)={-11}^{*}\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Use}\\phantom{\\rule{0.2em}{0ex}}-4,-7\\phantom{\\rule{0.2em}{0ex}}\\text{as the last terms of the binomials.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{7.6em}{0ex}}\\left(y-4\\right)\\left(y-7\\right)\\hfill \\\\ \\text{Check:}\\hfill &amp; &amp; &amp; \\\\ \\hfill \\left(y-4\\right)\\left(y-7\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill {y}^{2}-7y-4y+28\\hfill &amp; &amp; &amp; \\\\ \\hfill {y}^{2}-11y+28\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167824734057\"><p id=\"fs-id1167836448208\">Factor: \\({u}^{2}-9u+18.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829716542\"><p>\\(\\left(u-3\\right)\\left(u-6\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829688066\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836610152\"><div data-type=\"problem\" id=\"fs-id1167829694152\"><p>Factor: \\({y}^{2}-16y+63.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836529007\"><p id=\"fs-id1167833053900\">\\(\\left(y-7\\right)\\left(y-9\\right)\\)<\/p><\/div><\/div><\/div><p>Now, what if the last term in the trinomial is negative? Think about <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.<\/p><p>How do you get a <em data-effect=\"italics\">negative product<\/em> and a <em data-effect=\"italics\">positive sum<\/em>? We use one positive and one negative number.<\/p><p id=\"fs-id1167836389451\">When we factor trinomials, we must have the terms written in descending order\u2014in order from highest degree to lowest degree.<\/p><div data-type=\"example\" id=\"fs-id1167836399263\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836635213\"><div data-type=\"problem\" id=\"fs-id1167836357277\"><p id=\"fs-id1167836292963\">Factor: \\(2x+{x}^{2}-48.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829686027\"><p id=\"fs-id1167836287160\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}2x+{x}^{2}-48\\hfill \\\\ \\text{First we put the terms in decreasing degree order.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}{x}^{2}+2x-48\\hfill \\\\ \\text{Factors will be two binomials with first terms}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\hfill \\end{array}\\)<\/p><div data-type=\"newline\"><br><\/div><table class=\"unnumbered\" summary=\"This table has two columns showing factors of minus 48 and sum of the factors. Factors are: minus 1 and 48, whose sum is 47, minus 2 and 24, whose sum is 22, minus 3 and 16, whose sum is 13, minus 4 and 12, whose sum is 8, minus 6 and 8, whose sum is 2.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Factors of \\(-48\\)<\/th><th data-valign=\"middle\" data-align=\"left\">Sum of factors<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(-1,48\\)<div data-type=\"newline\"><br><\/div>\\(-2,24\\)<div data-type=\"newline\"><br><\/div>\\(-3,16\\)<div data-type=\"newline\"><br><\/div>\\(-4,12\\)<div data-type=\"newline\"><br><\/div>\\(-6,8\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(-1+48=47\\)<div data-type=\"newline\"><br><\/div>\\(-2+24=22\\)<div data-type=\"newline\"><br><\/div>\\(-3+16=13\\)<div data-type=\"newline\"><br><\/div>\\(-4+12=8\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.5em}{0ex}}-6+8={2}^{*}\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}\\text{Use}\\phantom{\\rule{0.2em}{0ex}}-6,8\\phantom{\\rule{0.2em}{0ex}}\\text{as the last terms of the binomials.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.7em}{0ex}}\\left(x-6\\right)\\left(x+8\\right)\\hfill \\\\ \\text{Check:}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill \\left(x-6\\right)\\left(x+8\\right)\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill {x}^{2}-6q+8q-48\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill {x}^{2}+2x-48\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836299692\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836362051\"><div data-type=\"problem\" id=\"fs-id1167836493252\"><p id=\"fs-id1167829714149\">Factor: \\(9m+{m}^{2}+18.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836571079\"><p id=\"fs-id1167836329246\">\\(\\left(m+3\\right)\\left(m+6\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829833335\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833055007\"><p id=\"fs-id1167836508324\">Factor: \\(-7n+12+{n}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836367004\"><p id=\"fs-id1167836415277\">\\(\\left(n-3\\right)\\left(n-4\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836537198\">Sometimes you\u2019ll need to factor trinomials of the form \\({x}^{2}+bxy+c{y}^{2}\\) with two variables, such as \\({x}^{2}+12xy+36{y}^{2}.\\) The first term, \\({x}^{2},\\) is the product of the first terms of the binomial factors, \\(x\u00b7x.\\) The \\({y}^{2}\\) in the last term means that the second terms of the binomial factors must each contain <em data-effect=\"italics\">y<\/em>. To get the coefficients <em data-effect=\"italics\">b<\/em> and <em data-effect=\"italics\">c<\/em>, you use the same process summarized in <a href=\"#fs-id1167836732813\">How To Factor trinomials<\/a>.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836612676\"><div data-type=\"problem\" id=\"fs-id1167829717306\"><p id=\"fs-id1167829899564\">Factor: \\({r}^{2}-8rs-9{s}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734295\"><p id=\"fs-id1167836481087\">We need <em data-effect=\"italics\">r<\/em> in the first term of each binomial and <em data-effect=\"italics\">s<\/em> in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; {r}^{2}-8rs-9{s}^{2}\\hfill \\\\ \\text{Note that the first terms are}\\phantom{\\rule{0.2em}{0ex}}r,\\phantom{\\rule{0.2em}{0ex}}\\text{last terms contain}\\phantom{\\rule{0.2em}{0ex}}s.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\left(r\\phantom{\\rule{1.2em}{0ex}}s\\right)\\left(r\\phantom{\\rule{1.2em}{0ex}}s\\right)\\hfill \\\\ \\text{Find the numbers that multiply to}\\phantom{\\rule{0.2em}{0ex}}-9\\phantom{\\rule{0.2em}{0ex}}\\text{and add to}\\phantom{\\rule{0.2em}{0ex}}-8.\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167829739852\" class=\"unnumbered\" summary=\"This table has 2 columns showing factors of minus 9 and sum of factors. The factors are: 1 and minus 9 whose sum is 8, minus 1 and 9 whose sum is minus 8, 3 and minus 3 whose sum is 0.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Factors of \\(-9\\)<\/th><th data-valign=\"middle\" data-align=\"left\">Sum of factors<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.65em}{0ex}}1,-9\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.65em}{0ex}}-1+9=8\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(-1,9\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(1+\\left(-9\\right)=\\text{\u2212}{8}^{*}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.65em}{0ex}}3,-3\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(3+\\left(-3\\right)=0\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Use}\\phantom{\\rule{0.2em}{0ex}}1,-9\\phantom{\\rule{0.2em}{0ex}}\\text{as coefficients of the last terms.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4.9em}{0ex}}\\left(r+s\\right)\\left(r-9s\\right)\\hfill \\\\ \\text{Check:}\\hfill &amp; &amp; &amp; \\\\ \\hfill \\left(r-9s\\right)\\left(r+s\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill {r}^{2}+rs-9rs-9{s}^{2}\\hfill &amp; &amp; &amp; \\\\ \\hfill {r}^{2}-8rs-9{s}^{2}\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832940558\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836547330\"><div data-type=\"problem\" id=\"fs-id1167829742871\"><p id=\"fs-id1167836689445\">Factor: \\({a}^{2}-11ab+10{b}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836434117\"><p id=\"fs-id1167829596516\">\\(\\left(a-b\\right)\\left(a-10b\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833263821\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836440155\"><div data-type=\"problem\" id=\"fs-id1167833129296\"><p id=\"fs-id1167833009882\">Factor: \\({m}^{2}-13mn+12{n}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836333459\"><p id=\"fs-id1167836525443\">\\(\\left(m-n\\right)\\left(m-12n\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836551828\">Some trinomials are prime. The only way to be certain a trinomial is <span data-type=\"term\" class=\"no-emphasis\">prime<\/span> is to list all the possibilities and show that none of them work.<\/p><div data-type=\"example\" id=\"fs-id1167836515109\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836522339\"><div data-type=\"problem\" id=\"fs-id1167836522341\"><p id=\"fs-id1167826102751\">Factor: \\({u}^{2}-9uv-12{v}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836606580\"><p id=\"fs-id1167836364044\">We need <em data-effect=\"italics\">u<\/em> in the first term of each binomial and <em data-effect=\"italics\">v<\/em> in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\hfill {u}^{2}-9uv-12{v}^{2}\\hfill \\\\ \\text{Note that the first terms are}\\phantom{\\rule{0.2em}{0ex}}u,\\phantom{\\rule{0.2em}{0ex}}\\text{last terms contain}\\phantom{\\rule{0.2em}{0ex}}v.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\left(u\\phantom{\\rule{1.2em}{0ex}}v\\right)\\left(u\\phantom{\\rule{1.2em}{0ex}}v\\right)\\hfill \\\\ \\text{Find the numbers that multiply to}\\phantom{\\rule{0.2em}{0ex}}-12\\phantom{\\rule{0.2em}{0ex}}\\text{and add to}\\phantom{\\rule{0.2em}{0ex}}-9.\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}\\) <table id=\"fs-id1167833051403\" class=\"unnumbered\" summary=\"This table has 2 columns showing factors of minus 12 and sum of factors. The factors are: 1 and minus 12 whose sum is minus 11, minus 1 and 12 whose sum is 11, 2 and minus 6 whose sum is minus 4, minus 2 and 6 whose sum is 4, 3 and minus 4 whose sum is minus 1, minus 3 and minus 3 and 4 whose sum is 1.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Factors of \\(\\text{\u2212}12\\)<\/th><th data-valign=\"top\" data-align=\"left\">Sum of factors<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.65em}{0ex}}1,-12\\)<div data-type=\"newline\"><br><\/div>\\(-1,12\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.65em}{0ex}}2,-6\\)<div data-type=\"newline\"><br><\/div>\\(-2,6\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.65em}{0ex}}3,-4\\)<div data-type=\"newline\"><br><\/div>\\(-3,4\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(1+\\left(-12\\right)=-11\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.65em}{0ex}}-1+12=11\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.5em}{0ex}}2+\\left(-6\\right)=-4\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.2em}{0ex}}-2+6=4\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.6em}{0ex}}3+\\left(-4\\right)=-1\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.2em}{0ex}}-3+4=1\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836547038\">Note there are no factor pairs that give us \\(-9\\) as a sum. The trinomial is prime.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836387300\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829738401\"><div data-type=\"problem\"><p id=\"fs-id1167833060906\">Factor: \\({x}^{2}-7xy-10{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836691877\"><p id=\"fs-id1167836444713\">prime<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836545726\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836652605\"><div data-type=\"problem\" id=\"fs-id1167836652607\"><p id=\"fs-id1167836516185\">Factor: \\({p}^{2}+15pq+20{q}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836487273\"><p id=\"fs-id1167833058791\">prime<\/p><\/div><\/div><\/div><p id=\"fs-id1167836387282\">Let\u2019s summarize the method we just developed to factor trinomials of the form \\({x}^{2}+bx+c.\\)<\/p><div data-type=\"note\" id=\"fs-id1167833046935\"><div data-type=\"title\">Strategy for Factoring Trinomials of the Form \\({x}^{2}+bx+c\\)<\/div><p id=\"fs-id1167836650044\">When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\begin{array}{c}\\hfill {x}^{2}+bx+c\\hfill \\\\ \\hfill \\left(x+m\\right)\\left(x+n\\right)\\hfill \\end{array}\\hfill \\\\ \\hfill \\mathbf{\\text{When}}\\phantom{\\rule{0.2em}{0ex}}{\\text{c}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{is positive,}}\\phantom{\\rule{0.2em}{0ex}}{\\text{m}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{and}}\\phantom{\\rule{0.2em}{0ex}}{\\text{n}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{have the same sign.}}\\hfill \\\\ \\hfill b\\phantom{\\rule{0.2em}{0ex}}\\text{positive}\\phantom{\\rule{16.5em}{0ex}}b\\phantom{\\rule{0.2em}{0ex}}\\text{negative}\\hfill \\\\ \\hfill m,n\\phantom{\\rule{0.2em}{0ex}}\\text{positive}\\phantom{\\rule{15em}{0ex}}m,n\\phantom{\\rule{0.2em}{0ex}}\\text{negative}\\hfill \\\\ \\hfill {x}^{2}+5x+6\\phantom{\\rule{16em}{0ex}}{x}^{2}-6x+8\\hfill \\\\ \\hfill \\left(x+2\\right)\\left(x+3\\right)\\phantom{\\rule{15em}{0ex}}\\left(x-4\\right)\\left(x-2\\right)\\hfill \\\\ \\hfill \\text{same signs}\\phantom{\\rule{16em}{0ex}}\\text{same signs}\\hfill \\\\ \\hfill \\mathbf{\\text{When}}\\phantom{\\rule{0.2em}{0ex}}{\\text{c}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{is negative,}}\\phantom{\\rule{0.2em}{0ex}}{\\text{m}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{and}}\\phantom{\\rule{0.2em}{0ex}}{\\text{n}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{have opposite signs.}}\\hfill \\\\ \\hfill {x}^{2}+x-12\\phantom{\\rule{16em}{0ex}}{x}^{2}-2x-15\\hfill \\\\ \\hfill \\left(x+4\\right)\\left(x-3\\right)\\phantom{\\rule{15em}{0ex}}\\left(x-5\\right)\\left(x+3\\right)\\hfill \\\\ \\hfill \\text{opposite signs}\\phantom{\\rule{15em}{0ex}}\\text{opposite signs}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836320419\">Notice that, in the case when <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> have opposite signs, the sign of the one with the larger absolute value matches the sign of <em data-effect=\"italics\">b<\/em>.<\/p><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833086888\"><h3 data-type=\"title\">Factor Trinomials of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> using Trial and Error<\/h3><p id=\"fs-id1167836613360\">Our next step is to factor trinomials whose leading coefficient is not 1, trinomials of the form \\(a{x}^{2}+bx+c.\\)<\/p><p id=\"fs-id1167829696932\">Remember to always check for a <span data-type=\"term\" class=\"no-emphasis\">GCF<\/span> first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods we\u2019ve used so far. Let\u2019s do an example to see how this works.<\/p><div data-type=\"example\" id=\"fs-id1167836415146\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836415148\"><div data-type=\"problem\" id=\"fs-id1167836456289\"><p id=\"fs-id1167836456291\">Factor completely: \\(4{x}^{3}+16{x}^{2}-20x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833053401\"><p id=\"fs-id1167833053404\">\\(\\begin{array}{cccc}\\text{Is there a greatest common factor?}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}4{x}^{3}+16{x}^{2}-20x\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{Yes, GCF}=4x.\\phantom{\\rule{0.2em}{0ex}}\\text{Factor it.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}4x\\left({x}^{2}+4x-5\\right)\\hfill \\\\ \\\\ \\\\ \\text{Binomial, trinomial, or more than three terms?}\\hfill &amp; &amp; &amp; \\\\ \\phantom{\\rule{2em}{0ex}}\\text{It is a trinomial. So \u201cundo FOIL.\u201d}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}4x\\left(x\\phantom{\\rule{1.8em}{0ex}}\\right)\\left(x\\phantom{\\rule{1.5em}{0ex}}\\right)\\hfill \\\\ \\\\ \\\\ \\text{Use a table like the one shown to find two numbers that}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}4x\\left(x-1\\right)\\left(x+5\\right)\\hfill \\\\ \\text{multiply to}\\phantom{\\rule{0.2em}{0ex}}-5\\phantom{\\rule{0.2em}{0ex}}\\text{and add to 4.}\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829784370\" class=\"unnumbered\" summary=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Factors of \\(\\text{\u2212}5\\)<\/th><th data-valign=\"top\" data-align=\"left\">Sum of factors<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\text{\u2212}1,5\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.6em}{0ex}}1,-5\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.6em}{0ex}}\\text{\u2212}1+5={4}^{*}\\)<div data-type=\"newline\"><br><\/div>\\(1+\\left(\\text{\u2212}5\\right)=-4\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Check:}\\hfill &amp; &amp; &amp; \\\\ &amp; &amp; &amp; \\hfill 4x\\left(x-1\\right)\\left(x+5\\right)\\hfill \\\\ &amp; &amp; &amp; \\hfill 4x\\left({x}^{2}+5x-x-5\\right)\\hfill \\\\ &amp; &amp; &amp; \\hfill 4x\\left({x}^{2}+4x-5\\right)\\hfill \\\\ &amp; &amp; &amp; \\hfill 4{x}^{3}+16{x}^{2}-20x\u2713\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829745918\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829595477\"><div data-type=\"problem\" id=\"fs-id1167829595479\"><p id=\"fs-id1167836378219\">Factor completely: \\(5{x}^{3}+15{x}^{2}-20x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836514323\"><p id=\"fs-id1167836514325\">\\(5x\\left(x-1\\right)\\left(x+4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829732136\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829732139\"><div data-type=\"problem\" id=\"fs-id1167829596896\"><p id=\"fs-id1167829596898\">Factor completely: \\(6{y}^{3}+18{y}^{2}-60y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836319758\"><p id=\"fs-id1167836319760\">\\(6y\\left(y-2\\right)\\left(y+5\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833009012\">What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.<\/p><p id=\"fs-id1167829712589\">Let\u2019s factor the trinomial \\(3{x}^{2}+5x+2.\\)<\/p><p id=\"fs-id1167836415401\">From our earlier work, we expect this will factor into two binomials.<\/p><div data-type=\"equation\" id=\"fs-id1167836415404\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{}\\\\ \\\\ \\hfill 3{x}^{2}+5x+2\\hfill \\\\ \\hfill \\left(\\phantom{\\rule{2em}{0ex}}\\right)\\left(\\phantom{\\rule{2em}{0ex}}\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167836692336\">We know the first terms of the binomial factors will multiply to give us \\(3{x}^{2}.\\) The only factors of \\(3{x}^{2}\\) are \\(1x,3x.\\) We can place them in the binomials.<\/p><span data-type=\"media\" id=\"fs-id1167833274184\" data-alt=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\"><\/span><p id=\"fs-id1167833054406\">Check: Does \\(1x\u00b73x=3{x}^{2}?\\)<\/p><p id=\"fs-id1167836774071\">We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1, 2. But we now have two cases to consider as it will make a difference if we write 1, 2 or 2, 1.<\/p><span data-type=\"media\" id=\"fs-id1167836548650\" data-alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses.\"><\/span><p id=\"fs-id1167829850087\">Which factors are correct? To decide that, we multiply the inner and outer terms.<\/p><span data-type=\"media\" id=\"fs-id1167836732179\" data-alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses. In each case, arrows are shown pairing the first term of the first factor with the last term of the second factor and the first term of the second factor with the last term of the first factor.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses. In each case, arrows are shown pairing the first term of the first factor with the last term of the second factor and the first term of the second factor with the last term of the first factor.\"><\/span><p id=\"fs-id1167836526346\">Since the middle term of the trinomial is \\(5x,\\) the factors in the first case will work. Let\u2019s use FOIL to check.<\/p><div data-type=\"equation\" id=\"fs-id1167836730749\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\left(x+1\\right)\\left(3x+2\\right)\\hfill \\\\ \\hfill 3{x}^{2}+2x+3x+2\\hfill \\\\ \\hfill 3{x}^{2}+5x+2\u2713\\hfill \\end{array}\\)<\/div><p>Our result of the factoring is:<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{}\\\\ \\hfill 3{x}^{2}+5x+2\\hfill \\\\ \\hfill \\left(x+1\\right)\\left(3x+2\\right)\\hfill \\end{array}\\)<\/div><div data-type=\"example\" id=\"fs-id1167829936476\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor a Trinomial Using Trial and Error<\/div><div data-type=\"exercise\" id=\"fs-id1167829936478\"><div data-type=\"problem\" id=\"fs-id1167829936480\"><p id=\"fs-id1167836492295\">Factor completely using trial and error: \\(3{y}^{2}+22y+7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829579867\"><span data-type=\"media\" id=\"fs-id1167829579869\" data-alt=\"Step 1 is to write the trinomial in descending order. The trinomial 3 y squared plus 22y plus 7 is already in descending order.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the trinomial in descending order. The trinomial 3 y squared plus 22y plus 7 is already in descending order.\"><\/span><span data-type=\"media\" id=\"fs-id1167833138154\" data-alt=\"Step 2 is to factor the GCF. Here, there is none.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to factor the GCF. Here, there is none.\"><\/span><span data-type=\"media\" id=\"fs-id1167836377144\" data-alt=\"Step 3 is Find all the factor pairs of the first term. The only factors here are 1y and 3y. Since there is only one pair, we can put each as the first term in the parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is Find all the factor pairs of the first term. The only factors here are 1y and 3y. Since there is only one pair, we can put each as the first term in the parentheses.\"><\/span><span data-type=\"media\" id=\"fs-id1167833055812\" data-alt=\"Step 4 is to find all the factor pairs of the third term. Here, the only pair is 1 and 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to find all the factor pairs of the third term. Here, the only pair is 1 and 7.\"><\/span><span data-type=\"media\" id=\"fs-id1167833024736\" data-alt=\"Step 5 is to test all the possible combinations of the factors until the correct product is found. For possible factors open parentheses y plus 1 close parentheses open parentheses 37 plus 7 close parentheses, the product is 3 y squared plus 10y plus 7. For the possible factors open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses, the product is 3 y squared plus 22y plus 7, which is the correct product. Hence, the correct factors are open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to test all the possible combinations of the factors until the correct product is found. For possible factors open parentheses y plus 1 close parentheses open parentheses 37 plus 7 close parentheses, the product is 3 y squared plus 10y plus 7. For the possible factors open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses, the product is 3 y squared plus 22y plus 7, which is the correct product. Hence, the correct factors are open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses.\"><\/span><span data-type=\"media\" id=\"fs-id1167836687917\" data-alt=\"Step 6 is to check by multiplying.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check by multiplying.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836732148\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836620110\"><div data-type=\"problem\" id=\"fs-id1167836620112\"><p id=\"fs-id1167836620115\">Factor completely using trial and error: \\(2{a}^{2}+5a+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836493740\"><p id=\"fs-id1167836571082\">\\(\\left(a+1\\right)\\left(2a+3\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836375598\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829692279\"><div data-type=\"problem\" id=\"fs-id1167829692281\"><p id=\"fs-id1167829692283\">Factor completely using trial and error: \\(4{b}^{2}+5b+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832999458\"><p id=\"fs-id1167832999460\">\\(\\left(b+1\\right)\\left(4b+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836325973\" class=\"howto\"><div data-type=\"title\">Factor trinomials of the form \\(a{x}^{2}+bx+c\\) using trial and error.<\/div><ol id=\"fs-id1167829740737\" type=\"1\" class=\"stepwise\"><li>Write the trinomial in descending order of degrees as needed.<\/li><li>Factor any GCF.<\/li><li>Find all the factor pairs of the first term.<\/li><li>Find all the factor pairs of the third term.<\/li><li>Test all the possible combinations of the factors until the correct product is found.<\/li><li>Check by multiplying.<\/li><\/ol><\/div><p id=\"fs-id1167836538544\">Remember, when the middle term is negative and the last term is positive, the signs in the binomials must both be negative.<\/p><div data-type=\"example\" id=\"fs-id1167829905361\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829905363\"><div data-type=\"problem\" id=\"fs-id1167829905366\"><p id=\"fs-id1167836667013\">Factor completely using trial and error: \\(6{b}^{2}-13b+5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832945624\"><table id=\"fs-id1167832945627\" class=\"unnumbered unstyled\" summary=\"The trinomial 6 b squared minus 13 b plus 5 is already in descending order. Factoring the first term, we get 1b times 6b and 2b times 3b. To find the factors of the last term, consider the signs. Since the last term, 5, is positive its factors must both be positive or both be negative. The coefficient of the middle term is negative, so we use the negative factors minus 1 and minus 5.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The trinomial is already in descending order.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829689434\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the factors of the first term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728970\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the factors of the last term. Consider the signs.<div data-type=\"newline\"><br><\/div>Since the last term, 5, is positive its factors must both be<div data-type=\"newline\"><br><\/div>positive or both be negative. The coefficient of the<div data-type=\"newline\"><br><\/div>middle term is negative, so we use the negative factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829746922\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829627623\">Consider all the combinations of factors.<\/p><table id=\"fs-id1167836313301\" class=\"unnumbered\" summary=\"This table shows the possible factors and corresponding products of 6 b squared minus 13 b plus 5. Factors: open parentheses b minus 1 close parentheses open parentheses 6b minus 5 close parentheses; product: 6 b squared minus 11 b plus 5. Factors: open parentheses b minus 5 close parentheses open parentheses 6b minus 1 close parentheses; product: 6 b squared minus 31 b plus 5. Factors: open parentheses 2b minus 1 close parentheses open parentheses 3b minus 5 close parentheses; product: 6 b squared minus 13b plus 5. This is the original trinomial. Factors: open parentheses 2b minus 5 close parentheses open parentheses 3b minus 1 close parentheses; product: 6 b squared minus 17b plus 5.\"><thead><tr valign=\"top\"><th colspan=\"2\" data-valign=\"top\" data-align=\"center\">\\(6{b}^{2}-13b+5\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Possible factors<\/th><th data-valign=\"top\" data-align=\"left\">Product<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(b-1\\right)\\left(6b-5\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(6{b}^{2}-11b+5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(b-5\\right)\\left(6b-1\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(6{b}^{2}-31b+5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(2b-1\\right)\\left(3b-5\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(6{b}^{2}-13b+{5}^{*}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\left(2b-5\\right)\\left(3b-1\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(6{b}^{2}-17b+5\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836503135\">\\(\\begin{array}{cccc}\\text{The correct factors are those whose product}\\hfill &amp; &amp; &amp; \\\\ \\text{is the original trinomial.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\left(2b-1\\right)\\left(3b-5\\right)\\hfill \\\\ \\text{Check by multiplying:}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\left(2b-1\\right)\\left(3b-5\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}6{b}^{2}-10b-3b+5\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}6{b}^{2}-13b+5\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833021860\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833021864\"><div data-type=\"problem\" id=\"fs-id1167829599560\"><p id=\"fs-id1167829599562\">Factor completely using trial and error: \\(8{x}^{2}-13x+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836612477\"><p id=\"fs-id1167836612479\">\\(\\left(2x-3\\right)\\left(4x-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836579458\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829894398\"><div data-type=\"problem\" id=\"fs-id1167829894400\"><p id=\"fs-id1167829872167\">Factor completely using trial and error: \\(10{y}^{2}-37y+7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836608328\"><p id=\"fs-id1167836608330\">\\(\\left(2y-7\\right)\\left(5y-1\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836533023\">When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.<\/p><div data-type=\"example\" id=\"fs-id1167836287935\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836287937\"><div data-type=\"problem\" id=\"fs-id1167836287939\"><p>Factor completely using trial and error: \\(18{x}^{2}-37xy+15{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825949241\"><table class=\"unnumbered unstyled\" summary=\"The trinomial 18 x squared minus 37xy plus 15y squared is already in descending order. Factoring the first term, we get 1x times 18x, 2x times 9x and 3x times 6x.To find the factors of the last term, consider the signs. Since 15 is positive and the coefficient of the middle term is negative, we use the negative factors minus 1, minus 5 and minus 5, minus 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The trinomial is already in descending order.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836800465\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the factors of the first term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833197132\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the factors of the last term. Consider the signs.<div data-type=\"newline\"><br><\/div>Since 15 is positive and the coefficient of the middle<div data-type=\"newline\"><br><\/div>term is negative, we use the negative factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836554867\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009c_img_Errata.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829789946\">Consider all the combinations of factors.<\/p><span data-type=\"media\" id=\"fs-id1167836521754\" data-alt=\"This table shows the possible factors and corresponding products of the trinomial 18 x squared minus 37xy plus 15 y squared. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: open parentheses x minus 1y close parentheses open parentheses 18x minus 15y close parentheses, highlighted. Factor, open parentheses x minus 15y close parentheses open parentheses 18x minus 1y close parentheses; product: 18 x squared minus 271xy plus 15 y squared. Factor open parentheses x minus 3y close parentheses open parentheses 18x minus 5 y close parentheses; product: 18 x squared minus 59xy plus 15 y squared. Factor: open parentheses x minus 5y close parentheses open parentheses 18x minus 3y close parentheses highlighted. Factor: open parentheses 2x minus 1y close parentheses open parentheses 9x minus 15y close parentheses highlighted. Factor: open parentheses 2x minus 15y close parentheses open parentheses 9x minus 1y close parentheses; product 18 x squared minus 137 xy plus 15y squared. Factor: open parentheses 2x minus 3y close parentheses open parentheses 9x minus 5y close parentheses; product: 18 x squared minus 37xy plus 15 y squared, which is the original trinomial. Factor: open parentheses 2x minus 57 close parentheses open parentheses 9x minus 3y close parentheses highlighted. Factor: open parentheses 3x minus 1y close parentheses open parentheses 6x minus 15y close parentheses highlighted. Factor: open parentheses 3x minus 15y close parentheses highlighted open parentheses 6x minus 1y close parentheses. Factor: open parentheses 3x minus 3y close parentheses highlighted open parentheses 6x minus 5y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table shows the possible factors and corresponding products of the trinomial 18 x squared minus 37xy plus 15 y squared. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: open parentheses x minus 1y close parentheses open parentheses 18x minus 15y close parentheses, highlighted. Factor, open parentheses x minus 15y close parentheses open parentheses 18x minus 1y close parentheses; product: 18 x squared minus 271xy plus 15 y squared. Factor open parentheses x minus 3y close parentheses open parentheses 18x minus 5 y close parentheses; product: 18 x squared minus 59xy plus 15 y squared. Factor: open parentheses x minus 5y close parentheses open parentheses 18x minus 3y close parentheses highlighted. Factor: open parentheses 2x minus 1y close parentheses open parentheses 9x minus 15y close parentheses highlighted. Factor: open parentheses 2x minus 15y close parentheses open parentheses 9x minus 1y close parentheses; product 18 x squared minus 137 xy plus 15y squared. Factor: open parentheses 2x minus 3y close parentheses open parentheses 9x minus 5y close parentheses; product: 18 x squared minus 37xy plus 15 y squared, which is the original trinomial. Factor: open parentheses 2x minus 57 close parentheses open parentheses 9x minus 3y close parentheses highlighted. Factor: open parentheses 3x minus 1y close parentheses open parentheses 6x minus 15y close parentheses highlighted. Factor: open parentheses 3x minus 15y close parentheses highlighted open parentheses 6x minus 1y close parentheses. Factor: open parentheses 3x minus 3y close parentheses highlighted open parentheses 6x minus 5y.\"><\/span><p id=\"fs-id1167829627829\">\\(\\begin{array}{cccc}\\text{The correct factors are those whose product is}\\hfill &amp; &amp; &amp; \\\\ \\text{the original trinomial.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\left(2x-3y\\right)\\left(9x-5y\\right)\\hfill \\\\ \\text{Check by multiplying:}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\left(2x-3y\\right)\\left(9x-5y\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}18{x}^{2}-10xy-27xy+15{y}^{2}\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}18{x}^{2}-37xy+15{y}^{2}\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829747027\"><div data-type=\"problem\" id=\"fs-id1167836512666\"><p>Factor completely using trial and error \\(18{x}^{2}-3xy-10{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833086736\"><p id=\"fs-id1167836551348\">\\(\\left(3x+2y\\right)\\left(6x-5y\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824755152\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833379398\"><div data-type=\"problem\" id=\"fs-id1167833379400\"><p id=\"fs-id1167833379402\">Factor completely using trial and error: \\(30{x}^{2}-53xy-21{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829790284\">\\(\\left(3x+y\\right)\\left(10x-21y\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167824765027\">Don\u2019t forget to look for a GCF first and remember if the leading coefficient is negative, so is the GCF.<\/p><div data-type=\"example\" id=\"fs-id1167836614266\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836614268\"><div data-type=\"problem\" id=\"fs-id1167833339780\"><p id=\"fs-id1167833339782\">Factor completely using trial and error: \\(-10{y}^{4}-55{y}^{3}-60{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825660234\"><table id=\"fs-id1167829621071\" class=\"unnumbered unstyled\" summary=\"The trinomial is minus 10 y to the power 4 minus 55 y cubed minus 60 y squared. Factoring the GCF, we get minus 5 y squared open parentheses 2 y squared plus 11y plus 12 close parentheses. The factors of the first term of the trinomial in the parentheses are y and 2y. The factor pairs of the last term are 1 and 12, 2 and 6, 3 and 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830123177\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Notice the greatest common factor, so factor it first.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025652\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833019180\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836534006\">Consider all the combinations.<\/p><span data-type=\"media\" id=\"fs-id1167836534010\" data-alt=\"This table shows the possible factors and product of the trinomial 2 y squared plus 11y plus 12. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: y plus 1, 2y plus 12 highlighted. Factor: y plus 12, 2y plus 1; product: 2 y squared plus 25y plus 12. Factor: y plus 2, 2y plus 6 highlighted. Factor: y plus 6, 2y plus 2 highlighted. Factor: y plus 3, 2y plus 4 highlighted. Factor: y plus 4, 2y plus 3; product: 2 y squared plus 11y plus 12. This is the original trinomial.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table shows the possible factors and product of the trinomial 2 y squared plus 11y plus 12. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: y plus 1, 2y plus 12 highlighted. Factor: y plus 12, 2y plus 1; product: 2 y squared plus 25y plus 12. Factor: y plus 2, 2y plus 6 highlighted. Factor: y plus 6, 2y plus 2 highlighted. Factor: y plus 3, 2y plus 4 highlighted. Factor: y plus 4, 2y plus 3; product: 2 y squared plus 11y plus 12. This is the original trinomial.\"><\/span><p id=\"fs-id1167824720957\">\\(\\begin{array}{cccc}\\text{The correct factors are those whose product}\\hfill &amp; &amp; &amp; \\\\ \\text{is the original trinomial. Remember to include}\\hfill &amp; &amp; &amp; \\\\ \\text{the factor}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}5{y}^{2}.\\hfill &amp; &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\text{\u2212}5{y}^{2}\\left(y+4\\right)\\left(2y+3\\right)\\hfill \\\\ \\text{Check by multiplying:}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\text{\u2212}5{y}^{2}\\left(y+4\\right)\\left(2y+3\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\text{\u2212}5{y}^{2}\\left(2{y}^{2}+8y+3y+12\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\text{\u2212}10{y}^{4}-55{y}^{3}-60{y}^{2}\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836550866\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836550870\"><div data-type=\"problem\" id=\"fs-id1167833061507\"><p id=\"fs-id1167833061509\">Factor completely using trial and error: \\(15{n}^{3}-85{n}^{2}+100n.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836391301\"><p id=\"fs-id1167836391303\">\\(5n\\left(n-4\\right)\\left(3n-5\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829830348\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829830352\"><div data-type=\"problem\" id=\"fs-id1167836557643\"><p id=\"fs-id1167836557645\">Factor completely using trial and error: \\(56{q}^{3}+320{q}^{2}-96q.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833369061\"><p id=\"fs-id1167833369063\">\\(8q\\left(q+6\\right)\\left(7q-2\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833135490\"><h3 data-type=\"title\">Factor Trinomials of the Form \\(a{x}^{2}+bx+c\\) using the \u201cac\u201d Method<\/h3><p id=\"fs-id1167836487108\">Another way to factor trinomials of the form \\(a{x}^{2}+bx+c\\) is the \u201cac\u201d method. (The \u201cac\u201d method is sometimes called the grouping method.) The \u201cac\u201d method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!<\/p><div data-type=\"example\" id=\"fs-id1167829812485\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor Trinomials using the \u201cac\u201d Method<\/div><div data-type=\"exercise\" id=\"fs-id1167829812487\"><div data-type=\"problem\" id=\"fs-id1167829812489\"><p id=\"fs-id1167833018423\">Factor using the <em data-effect=\"italics\">\u2018ac\u2019<\/em> method: \\(6{x}^{2}+7x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836553592\"><span data-type=\"media\" id=\"fs-id1167836553594\" data-alt=\"Step 1 is to factor the GCF. There is none in 6 x squared plus 7x plus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to factor the GCF. There is none in 6 x squared plus 7x plus 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167824763512\" data-alt=\"Step 2 is to find the product of a and c. The product of 6 and 2 is 12.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the product of a and c. The product of 6 and 2 is 12.\"><\/span><span data-type=\"media\" id=\"fs-id1167824732580\" data-alt=\"Step 3 is to find 2 numbers m and n such that mn is ac and m plus n is b. So we need to numbers that multiply to 12 and add to 7. Both factors must be positive. 3 times 4 is 12 and 3 plus 4 is 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find 2 numbers m and n such that mn is ac and m plus n is b. So we need to numbers that multiply to 12 and add to 7. Both factors must be positive. 3 times 4 is 12 and 3 plus 4 is 7.\"><\/span><span data-type=\"media\" id=\"fs-id1167833022512\" data-alt=\"Step 4 is to split the middle term using m and n. So we rewrite 7 x as 3x plus 4x. It would give the same result if we used 4x plus 3x. Rewriting, we get 6 x squared plus 3x plus 4x plus 2. Notice that this is the same as the original polynomial. We just split the middle term to get a more useful form\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to split the middle term using m and n. So we rewrite 7 x as 3x plus 4x. It would give the same result if we used 4x plus 3x. Rewriting, we get 6 x squared plus 3x plus 4x plus 2. Notice that this is the same as the original polynomial. We just split the middle term to get a more useful form\"><\/span><span data-type=\"media\" id=\"fs-id1167829713653\" data-alt=\"Step 5 is to factor by grouping. So, we get, 3x open parentheses 2x plus 1 close parentheses plus 2 open parentheses 2x plus 1 close parentheses. This is equal to 2x plus 1, 3x plus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to factor by grouping. So, we get, 3x open parentheses 2x plus 1 close parentheses plus 2 open parentheses 2x plus 1 close parentheses. This is equal to 2x plus 1, 3x plus 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167836392737\" data-alt=\"Step 6 is to check by multiplying the factors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check by multiplying the factors.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833050735\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833050739\"><div data-type=\"problem\" id=\"fs-id1167833050741\"><p id=\"fs-id1167833018138\">Factor using the \u2018ac\u2019 method: \\(6{x}^{2}+13x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836548815\"><p id=\"fs-id1167836548817\">\\(\\left(x+2\\right)\\left(6x+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836387643\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836387646\"><div data-type=\"problem\" id=\"fs-id1167836387648\"><p id=\"fs-id1167836728730\">Factor using the \u2018ac\u2019 method: \\(4{y}^{2}+8y+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836576087\"><p id=\"fs-id1167836576089\">\\(\\left(2y+1\\right)\\left(2y+3\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836629022\">The \u201cac\u201d method is summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167833082444\" class=\"howto\"><div data-type=\"title\">Factor trinomials of the form \\(a{x}^{2}+bx+c\\) using the \u201cac\u201d method.<\/div><ol id=\"fs-id1167824735184\" type=\"1\" class=\"stepwise\"><li>Factor any GCF.<\/li><li>Find the product <em data-effect=\"italics\">ac<\/em>.<\/li><li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that:<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}\\text{Multiply to}\\phantom{\\rule{0.2em}{0ex}}ac\\hfill &amp; &amp; &amp; &amp; &amp; m\u00b7n=a\u00b7c\\hfill \\\\ \\text{Add to}\\phantom{\\rule{0.2em}{0ex}}b\\hfill &amp; &amp; &amp; &amp; &amp; m+n=b\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; a{x}^{2}+bx+c\\hfill \\end{array}\\)<\/li><li>Split the middle term using <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>. \\(\\phantom{\\rule{5em}{0ex}}a{x}^{2}+mx+nx+c\\)<\/li><li>Factor by grouping.<\/li><li>Check by multiplying the factors.<\/li><\/ol><\/div><p id=\"fs-id1167836614469\">Don\u2019t forget to look for a common factor!<\/p><div data-type=\"example\" id=\"fs-id1167832926004\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832926007\"><div data-type=\"problem\" id=\"fs-id1167832926009\"><p id=\"fs-id1167829810980\">Factor using the <em data-effect=\"italics\">\u2018ac\u2019<\/em> method: \\(10{y}^{2}-55y+70.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829749936\"><table id=\"fs-id1167833257770\" class=\"unnumbered unstyled\" summary=\"The GCF in 10 y squared minus 55y plus 70 is 5. Factoring this, we get 5 open parentheses 2 y squared minus 11 y plus 14. The trinomial inside the parentheses has a leading coefficient that is not 1. So we find the product ac, which is 28. Now we find two numbers that multiply to ac and add to b. Minus 4 times minus 7 is 28 and minus 4 minus 7 is minus 11. Splitting the middle term of the trinomial, we get, 5 open parentheses 2 y squared minus 7y minus 4 y plus 14 close parentheses. We factor by grouping to get 5 open parentheses y minus 2 close parentheses open parentheses 2y minus 7 close parentheses. Now we check by multiplying all three factors to get the original polynomial.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is there a greatest common factor?<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Yes. The GCF is 5.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586669\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor it.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833057789\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The trinomial inside the parentheses has a<div data-type=\"newline\"><br><\/div>leading coefficient that is not 1.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829861944\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the product \\(ac.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{3.3em}{0ex}}ac=28\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find two numbers that multiply to \\(ac\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.5em}{0ex}}\\left(-4\\right)\\left(-7\\right)=28\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">and add to <em data-effect=\"italics\">b<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(-4+\\left(-7\\right)=-11\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Split the middle term.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477566\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/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_06_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial by grouping.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693285\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836747903\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check by multiplying all three factors.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\hfill 5\\left(y-2\\right)\\left(2y-7\\right)\\hfill \\\\ \\hfill 5\\left(2{y}^{2}-7y-4y+14\\right)\\hfill \\\\ \\hfill 5\\left(2{y}^{2}-11y+14\\right)\\hfill \\\\ \\hfill 10{y}^{2}-55y+70\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836356159\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836356163\"><div data-type=\"problem\" id=\"fs-id1167836356165\"><p id=\"fs-id1167829720098\">Factor using the \u2018ac\u2019 method: \\(16{x}^{2}-32x+12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833048819\"><p id=\"fs-id1167833048821\">\\(4\\left(2x-3\\right)\\left(2x-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836539503\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836539508\"><div data-type=\"problem\" id=\"fs-id1167836539510\"><p id=\"fs-id1167829621078\">Factor using the \u2018ac\u2019 method: \\(18{w}^{2}-39w+18.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833274653\"><p>\\(3\\left(3w-2\\right)\\left(2w-3\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836440705\"><h3 data-type=\"title\">Factor Using Substitution<\/h3><p id=\"fs-id1167836440710\">Sometimes a trinomial does not appear to be in the \\(a{x}^{2}+bx+c\\) form. However, we can often make a thoughtful substitution that will allow us to make it fit the \\(a{x}^{2}+bx+c\\) form. This is called <span data-type=\"term\" class=\"no-emphasis\">factoring by substitution<\/span>. It is standard to use <em data-effect=\"italics\">u<\/em> for the substitution.<\/p><p id=\"fs-id1167833051242\">In the \\(a{x}^{2}+bx+c,\\) the middle term has a variable, <em data-effect=\"italics\">x<\/em>, and its square, \\({x}^{2},\\) is the variable part of the first term. Look for this relationship as you try to find a substitution.<\/p><div data-type=\"example\" id=\"fs-id1167829908071\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829908073\"><div data-type=\"problem\" id=\"fs-id1167836743430\"><p id=\"fs-id1167836743432\">Factor by substitution: \\({x}^{4}-4{x}^{2}-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836649904\"><p id=\"fs-id1167836649906\">The variable part of the middle term is \\({x}^{2}\\) and its square, \\({x}^{4},\\) is the variable part of the first term. (We know \\({\\left({x}^{2}\\right)}^{2}={x}^{4}\\right).\\) If we let \\(u={x}^{2},\\) we can put our trinomial in the \\(a{x}^{2}+bx+c\\) form we need to factor it.<\/p><table id=\"fs-id1167836615497\" class=\"unnumbered unstyled\" summary=\"In the polynomial x to the power 4 minus 4 x squared minus 5, substitute x squared with u. We get the trinomial u squared minus 4u minus 5. We factor this to get u plus 1, u minus 5. Replacing u with x squared, we get x squared plus 1, x squared minus 5. We check by multiplying the factors to get the original polynomial.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833271796\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832971395\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(u={x}^{2}\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833053520\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833024758\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\({x}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756401\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{3em}{0ex}}\\begin{array}{c}\\hfill \\left({x}^{2}+1\\right)\\left({x}^{2}-5\\right)\\hfill \\\\ \\hfill {x}^{4}-5{x}^{2}+{x}^{2}-5\\hfill \\\\ \\hfill {x}^{4}-4{x}^{2}-5\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824735528\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824735532\"><div data-type=\"problem\" id=\"fs-id1167824735534\"><p id=\"fs-id1167832971372\">Factor by substitution: \\({h}^{4}+4{h}^{2}-12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829789387\"><p id=\"fs-id1167829789389\">\\(\\left({h}^{2}-2\\right)\\left({h}^{2}+6\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829597653\"><div data-type=\"problem\" id=\"fs-id1167829597655\"><p id=\"fs-id1167829741295\">Factor by substitution: \\({y}^{4}-{y}^{2}-20.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836508040\"><p id=\"fs-id1167836508042\">\\(\\left({y}^{2}+4\\right)\\left({y}^{2}-5\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1171791549871\">Sometimes the expression to be substituted is not a monomial.<\/p><div data-type=\"example\" id=\"fs-id1167836320647\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836320649\"><div data-type=\"problem\" id=\"fs-id1167836320651\"><p id=\"fs-id1167836320653\">Factor by substitution: \\({\\left(x-2\\right)}^{2}+7\\left(x-2\\right)+12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836319286\"><p id=\"fs-id1167836319288\">The binomial in the middle term, \\(\\left(x-2\\right)\\) is squared in the first term. If we let \\(u=x-2\\) and substitute, our trinomial will be in \\(a{x}^{2}+bx+c\\) form.<\/p><table id=\"fs-id1167836398883\" class=\"unnumbered unstyled\" summary=\"The polynomial is open parentheses x minus 2 close parentheses squared plus 7 open parentheses x minus 2 close parentheses plus 12. Substituting x minus 2 with u, we get u squared plus 7u plus 12. We factor this to get u plus 3, u plus 4. Replacing u with x minus 2, we get open parentheses x minus 2 plus 3 close parentheses open parentheses x minus 2 plus 4 close parentheses. Simplifying, we get x plus 1, x plus 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748361\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691720\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(u=x-2\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829579793\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836699186\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\(x-2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829712623\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify inside the parentheses.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408611\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167833054731\">This could also be factored by first multiplying out the \\({\\left(x-2\\right)}^{2}\\) and the \\(7\\left(x-2\\right)\\) and then combining like terms and then factoring. Most students prefer the substitution method.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829849379\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829589752\"><div data-type=\"problem\" id=\"fs-id1167829589754\"><p id=\"fs-id1167829589756\">Factor by substitution: \\({\\left(x-5\\right)}^{2}+6\\left(x-5\\right)+8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826025462\"><p id=\"fs-id1167826025464\">\\(\\left(x-3\\right)\\left(x-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829614310\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829614315\"><div data-type=\"problem\" id=\"fs-id1167829979385\"><p id=\"fs-id1167829979387\">Factor by substitution: \\({\\left(y-4\\right)}^{2}+8\\left(y-4\\right)+15.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836386289\"><p id=\"fs-id1167836440365\">\\(\\left(y-1\\right)\\left(y+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832977093\" class=\"media-2\"><p id=\"fs-id1167832977097\">Access this online resource for additional instruction and practice with factoring.<\/p><ul id=\"fs-id1167832977100\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37ACmethod\">Factor a trinomial using the AC method<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167824737671\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836545407\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to factor trinomials of the form \\({x}^{2}+bx+c.\\)<\/strong><ol id=\"fs-id1167836521132\" type=\"1\" class=\"stepwise\"><li>Write the factors as two binomials with first terms <em data-effect=\"italics\">x<\/em>. \\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\hfill {x}^{2}+bx+c\\hfill \\\\ \\hfill \\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\left(x\\phantom{\\rule{1.2em}{0ex}}\\right)\\hfill \\end{array}\\)<\/li><li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccc}\\text{multiply to}\\hfill &amp; &amp; c,m\u00b7n=c\\hfill \\\\ \\text{add to}\\hfill &amp; &amp; b,m+n=b\\hfill \\end{array}\\)<\/li><li>Use <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> as the last terms of the factors. \\(\\phantom{\\rule{8em}{0ex}}\\left(x+m\\right)\\left(x+n\\right)\\)<div data-type=\"newline\"><br><\/div><\/li><li>Check by multiplying the factors.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Strategy for Factoring Trinomials of the Form \\({x}^{2}+bx+c\\):<\/strong> When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{c}\\hfill \\begin{array}{c}\\hfill {x}^{2}+bx+c\\hfill \\\\ \\hfill \\left(x+m\\right)\\left(x+n\\right)\\hfill \\end{array}\\hfill \\\\ \\hfill \\mathbf{\\text{When}}\\phantom{\\rule{0.2em}{0ex}}{\\text{c}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{is positive,}}\\phantom{\\rule{0.2em}{0ex}}{\\text{m}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{and}}\\phantom{\\rule{0.2em}{0ex}}{\\text{n}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{have the same sign.}}\\hfill \\\\ \\hfill b\\phantom{\\rule{0.2em}{0ex}}\\text{positive}\\phantom{\\rule{16.5em}{0ex}}b\\phantom{\\rule{0.2em}{0ex}}\\text{negative}\\hfill \\\\ \\hfill m,n\\phantom{\\rule{0.2em}{0ex}}\\text{positive}\\phantom{\\rule{15em}{0ex}}m,n\\phantom{\\rule{0.2em}{0ex}}\\text{negative}\\hfill \\\\ \\hfill {x}^{2}+5x+6\\phantom{\\rule{16em}{0ex}}{x}^{2}-6x+8\\hfill \\\\ \\hfill \\left(x+2\\right)\\left(x+3\\right)\\phantom{\\rule{15em}{0ex}}\\left(x-4\\right)\\left(x-2\\right)\\hfill \\\\ \\hfill \\text{same signs}\\phantom{\\rule{16em}{0ex}}\\text{same signs}\\hfill \\\\ \\hfill \\mathbf{\\text{When}}\\phantom{\\rule{0.2em}{0ex}}{\\text{c}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{is negative,}}\\phantom{\\rule{0.2em}{0ex}}{\\text{m}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{and}}\\phantom{\\rule{0.2em}{0ex}}{\\text{n}}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{have opposite signs.}}\\hfill \\\\ \\hfill {x}^{2}+x-12\\phantom{\\rule{15em}{0ex}}{x}^{2}-2x-15\\hfill \\\\ \\hfill \\left(x+4\\right)\\left(x-3\\right)\\phantom{\\rule{15em}{0ex}}\\left(x-5\\right)\\left(x+3\\right)\\hfill \\\\ \\hfill \\text{opposite signs}\\phantom{\\rule{15em}{0ex}}\\text{opposite signs}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> Notice that, in the case when <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> have opposite signs, the sign of the one with the larger absolute value matches the sign of <em data-effect=\"italics\">b<\/em>.<\/li><li><strong data-effect=\"bold\">How to factor trinomials of the form \\(a{x}^{2}+bx+c\\) using trial and error.<\/strong><ol id=\"fs-id1167833086362\" type=\"1\" class=\"stepwise\"><li>Write the trinomial in descending order of degrees as needed.<\/li><li>Factor any GCF.<\/li><li>Find all the factor pairs of the first term.<\/li><li>Find all the factor pairs of the third term.<\/li><li>Test all the possible combinations of the factors until the correct product is found.<\/li><li>Check by multiplying.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to factor trinomials of the form \\(a{x}^{2}+bx+c\\) using the \u201cac\u201d method.<\/strong><ol id=\"fs-id1167829696444\" type=\"1\" class=\"stepwise\"><li>Factor any GCF.<\/li><li>Find the product <em data-effect=\"italics\">ac<\/em>.<\/li><li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that:<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}\\text{Multiply to}\\phantom{\\rule{0.2em}{0ex}}ac.\\hfill &amp; &amp; &amp; &amp; &amp; m\u00b7n=a\u00b7c\\hfill \\\\ \\text{Add to}\\phantom{\\rule{0.2em}{0ex}}b.\\hfill &amp; &amp; &amp; &amp; &amp; m+n=b\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; a{x}^{2}+bx+c\\hfill \\end{array}\\)<\/li><li>Split the middle term using <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>. \\(\\phantom{\\rule{4em}{0ex}}a{x}^{2}+mx+nx+c\\)<\/li><li>Factor by grouping.<\/li><li>Check by multiplying the factors.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836487152\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829785803\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167829579829\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\({x}^{2}+bx+c\\)<\/strong><\/p><p id=\"fs-id1167836409781\">In the following exercises, factor each trinomial of the form \\({x}^{2}+bx+c.\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167836504047\"><div data-type=\"problem\" id=\"fs-id1167836504049\"><p id=\"fs-id1167836504051\">\\({p}^{2}+11p+30\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833356059\"><p id=\"fs-id1167836625804\">\\(\\left(p+5\\right)\\left(p+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829931154\"><div data-type=\"problem\" id=\"fs-id1167829931156\"><p id=\"fs-id1167829931158\">\\({w}^{2}+10x+21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836712400\"><div data-type=\"problem\"><p id=\"fs-id1167836409429\">\\({n}^{2}+19n+48\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833380083\"><p id=\"fs-id1167833380085\">\\(\\left(n+3\\right)\\left(n+16\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624541\"><div data-type=\"problem\" id=\"fs-id1167829624543\"><p id=\"fs-id1167829624545\">\\({b}^{2}+14b+48\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824754942\"><div data-type=\"problem\"><p id=\"fs-id1167824735504\">\\({a}^{2}+25a+100\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833256132\"><p id=\"fs-id1167833256134\">\\(\\left(a+5\\right)\\left(a+20\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836667054\"><div data-type=\"problem\" id=\"fs-id1167836667056\"><p id=\"fs-id1167836667058\">\\({u}^{2}+101u+100\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082098\"><div data-type=\"problem\" id=\"fs-id1167833082100\"><p id=\"fs-id1167833082102\">\\({x}^{2}-8x+12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829719705\"><p id=\"fs-id1167829719708\">\\(\\left(x-2\\right)\\left(x-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836578775\"><div data-type=\"problem\" id=\"fs-id1167836578778\"><p id=\"fs-id1167836578780\">\\({q}^{2}-13q+36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836579148\"><div data-type=\"problem\" id=\"fs-id1167833008962\"><p id=\"fs-id1167833008964\">\\({y}^{2}-18y+45\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829877863\"><p id=\"fs-id1167829877865\">\\(\\left(y-3\\right)\\left(y-15\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836662669\"><div data-type=\"problem\" id=\"fs-id1167836707254\"><p id=\"fs-id1167836707256\">\\({m}^{2}-13m+30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836600208\"><div data-type=\"problem\" id=\"fs-id1167829651488\"><p id=\"fs-id1167829651490\">\\({x}^{2}-8x+7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836485691\"><p id=\"fs-id1167836485694\">\\(\\left(x-1\\right)\\left(x-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836619465\"><div data-type=\"problem\" id=\"fs-id1167836409749\"><p id=\"fs-id1167836409751\">\\({y}^{2}-5y+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833142068\"><div data-type=\"problem\" id=\"fs-id1167836578916\"><p id=\"fs-id1167836578918\">\\(5p-6+{p}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832926693\"><p id=\"fs-id1167832926695\">\\(\\left(p-1\\right)\\left(p+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836447414\"><div data-type=\"problem\" id=\"fs-id1167836546028\"><p id=\"fs-id1167836546030\">\\(6n-7+{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836597968\"><div data-type=\"problem\" id=\"fs-id1167836691283\"><p id=\"fs-id1167836691285\">\\(8-6x+{x}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836650093\"><p id=\"fs-id1167836650096\">\\(\\left(x-4\\right)\\left(x-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829686747\"><div data-type=\"problem\" id=\"fs-id1167836743445\"><p id=\"fs-id1167836743448\">\\(7x+{x}^{2}+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836704362\"><div data-type=\"problem\" id=\"fs-id1167829833455\"><p id=\"fs-id1167829833457\">\\({x}^{2}-12-11x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836731142\"><p id=\"fs-id1167836731144\">\\(\\left(x-12\\right)\\left(x+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836662752\"><div data-type=\"problem\"><p id=\"fs-id1167829714524\">\\(-11-10x+{x}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167836701267\">In the following exercises, factor each trinomial of the form \\({x}^{2}+bxy+c{y}^{2}.\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167836409536\"><div data-type=\"problem\"><p>\\({x}^{2}-2xy-80{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836792241\"><p id=\"fs-id1167836792243\">\\(\\left(x+8y\\right)\\left(x-10y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824720795\"><div data-type=\"problem\" id=\"fs-id1167824720798\"><p id=\"fs-id1167824720800\">\\({p}^{2}-8pq-65{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829791592\"><div data-type=\"problem\" id=\"fs-id1167829791595\"><p id=\"fs-id1167829791597\">\\({m}^{2}-64mn-65{n}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826025369\"><p id=\"fs-id1167826025371\">\\(\\left(m+n\\right)\\left(m-65n\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830123538\"><div data-type=\"problem\" id=\"fs-id1167830123540\"><p id=\"fs-id1167830123542\">\\({p}^{2}-2pq-35{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833366578\"><div data-type=\"problem\" id=\"fs-id1167833366580\"><p id=\"fs-id1167833366582\">\\({a}^{2}+5ab-24{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829744468\"><p id=\"fs-id1167829744470\">\\(\\left(a+8b\\right)\\left(a-3b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082568\"><div data-type=\"problem\" id=\"fs-id1167833082570\"><p id=\"fs-id1167833082572\">\\({r}^{2}+3rs-28{s}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829620568\"><div data-type=\"problem\" id=\"fs-id1167826170218\"><p id=\"fs-id1167826170220\">\\({x}^{2}-3xy-14{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829690069\"><p id=\"fs-id1167829690071\">Prime<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829789140\"><p id=\"fs-id1167829789142\">\\({u}^{2}-8uv-24{v}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833339029\"><div data-type=\"problem\" id=\"fs-id1167833339031\"><p>\\({m}^{2}-5mn+30{n}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836667115\"><p id=\"fs-id1167836667117\">Prime<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836738050\"><div data-type=\"problem\" id=\"fs-id1167836738052\"><p id=\"fs-id1167836738054\">\\({c}^{2}-7cd+18{d}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167836691590\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\(a{x}^{2}+bx+c\\) Using Trial and Error<\/strong><\/p><p id=\"fs-id1167824617569\">In the following exercises, factor completely using trial and error.<\/p><div data-type=\"exercise\" id=\"fs-id1167824617573\"><div data-type=\"problem\" id=\"fs-id1167824617575\"><p id=\"fs-id1167824617577\">\\({p}^{3}-8{p}^{2}-20p\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833025378\"><p id=\"fs-id1167836705241\">\\(p\\left(p-10\\right)\\left(p+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836635464\"><div data-type=\"problem\" id=\"fs-id1167836635466\"><p id=\"fs-id1167836635468\">\\({q}^{3}-5{q}^{2}-24q\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833051603\"><div data-type=\"problem\" id=\"fs-id1167833051605\"><p id=\"fs-id1167833051607\">\\(3{m}^{3}-21{m}^{2}+30m\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836790669\"><p id=\"fs-id1167836790671\">\\(3m\\left(m-5\\right)\\left(m-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829696647\"><div data-type=\"problem\" id=\"fs-id1167836487008\"><p id=\"fs-id1167836487011\">\\(11{n}^{3}-55{n}^{2}+44n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824734155\"><div data-type=\"problem\" id=\"fs-id1167836513313\"><p id=\"fs-id1167836513315\">\\(5{x}^{4}+10{x}^{3}-75{x}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829905215\"><p id=\"fs-id1167829905218\">\\(5{x}^{2}\\left(x-3\\right)\\left(x+5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836621263\"><p id=\"fs-id1167836621266\">\\(6{y}^{4}+12{y}^{3}-48{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836515214\"><div data-type=\"problem\" id=\"fs-id1167836515216\"><p id=\"fs-id1167836515218\">\\(2{t}^{2}+7t+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833328966\"><p id=\"fs-id1167833328968\">\\(\\left(2t+5\\right)\\left(t+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836511666\"><div data-type=\"problem\" id=\"fs-id1167836511668\"><p id=\"fs-id1167829731913\">\\(5{y}^{2}+16y+11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833114742\"><div data-type=\"problem\" id=\"fs-id1167833114744\"><p id=\"fs-id1167833114746\">\\(11{x}^{2}+34x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734715\"><p id=\"fs-id1167824734718\">\\(\\left(11x+1\\right)\\left(x+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836623750\"><div data-type=\"problem\" id=\"fs-id1167836623752\"><p id=\"fs-id1167836623754\">\\(7{b}^{2}+50b+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836627631\"><div data-type=\"problem\" id=\"fs-id1167836627633\"><p id=\"fs-id1167836627635\">\\(4{w}^{2}-5w+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829749988\"><p id=\"fs-id1167829749990\">\\(\\left(4w-1\\right)\\left(w-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836635483\"><div data-type=\"problem\" id=\"fs-id1167836635485\"><p id=\"fs-id1167836635487\">\\(5{x}^{2}-17x+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836532737\"><div data-type=\"problem\" id=\"fs-id1167836532740\"><p id=\"fs-id1167836532742\">\\(4{q}^{2}-7q-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836599429\"><p>\\(\\left(4q+1\\right)\\left(q-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833023066\"><div data-type=\"problem\" id=\"fs-id1167833023068\"><p id=\"fs-id1167833023070\">\\(10{y}^{2}-53y-11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381421\"><div data-type=\"problem\" id=\"fs-id1167833381423\"><p id=\"fs-id1167833381425\">\\(6{p}^{2}-19pq+10{q}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836349251\"><p id=\"fs-id1167836349253\">\\(\\left(2p-5q\\right)\\left(3p-2q\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836524333\"><div data-type=\"problem\" id=\"fs-id1167829930941\"><p id=\"fs-id1167829930943\">\\(21{m}^{2}-29mn+10{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833224665\"><div data-type=\"problem\" id=\"fs-id1167833224667\"><p>\\(4{a}^{2}+17ab-15{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836356298\"><p id=\"fs-id1167836356300\">\\(\\left(4a-3b\\right)\\left(a+5b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p>\\(6{u}^{2}+5uv-14{v}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624130\"><div data-type=\"problem\" id=\"fs-id1167829624132\"><p id=\"fs-id1167829624134\">\\(-16{x}^{2}-32x-16\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734903\"><p id=\"fs-id1167824734905\">\\(-16\\left(x+1\\right)\\left(x+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833412554\"><div data-type=\"problem\" id=\"fs-id1167833412556\"><p id=\"fs-id1167833412558\">\\(-81{a}^{2}+153a+18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829619774\"><div data-type=\"problem\" id=\"fs-id1167829619776\"><p id=\"fs-id1167829619779\">\\(-30{q}^{3}-140{q}^{2}-80q\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752404\"><p id=\"fs-id1167829752406\">\\(10q\\left(3q+2\\right)\\left(q+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829908109\"><div data-type=\"problem\" id=\"fs-id1167832940635\"><p id=\"fs-id1167832940637\">\\(-5{y}^{3}-30{y}^{2}+35y\\)<\/p><\/div><\/div><p id=\"fs-id1167833036714\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\(a{x}^{2}+bx+c\\) using the \u2018ac\u2019 Method<\/strong><\/p><p id=\"fs-id1167833023204\">In the following exercises, factor using the \u2018ac\u2019 method.<\/p><div data-type=\"exercise\" id=\"fs-id1167829752649\"><div data-type=\"problem\" id=\"fs-id1167829752651\"><p id=\"fs-id1167829752653\">\\(5{n}^{2}+21n+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829624525\"><p id=\"fs-id1167829624527\">\\(\\left(5n+1\\right)\\left(n+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829719186\"><div data-type=\"problem\" id=\"fs-id1167829719188\"><p id=\"fs-id1167824733201\">\\(8{w}^{2}+25w+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829784219\"><div data-type=\"problem\" id=\"fs-id1167829784221\"><p id=\"fs-id1167829784223\">\\(4{k}^{2}-16k+15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836790645\"><p id=\"fs-id1167836790647\">\\(\\left(2k-3\\right)\\left(2k-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836429494\"><div data-type=\"problem\" id=\"fs-id1167836429496\"><p>\\(5{s}^{2}-9s+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829789158\"><div data-type=\"problem\" id=\"fs-id1167829789160\"><p id=\"fs-id1167833239776\">\\(6{y}^{2}+y-15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829614527\"><p id=\"fs-id1167829614529\">\\(\\left(3y+5\\right)\\left(2y-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167824734630\">\\(6{p}^{2}+p-22\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833085475\"><div data-type=\"problem\" id=\"fs-id1167833085477\"><p id=\"fs-id1167833024802\">\\(2{n}^{2}-27n-45\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747883\"><p id=\"fs-id1167829747885\">\\(\\left(2n+3\\right)\\left(n-15\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836398908\"><div data-type=\"problem\" id=\"fs-id1167836398910\"><p id=\"fs-id1167836398912\">\\(12{z}^{2}-41z-11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836731922\"><div data-type=\"problem\" id=\"fs-id1167836731924\"><p id=\"fs-id1167836731926\">\\(60{y}^{2}+290y-50\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836519248\"><p id=\"fs-id1167836333653\">\\(10\\left(6y-1\\right)\\left(y+5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836532781\"><div data-type=\"problem\" id=\"fs-id1167836532783\"><p id=\"fs-id1167836508660\">\\(6{u}^{2}-46u-16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833087032\"><div data-type=\"problem\" id=\"fs-id1167833087034\"><p id=\"fs-id1167833087036\">\\(48{z}^{3}-102{z}^{2}-45z\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836537843\"><p id=\"fs-id1167836537845\">\\(3z\\left(8z+3\\right)\\left(2z-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829908836\"><div data-type=\"problem\" id=\"fs-id1167829908838\"><p id=\"fs-id1167836697363\">\\(90{n}^{3}+42{n}^{2}-216n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829783678\"><div data-type=\"problem\" id=\"fs-id1167829783680\"><p id=\"fs-id1167829783682\">\\(16{s}^{2}+40s+24\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836791403\"><p id=\"fs-id1167836791405\">\\(8\\left(2s+3\\right)\\left(s+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833023438\"><div data-type=\"problem\" id=\"fs-id1167833023440\"><p id=\"fs-id1167833023442\">\\(24{p}^{2}+160p+96\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829712919\"><div data-type=\"problem\" id=\"fs-id1167829712921\"><p id=\"fs-id1167829712923\">\\(48{y}^{2}+12y-36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829791297\"><p id=\"fs-id1167836485977\">\\(12\\left(4y-3\\right)\\left(y+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836585285\"><div data-type=\"problem\" id=\"fs-id1167836585287\"><p id=\"fs-id1167836585289\">\\(30{x}^{2}+105x-60\\)<\/p><\/div><\/div><p id=\"fs-id1167832925672\"><strong data-effect=\"bold\">Factor Using Substitution<\/strong><\/p><p id=\"fs-id1167832925678\">In the following exercises, factor using substitution.<\/p><div data-type=\"exercise\" id=\"fs-id1167836705253\"><div data-type=\"problem\" id=\"fs-id1167836705255\"><p id=\"fs-id1167836705257\">\\({x}^{4}-{x}^{2}-12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833136734\"><p id=\"fs-id1167833136736\">\\(\\left({x}^{2}+3\\right)\\left({x}^{2}-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836535867\"><div data-type=\"problem\" id=\"fs-id1167836535869\"><p id=\"fs-id1167836535872\">\\({x}^{4}+2{x}^{2}-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833224803\"><div data-type=\"problem\" id=\"fs-id1167833224805\"><p id=\"fs-id1167833224807\">\\({x}^{4}-3{x}^{2}-28\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824735216\"><p id=\"fs-id1167824735218\">\\(\\left({x}^{2}-7\\right)\\left({x}^{2}+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829894287\"><div data-type=\"problem\" id=\"fs-id1167829894289\"><p id=\"fs-id1167829894292\">\\({x}^{4}-13{x}^{2}-30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833050525\"><div data-type=\"problem\" id=\"fs-id1167833050527\"><p id=\"fs-id1167833050529\">\\({\\left(x-3\\right)}^{2}-5\\left(x-3\\right)-36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833386411\"><p>\\(\\left(x-12\\right)\\left(x+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752883\"><div data-type=\"problem\" id=\"fs-id1167829752885\"><p id=\"fs-id1167833396915\">\\({\\left(x-2\\right)}^{2}-3\\left(x-2\\right)-54\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833041786\"><div data-type=\"problem\"><p id=\"fs-id1167836727764\">\\({\\left(3y-2\\right)}^{2}-\\left(3y-2\\right)-2\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167826025137\">\\(\\left(3y-4\\right)\\left(3y-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832937141\"><div data-type=\"problem\" id=\"fs-id1167832937143\"><p id=\"fs-id1167832937145\">\\({\\left(5y-1\\right)}^{2}-3\\left(5y-1\\right)-18\\)<\/p><\/div><\/div><p id=\"fs-id1167833339369\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p id=\"fs-id1167833339375\">In the following exercises, factor each expression using any method.<\/p><div data-type=\"exercise\" id=\"fs-id1167833339378\"><div data-type=\"problem\" id=\"fs-id1167833082156\"><p id=\"fs-id1167833082158\">\\({u}^{2}-12u+36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829942496\"><p id=\"fs-id1167829942498\">\\(\\left(u-6\\right)\\left(u-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836652948\"><div data-type=\"problem\" id=\"fs-id1167836652950\"><p id=\"fs-id1167836652952\">\\({x}^{2}-14x-32\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836732422\"><div data-type=\"problem\" id=\"fs-id1167836732424\"><p id=\"fs-id1167836732426\">\\({r}^{2}-20rs+64{s}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833082178\"><p id=\"fs-id1167833082180\">\\(\\left(r-4s\\right)\\left(r-16s\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829691171\"><div data-type=\"problem\"><p id=\"fs-id1167829691175\">\\({q}^{2}-29qr-96{r}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836664187\"><div data-type=\"problem\" id=\"fs-id1167836664189\"><p id=\"fs-id1167836664191\">\\(12{y}^{2}-29y+14\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833086526\"><p id=\"fs-id1167833086528\">\\(\\left(4y-7\\right)\\left(3y-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832936294\"><div data-type=\"problem\" id=\"fs-id1167832936296\"><p id=\"fs-id1167832936298\">\\(12{x}^{2}+36y-24z\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836614302\"><div data-type=\"problem\" id=\"fs-id1167829696487\"><p id=\"fs-id1167829696489\">\\(6{n}^{2}+5n-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829851524\"><p id=\"fs-id1167829851526\">\\(\\left(2n-1\\right)\\left(3n+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829619062\"><div data-type=\"problem\" id=\"fs-id1167829619064\"><p id=\"fs-id1167829619067\">\\(3{q}^{2}+6q+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836363546\"><div data-type=\"problem\" id=\"fs-id1167836363548\"><p id=\"fs-id1167836363550\">\\(13{z}^{2}+39z-26\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829924882\"><p>\\(13\\left({z}^{2}+3z-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836662701\"><div data-type=\"problem\" id=\"fs-id1167836662703\"><p id=\"fs-id1167836662705\">\\(5{r}^{2}+25r+30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836598121\"><div data-type=\"problem\" id=\"fs-id1167829713020\"><p id=\"fs-id1167829713022\">\\(3{p}^{2}+21p\\)<\/p><\/div><div data-type=\"solution\"><p>\\(3p\\left(p+7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829594556\"><div data-type=\"problem\"><p id=\"fs-id1167829594561\">\\(7{x}^{2}-21x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833053777\"><div data-type=\"problem\" id=\"fs-id1167833053780\"><p id=\"fs-id1167833053782\">\\(6{r}^{2}+30r+36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836614430\"><p id=\"fs-id1167836614432\">\\(6\\left(r+2\\right)\\left(r+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836578943\"><div data-type=\"problem\" id=\"fs-id1167836578945\"><p id=\"fs-id1167836578947\">\\(18{m}^{2}+15m+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836625666\"><div data-type=\"problem\" id=\"fs-id1167836789914\"><p id=\"fs-id1167836789916\">\\(24{n}^{2}+20n+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750344\"><p id=\"fs-id1167829750346\">\\(4\\left(2n+1\\right)\\left(3n+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409387\"><div data-type=\"problem\" id=\"fs-id1167836409389\"><p id=\"fs-id1167836409392\">\\(4{a}^{2}+5a+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833004862\"><div data-type=\"problem\" id=\"fs-id1167833004864\"><p id=\"fs-id1167833004866\">\\({x}^{4}-4{x}^{2}-12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836574849\"><p id=\"fs-id1167836574851\">\\(\\left({x}^{2}+2\\right)\\left({x}^{2}-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832980660\"><div data-type=\"problem\" id=\"fs-id1167832980662\"><p id=\"fs-id1167832980664\">\\({x}^{4}-7{x}^{2}-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829716233\"><div data-type=\"problem\" id=\"fs-id1167829716235\"><p id=\"fs-id1167829716237\">\\({\\left(x+3\\right)}^{2}-9\\left(x+3\\right)-36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829686722\"><p id=\"fs-id1167829686724\">\\(\\left(x-9\\right)\\left(x+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832936260\"><div data-type=\"problem\" id=\"fs-id1167832936262\"><p id=\"fs-id1167832936264\">\\({\\left(x+2\\right)}^{2}-25\\left(x+2\\right)-54\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167824763176\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829752897\"><div data-type=\"problem\" id=\"fs-id1167829752899\"><p id=\"fs-id1167829752901\">Many trinomials of the form \\({x}^{2}+bx+c\\) factor into the product of two binomials \\(\\left(x+m\\right)\\left(x+n\\right).\\) Explain how you find the values of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829761405\"><p id=\"fs-id1167829761407\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836790885\"><div data-type=\"problem\" id=\"fs-id1167836790887\"><p id=\"fs-id1167836790889\">Tommy factored \\({x}^{2}-x-20\\) as \\(\\left(x+5\\right)\\left(x-4\\right).\\) Sara factored it as \\(\\left(x+4\\right)\\left(x-5\\right).\\) Ernesto factored it as \\(\\left(x-5\\right)\\left(x-4\\right).\\) Who is correct? Explain why the other two are wrong.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836305370\"><div data-type=\"problem\" id=\"fs-id1167836305373\"><p id=\"fs-id1167836305375\">List, in order, all the steps you take when using the \u201cac\u201d method to factor a trinomial of the form \\(a{x}^{2}+bx+c.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829785037\"><p id=\"fs-id1167836391259\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836391265\"><div data-type=\"problem\" id=\"fs-id1167836391267\"><p id=\"fs-id1167836391269\">How is the \u201cac\u201d method similar to the \u201cundo FOIL\u201d method? How is it different?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836399159\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836399164\"><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-id1167836399173\" data-alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the \u201cac\u201d method, factor using substitution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the \u201cac\u201d method, factor using substitution.\"><\/span><p id=\"fs-id1167833380224\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/li>\n<li>Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using trial and error<\/li>\n<li>Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u2018ac\u2019 method<\/li>\n<li>Factor using substitution<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833052618\" class=\"be-prepared\">\n<p id=\"fs-id1167836363691\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167833047839\" type=\"1\">\n<li>Find all the factors of 72.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167829937177\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-557295d6e4f8eda2e2cf631a4c579529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836544266\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7ffc80c0e5d1e087a7fac7d4f8f6d82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede19421341c3b998e627f0339dedf45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536325\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836601351\">\n<h3 data-type=\"title\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/h3>\n<p id=\"fs-id1167824578628\">You have already learned how to multiply binomials using <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>. Now you\u2019ll need to \u201cundo\u201d this multiplication. To factor the trinomial means to start with the product, and end with the factors.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836508876\" data-alt=\"Figure shows the equation open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 5 x plus 6. The left side of the equation is labeled factors and the right is labeled product. An arrow pointing right is labeled multiply. An arrow pointing left is labeled factor.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the equation open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 5 x plus 6. The left side of the equation is labeled factors and the right is labeled product. An arrow pointing right is labeled multiply. An arrow pointing left is labeled factor.\" \/><\/span><\/p>\n<p id=\"fs-id1167836545205\">To figure out how we would factor a <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f7e667d716fddce00c27d4e71431539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/> such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-796dbf4877c44d45b95e2f7845612068_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/> and factor it to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db720eba867c5f81598a9aa026df0cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/> let\u2019s start with two general binomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ae1e411e03346a32e77563c00d91b40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6800e6de9c26568fcd5658aaad95c73c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"fs-id1167836549815\" class=\"unnumbered unstyled\" summary=\"We have open parentheses x plus m close parentheses open parentheses x plus n close parentheses. Foil to find the product x squared plus mx plus nx plus mn. Factor the GCF from the middle terms x squared plus open parentheses m plus n close parentheses x plus mn. Now our trinomial is of the form x squared plus bx plus c, where b is m plus n and c is mn\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836444935\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Foil to find the product.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836366433\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the GCF from the middle 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_06_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Our trinomial is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056814\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836625705\">This tells us that to factor a trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f7e667d716fddce00c27d4e71431539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/> we need two factors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ae1e411e03346a32e77563c00d91b40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-980ac67ad78a228834b3ce882d00a53f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> where the two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> multiply to <em data-effect=\"italics\">c<\/em> and add to <em data-effect=\"italics\">b<\/em>.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor a Trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833386897\">\n<div data-type=\"problem\" id=\"fs-id1167836525654\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0de28e76879bf996d365603691f85084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#50;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167829714211\" data-alt=\"Step 1 is to write the factors of x squared plus 11x plus 24 as two binomials with first terms x. Write two sets of parentheses and put x as the first term.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the factors of x squared plus 11x plus 24 as two binomials with first terms x. Write two sets of parentheses and put x as the first term.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829676303\" data-alt=\"Step 2 is to find two numbers m and n that multiply to c, m times n is c and add to b, m plus n is b. So, find two numbers that multiply to 24 and add to 11. Factors of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6. Sum of factors: 1 plus 24 is 25, 2 plus 12 is 14, 3 plus 8 is 11 and 4 plus 6 is 10.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find two numbers m and n that multiply to c, m times n is c and add to b, m plus n is b. So, find two numbers that multiply to 24 and add to 11. Factors of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6. Sum of factors: 1 plus 24 is 25, 2 plus 12 is 14, 3 plus 8 is 11 and 4 plus 6 is 10.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 3 is to use m and n, in this case, 3 and 8, as the last terms of the binomials. So we get open parentheses x plus 3 close parentheses open parentheses x plus 8 close parentheses\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use m and n, in this case, 3 and 8, as the last terms of the binomials. So we get open parentheses x plus 3 close parentheses open parentheses x plus 8 close parentheses\" \/><\/span><span data-type=\"media\" data-alt=\"Step 4 is to check by multiplying the factors to get the original polynomial.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_003d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check by multiplying the factors to get the original polynomial.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836377040\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836538765\">\n<div data-type=\"problem\" id=\"fs-id1167836543830\">\n<p id=\"fs-id1167836449827\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8efba0bd46fcd0c1756c750115fc86fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#113;&#43;&#50;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512368\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17cddfa10970b6bde868ec6f0cb4e68d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#54;&#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>\n<div data-type=\"note\" id=\"fs-id1167836609921\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836547465\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e30d33970110ae6503ba7fd6e8b40713_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#116;&#43;&#50;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836282499\">\n<p id=\"fs-id1167836360137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d011056a166f607a906713a2d7fb26bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836545745\">Let\u2019s summarize the steps we used to find the factors.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836732813\" class=\"howto\">\n<div data-type=\"title\">Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/div>\n<ol type=\"1\" class=\"stepwise\">\n<li>Write the factors as two binomials with first terms <em data-effect=\"italics\">x<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb6a41c56ceaad1afc65789fde64fd4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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=\"41\" width=\"92\" style=\"vertical-align: -15px;\" \/><\/li>\n<li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that\n<ul id=\"fs-id1167825908932\" data-bullet-style=\"bullet\">\n<li>multiply to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82f7c35624a74dee9ca64ddf721098b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#44;&#109;&middot;&#110;&#61;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>add to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57ad0d3b879dc5a5a8477b5783ceabb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#44;&#109;&#43;&#110;&#61;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li>Use <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> as the last terms of the factors. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aa3be8884cd377e68fd41293e96e334_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;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836419009\">In the first example, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let\u2019s look first at trinomials with only the middle term negative.<\/p>\n<p>How do you get a <em data-effect=\"italics\">positive product<\/em> and a <em data-effect=\"italics\">negative sum<\/em>? We use two negative numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829716451\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836524746\">\n<div data-type=\"problem\" id=\"fs-id1167832945802\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d039080291415a1c06afe371fe4dad08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#121;&#43;&#50;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836516109\">\n<p id=\"fs-id1167836544225\">Again, with the positive last term, 28, and the negative middle term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee33c57888c459335788580c177a706c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#121;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -4px;\" \/> we need two negative factors. Find two numbers that multiply 28 and add to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-502fc393fd35fe09cdcdd851dc374afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/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-15ea5a90769603a9f3ac1e4ba2740b1c_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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#121;&#43;&#50;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#97;&#115;&#32;&#116;&#119;&#111;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#32;&#119;&#105;&#116;&#104;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#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;&#121;&#46;&#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;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#119;&#111;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#116;&#104;&#97;&#116;&#58;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#32;&#50;&#56;&#32;&#97;&#110;&#100;&#32;&#97;&#100;&#100;&#32;&#116;&#111;&#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;&#49;&#49;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"666\" style=\"vertical-align: -25px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167830123906\" class=\"unnumbered\" summary=\"This table has two columns showing factors of 28 and sum of the factors. Factors are: minus 1 and minus 28, whose sum is minus 29, minus 2 and minus 14 whose sum is minus 16, minus 4 and minus 7 whose sum is minus 11.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80588c2af70b499bde2614a130c98b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"center\">Sum of factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2334c44f8ac5d99ce9fb8b4155a112_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#44;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\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-2331bffdcc4812d8df2c6c755462fd2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#44;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\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-d88271966832a1ffc2cb7df09dd5d617_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#44;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-377d686fc257ccadbb29422bc5b4d15b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\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-0a90a716eb60d3bdba0babac5c0f8550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\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-12216fa743494bfc15494ea669024664_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;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#45;&#49;&#49;&#125;&#94;&#123;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-450aa4f9c434e78646f5b81fd111f92a_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;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#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;&#45;&#52;&#44;&#45;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#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;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#121;&#45;&#52;&#121;&#43;&#50;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#121;&#43;&#50;&#56;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"657\" style=\"vertical-align: -48px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167824734057\">\n<p id=\"fs-id1167836448208\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aabf04489f2da150f66ad4c4cea1130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#117;&#43;&#49;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829716542\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53be6e377ffa29cef62526b5a51c1c23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#54;&#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>\n<div data-type=\"note\" id=\"fs-id1167829688066\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836610152\">\n<div data-type=\"problem\" id=\"fs-id1167829694152\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31182724f9d51590e1d37a2807a0bf64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#121;&#43;&#54;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836529007\">\n<p id=\"fs-id1167833053900\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60b9067b1e73ee9109767625d9669cf8_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;&#45;&#57;&#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>\n<p>Now, what if the last term in the trinomial is negative? Think about <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.<\/p>\n<p>How do you get a <em data-effect=\"italics\">negative product<\/em> and a <em data-effect=\"italics\">positive sum<\/em>? We use one positive and one negative number.<\/p>\n<p id=\"fs-id1167836389451\">When we factor trinomials, we must have the terms written in descending order\u2014in order from highest degree to lowest degree.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836399263\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836635213\">\n<div data-type=\"problem\" id=\"fs-id1167836357277\">\n<p id=\"fs-id1167836292963\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f13ba71547ce5a6d4bf4c5852e89380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829686027\">\n<p id=\"fs-id1167836287160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de755734d161a7da8b9d2700d599b08_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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#120;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#114;&#115;&#116;&#32;&#119;&#101;&#32;&#112;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#105;&#110;&#32;&#100;&#101;&#99;&#114;&#101;&#97;&#115;&#105;&#110;&#103;&#32;&#100;&#101;&#103;&#114;&#101;&#101;&#32;&#111;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#32;&#116;&#119;&#111;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#32;&#119;&#105;&#116;&#104;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#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;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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=\"63\" width=\"607\" style=\"vertical-align: -26px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered\" summary=\"This table has two columns showing factors of minus 48 and sum of the factors. Factors are: minus 1 and 48, whose sum is 47, minus 2 and 24, whose sum is 22, minus 3 and 16, whose sum is 13, minus 4 and 12, whose sum is 8, minus 6 and 8, whose sum is 2.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08ed8320699be2770837da731d65c7c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\">Sum of factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bac41d333e4a6d567447dbd910611b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43a37d68cc55d87d54bf5c7a057d9ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c06f323b87101b1292bc24ce6f4e8d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d742fba003c9601d8d4259d9ac676e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#44;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0feadae254325e6173e0c16dafd0d2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#44;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7ee95e2a099808de59dd9f2826b59b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#52;&#56;&#61;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/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-b2bb50043af5215cee90183cea87327a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#50;&#52;&#61;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/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-0cd5f0639c8774bc4e2e2418dae9f2d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#43;&#49;&#54;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/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-e2dfe13d23c080ea6b03a26a5b3aede9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#43;&#49;&#50;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -2px;\" \/><\/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-a1f5b74f4c30d6264d54e9563d1ebce3_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;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#54;&#43;&#56;&#61;&#123;&#50;&#125;&#94;&#123;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-0fa2da11039354a4465f6fb661c545de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#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;&#45;&#54;&#44;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#113;&#43;&#56;&#113;&#45;&#52;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;&#56;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"626\" style=\"vertical-align: -46px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836299692\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836362051\">\n<div data-type=\"problem\" id=\"fs-id1167836493252\">\n<p id=\"fs-id1167829714149\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af5b9567b1d701bf2e8dd0a4ff34ba14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#109;&#43;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836571079\">\n<p id=\"fs-id1167836329246\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02a062d49bf775762aecec8657e8b7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829833335\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833055007\">\n<p id=\"fs-id1167836508324\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-613f3651ef0ec729a5f840eb1938551a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#110;&#43;&#49;&#50;&#43;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836367004\">\n<p id=\"fs-id1167836415277\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8141f2c9cc8a866409b13fe77def73e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836537198\">Sometimes you\u2019ll need to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02db4b17730ec637e3f4c699969ade06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#121;&#43;&#99;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/> with two variables, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-882ec712b3899c9a70f9d429aef0943d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#121;&#43;&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/> The first term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f82905c002b530c14921e8d459fe64b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> is the product of the first terms of the binomial factors, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-733d7d221a314b4af0363c8738e106e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&middot;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"24\" style=\"vertical-align: 0px;\" \/> The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> in the last term means that the second terms of the binomial factors must each contain <em data-effect=\"italics\">y<\/em>. To get the coefficients <em data-effect=\"italics\">b<\/em> and <em data-effect=\"italics\">c<\/em>, you use the same process summarized in <a href=\"#fs-id1167836732813\">How To Factor trinomials<\/a>.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836612676\">\n<div data-type=\"problem\" id=\"fs-id1167829717306\">\n<p id=\"fs-id1167829899564\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba34207435a5f04c0c3ab6e2375ebd45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#114;&#115;&#45;&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"114\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734295\">\n<p id=\"fs-id1167836481087\">We need <em data-effect=\"italics\">r<\/em> in the first term of each binomial and <em data-effect=\"italics\">s<\/em> in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.<\/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-1fc136faa34bb53505cb237bced67982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#114;&#115;&#45;&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#116;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#114;&#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;&#114;&#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;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#99;&#111;&#110;&#116;&#97;&#105;&#110;&#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;&#115;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#116;&#104;&#97;&#116;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#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;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#97;&#100;&#100;&#32;&#116;&#111;&#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;&#56;&#46;&#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=\"621\" style=\"vertical-align: -25px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829739852\" class=\"unnumbered\" summary=\"This table has 2 columns showing factors of minus 9 and sum of factors. The factors are: 1 and minus 9 whose sum is 8, minus 1 and 9 whose sum is minus 8, 3 and minus 3 whose sum is 0.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\">Sum of factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce83e3174fa08b648c0fbc10b9897510_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65775d74bd4abd115f83c1a6ff465ab6_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#43;&#57;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc3e9f9c6cf0cbd0eb1b78d7afa30b7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edfb009e9b7d8f149f94ba6e1aa37131_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#56;&#125;&#94;&#123;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772c2450bc689110c781ec2ed6dc0a41_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#44;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73e86a59d27c4436a13252059c8b5ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-3d43814865fa2569bb12f3ea3248e5e1_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;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#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;&#49;&#44;&#45;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#32;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#57;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#57;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#114;&#115;&#45;&#57;&#114;&#115;&#45;&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#114;&#115;&#45;&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"573\" style=\"vertical-align: -44px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832940558\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836547330\">\n<div data-type=\"problem\" id=\"fs-id1167829742871\">\n<p id=\"fs-id1167836689445\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12184ed4daad6f0f5de198129678c60d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#97;&#98;&#43;&#49;&#48;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836434117\">\n<p id=\"fs-id1167829596516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-207d840c06582684b2cc8b378547fa67_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;&#45;&#49;&#48;&#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>\n<div data-type=\"note\" id=\"fs-id1167833263821\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836440155\">\n<div data-type=\"problem\" id=\"fs-id1167833129296\">\n<p id=\"fs-id1167833009882\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a39a08c7c397315c1b72f7f0feb5e2b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#109;&#110;&#43;&#49;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"151\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836333459\">\n<p id=\"fs-id1167836525443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-558a1566c03a7d50c2ab1af5e93c1b57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#50;&#110;&#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>\n<p id=\"fs-id1167836551828\">Some trinomials are prime. The only way to be certain a trinomial is <span data-type=\"term\" class=\"no-emphasis\">prime<\/span> is to list all the possibilities and show that none of them work.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836515109\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836522339\">\n<div data-type=\"problem\" id=\"fs-id1167836522341\">\n<p id=\"fs-id1167826102751\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b77d47a5a534a10f9b6a3f6845d05fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#117;&#118;&#45;&#49;&#50;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"128\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836606580\">\n<p id=\"fs-id1167836364044\">We need <em data-effect=\"italics\">u<\/em> in the first term of each binomial and <em data-effect=\"italics\">v<\/em> in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.<\/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-3dac37a3a98533a19e599483d03e6ff1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#117;&#118;&#45;&#49;&#50;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#116;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#114;&#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;&#117;&#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;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#99;&#111;&#110;&#116;&#97;&#105;&#110;&#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;&#118;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#116;&#104;&#97;&#116;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#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;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#97;&#100;&#100;&#32;&#116;&#111;&#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;&#57;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"644\" style=\"vertical-align: -25px;\" \/> <\/p>\n<table id=\"fs-id1167833051403\" class=\"unnumbered\" summary=\"This table has 2 columns showing factors of minus 12 and sum of factors. The factors are: 1 and minus 12 whose sum is minus 11, minus 1 and 12 whose sum is 11, 2 and minus 6 whose sum is minus 4, minus 2 and 6 whose sum is 4, 3 and minus 4 whose sum is minus 1, minus 3 and minus 3 and 4 whose sum is 1.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed0df082619b14cf697a0cc2805abd5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Sum of factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cea795e3997c15b240edd7c9a1fa3e07_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bdb59cc4980dec0738c7f5a8ffb793a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfb3e1a2484374a16dbd814e910f14e1_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#44;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b6cb7bd123df87bfa34d0f1d2da73de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1192064e26d969ac8e9b38b05b923953_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#44;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb23bbfc0807f7a62700e62cf257910b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9001355fd34938476ac0c80f25c79022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b79c363ee329114f7d2492f1ce368886_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;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#43;&#49;&#50;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/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-e289f96a0a859fed5efc4e665122af9f_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;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0366cce9f223c93275f4ef4b35ecde5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#43;&#54;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/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-4b798acfb97ba773152a611010aab3dd_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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68594a439484d851c65407c8e0fe7874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#43;&#52;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836547038\">Note there are no factor pairs that give us <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> as a sum. The trinomial is prime.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836387300\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829738401\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833060906\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34322eade53ab1db1635594c89a9cea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#121;&#45;&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836691877\">\n<p id=\"fs-id1167836444713\">prime<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836545726\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836652605\">\n<div data-type=\"problem\" id=\"fs-id1167836652607\">\n<p id=\"fs-id1167836516185\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc140836084b4eb42f57b07243f788bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#112;&#113;&#43;&#50;&#48;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836487273\">\n<p id=\"fs-id1167833058791\">prime<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836387282\">Let\u2019s summarize the method we just developed to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167833046935\">\n<div data-type=\"title\">Strategy for Factoring Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/div>\n<p id=\"fs-id1167836650044\">When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b6084a1bc7c897284e7bc4ca7333936_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#101;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#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;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#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;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#44;&#110;&#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;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;&#44;&#110;&#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;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#101;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#118;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#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=\"261\" width=\"494\" style=\"vertical-align: -124px;\" \/><\/div>\n<p id=\"fs-id1167836320419\">Notice that, in the case when <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> have opposite signs, the sign of the one with the larger absolute value matches the sign of <em data-effect=\"italics\">b<\/em>.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833086888\">\n<h3 data-type=\"title\">Factor Trinomials of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> using Trial and Error<\/h3>\n<p id=\"fs-id1167836613360\">Our next step is to factor trinomials whose leading coefficient is not 1, trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13fc8234342051cf7b36d52613e9e1f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167829696932\">Remember to always check for a <span data-type=\"term\" class=\"no-emphasis\">GCF<\/span> first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods we\u2019ve used so far. Let\u2019s do an example to see how this works.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836415146\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836415148\">\n<div data-type=\"problem\" id=\"fs-id1167836456289\">\n<p id=\"fs-id1167836456291\">Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87c3a6798694af09942a349e716dd1df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833053401\">\n<p id=\"fs-id1167833053404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae8d0c7cedf81c518b6744d6e56f6c4e_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;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#114;&#101;&#32;&#97;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#115;&#116;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#116;&#101;&#120;&#116;&#123;&#89;&#101;&#115;&#44;&#32;&#71;&#67;&#70;&#125;&#61;&#52;&#120;&#46;&#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;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#105;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#44;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#44;&#32;&#111;&#114;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#116;&#104;&#114;&#101;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#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;&#116;&#101;&#120;&#116;&#123;&#73;&#116;&#32;&#105;&#115;&#32;&#97;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#32;&#83;&#111;&#32;&#96;&#96;&#117;&#110;&#100;&#111;&#32;&#70;&#79;&#73;&#76;&#46;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#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;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#97;&#32;&#116;&#97;&#98;&#108;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#104;&#101;&#32;&#111;&#110;&#101;&#32;&#115;&#104;&#111;&#119;&#110;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#119;&#111;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#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;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#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;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#97;&#100;&#100;&#32;&#116;&#111;&#32;&#52;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"217\" width=\"690\" style=\"vertical-align: -102px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829784370\" class=\"unnumbered\" summary=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84de678497718651381f1800cefca350_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Sum of factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d06b0ed4d47f4319c26dcc04e586f0e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#44;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"24\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ca131399c8fae94c14716a8f86c6c62_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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbce9198e9879ae53e0c87f3dca6c79c_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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#43;&#53;&#61;&#123;&#52;&#125;&#94;&#123;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"77\" style=\"vertical-align: -2px;\" \/><\/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-e7e149d64bf93df2e64866d93ac5b83b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-101323fe74f1d9be11b92b5b9ea56dfa_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;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" width=\"258\" style=\"vertical-align: -46px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829745918\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829595477\">\n<div data-type=\"problem\" id=\"fs-id1167829595479\">\n<p id=\"fs-id1167836378219\">Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebf399b0109d7d5f06b371d827f48263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836514323\">\n<p id=\"fs-id1167836514325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56534acb1d0b2275eed8afef28ebf0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#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>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829732136\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829732139\">\n<div data-type=\"problem\" id=\"fs-id1167829596896\">\n<p id=\"fs-id1167829596898\">Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2674985d1fb358aaca3b48e4fdac9921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836319758\">\n<p id=\"fs-id1167836319760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52ff1f680accdcd212290dfa7bed5ddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833009012\">What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.<\/p>\n<p id=\"fs-id1167829712589\">Let\u2019s factor the trinomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1271de622ef1ec18ed7fdc8394061e8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167836415401\">From our earlier work, we expect this will factor into two binomials.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836415404\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66f2b43c122dd8fa375582316cc74cfa_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;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#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;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#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;&#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=\"41\" width=\"100\" style=\"vertical-align: -37px;\" \/><\/div>\n<p id=\"fs-id1167836692336\">We know the first terms of the binomial factors will multiply to give us <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f5504a5293c734491a54d004cbff27b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: 0px;\" \/> The only factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9608403b176cb023606ca01493d9f883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/> are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a22d8dd6da08eae0c39d4dc32f19e028_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#120;&#44;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\" \/> We can place them in the binomials.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833274184\" data-alt=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.\" \/><\/span><\/p>\n<p id=\"fs-id1167833054406\">Check: Does <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a641a657cf51e1a28ab8b54a051da35b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#120;&middot;&#51;&#120;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167836774071\">We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1, 2. But we now have two cases to consider as it will make a difference if we write 1, 2 or 2, 1.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836548650\" data-alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses.\" \/><\/span><\/p>\n<p id=\"fs-id1167829850087\">Which factors are correct? To decide that, we multiply the inner and outer terms.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836732179\" data-alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses. In each case, arrows are shown pairing the first term of the first factor with the last term of the second factor and the first term of the second factor with the last term of the first factor.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses. In each case, arrows are shown pairing the first term of the first factor with the last term of the second factor and the first term of the second factor with the last term of the first factor.\" \/><\/span><\/p>\n<p id=\"fs-id1167836526346\">Since the middle term of the trinomial is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59a5e4d64a4725f57ec508f38ec1b820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"23\" style=\"vertical-align: -4px;\" \/> the factors in the first case will work. Let\u2019s use FOIL to check.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836730749\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f36dda74eea7194f9a1f2380fcc25c5_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#51;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"138\" style=\"vertical-align: -24px;\" \/><\/div>\n<p>Our result of the factoring is:<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d5bdfcbcb52dd4c0667c743e9b09083_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;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#50;&#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=\"41\" width=\"119\" style=\"vertical-align: -26px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167829936476\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor a Trinomial Using Trial and Error<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829936478\">\n<div data-type=\"problem\" id=\"fs-id1167829936480\">\n<p id=\"fs-id1167836492295\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a4b321e9c94d89c08714c6188ec2437_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#50;&#121;&#43;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829579867\"><span data-type=\"media\" id=\"fs-id1167829579869\" data-alt=\"Step 1 is to write the trinomial in descending order. The trinomial 3 y squared plus 22y plus 7 is already in descending order.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the trinomial in descending order. The trinomial 3 y squared plus 22y plus 7 is already in descending order.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833138154\" data-alt=\"Step 2 is to factor the GCF. Here, there is none.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to factor the GCF. Here, there is none.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836377144\" data-alt=\"Step 3 is Find all the factor pairs of the first term. The only factors here are 1y and 3y. Since there is only one pair, we can put each as the first term in the parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is Find all the factor pairs of the first term. The only factors here are 1y and 3y. Since there is only one pair, we can put each as the first term in the parentheses.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833055812\" data-alt=\"Step 4 is to find all the factor pairs of the third term. Here, the only pair is 1 and 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to find all the factor pairs of the third term. Here, the only pair is 1 and 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833024736\" data-alt=\"Step 5 is to test all the possible combinations of the factors until the correct product is found. For possible factors open parentheses y plus 1 close parentheses open parentheses 37 plus 7 close parentheses, the product is 3 y squared plus 10y plus 7. For the possible factors open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses, the product is 3 y squared plus 22y plus 7, which is the correct product. Hence, the correct factors are open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to test all the possible combinations of the factors until the correct product is found. For possible factors open parentheses y plus 1 close parentheses open parentheses 37 plus 7 close parentheses, the product is 3 y squared plus 10y plus 7. For the possible factors open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses, the product is 3 y squared plus 22y plus 7, which is the correct product. Hence, the correct factors are open parentheses y plus 7 close parentheses open parentheses 3y plus 1 close parentheses.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836687917\" data-alt=\"Step 6 is to check by multiplying.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_007f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check by multiplying.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836732148\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836620110\">\n<div data-type=\"problem\" id=\"fs-id1167836620112\">\n<p id=\"fs-id1167836620115\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-288f57f6f2f308b7ddd17dd32aecdbae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#97;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836493740\">\n<p id=\"fs-id1167836571082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48581809c50851cfc75dd99e4b146387_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#43;&#51;&#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>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836375598\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829692279\">\n<div data-type=\"problem\" id=\"fs-id1167829692281\">\n<p id=\"fs-id1167829692283\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45defcceeb7d5def977aabd606e855fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#98;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"97\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832999458\">\n<p id=\"fs-id1167832999460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb3da2b9cb967512b5c4e0b3ad0435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836325973\" class=\"howto\">\n<div data-type=\"title\">Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using trial and error.<\/div>\n<ol id=\"fs-id1167829740737\" type=\"1\" class=\"stepwise\">\n<li>Write the trinomial in descending order of degrees as needed.<\/li>\n<li>Factor any GCF.<\/li>\n<li>Find all the factor pairs of the first term.<\/li>\n<li>Find all the factor pairs of the third term.<\/li>\n<li>Test all the possible combinations of the factors until the correct product is found.<\/li>\n<li>Check by multiplying.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836538544\">Remember, when the middle term is negative and the last term is positive, the signs in the binomials must both be negative.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829905361\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829905363\">\n<div data-type=\"problem\" id=\"fs-id1167829905366\">\n<p id=\"fs-id1167836667013\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91e377837ba84f7457d088e3d664a6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#98;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832945624\">\n<table id=\"fs-id1167832945627\" class=\"unnumbered unstyled\" summary=\"The trinomial 6 b squared minus 13 b plus 5 is already in descending order. Factoring the first term, we get 1b times 6b and 2b times 3b. To find the factors of the last term, consider the signs. Since the last term, 5, is positive its factors must both be positive or both be negative. The coefficient of the middle term is negative, so we use the negative factors minus 1 and minus 5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The trinomial is already in descending order.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829689434\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the factors of the first term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728970\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the factors of the last term. Consider the signs.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since the last term, 5, is positive its factors must both be<\/p>\n<div data-type=\"newline\"><\/div>\n<p>positive or both be negative. The coefficient of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>middle term is negative, so we use the negative factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829746922\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829627623\">Consider all the combinations of factors.<\/p>\n<table id=\"fs-id1167836313301\" class=\"unnumbered\" summary=\"This table shows the possible factors and corresponding products of 6 b squared minus 13 b plus 5. Factors: open parentheses b minus 1 close parentheses open parentheses 6b minus 5 close parentheses; product: 6 b squared minus 11 b plus 5. Factors: open parentheses b minus 5 close parentheses open parentheses 6b minus 1 close parentheses; product: 6 b squared minus 31 b plus 5. Factors: open parentheses 2b minus 1 close parentheses open parentheses 3b minus 5 close parentheses; product: 6 b squared minus 13b plus 5. This is the original trinomial. Factors: open parentheses 2b minus 5 close parentheses open parentheses 3b minus 1 close parentheses; product: 6 b squared minus 17b plus 5.\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"2\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95ea8added9c21b552fafffc3ef6c9ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#98;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Possible factors<\/th>\n<th data-valign=\"top\" data-align=\"left\">Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa917843f8b227b49d9c2db7ef6fd749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35e2a9617ccefaf820930e5c66eef071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#98;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9144e0b20a551797a84068a9a06ee15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b9e7ec20ae13b2723546d0b2dc560b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#49;&#98;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91d17639408abc0562d9360dadee23d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f152d2f02980f99842c93e0629c0d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#98;&#43;&#123;&#53;&#125;&#94;&#123;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"108\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21b42249df5352e56378141c6fb9166b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cece205ff6f42f5f0758b0229040b5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#55;&#98;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836503135\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74513acbc23405748a61fc31e41f9f72_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;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#111;&#114;&#114;&#101;&#99;&#116;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#111;&#115;&#101;&#32;&#119;&#104;&#111;&#115;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#114;&#105;&#103;&#105;&#110;&#97;&#108;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#98;&#45;&#51;&#98;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#98;&#43;&#53;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"169\" width=\"589\" style=\"vertical-align: -79px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833021860\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833021864\">\n<div data-type=\"problem\" id=\"fs-id1167829599560\">\n<p id=\"fs-id1167829599562\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1eb4e740b043ece65f4b55b63feb2fea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836612477\">\n<p id=\"fs-id1167836612479\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59bee36c1fdea60d3fd6c8838f987c1b_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;&#52;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836579458\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829894398\">\n<div data-type=\"problem\" id=\"fs-id1167829894400\">\n<p id=\"fs-id1167829872167\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04d7ec788896afce80556d7be387cb72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#55;&#121;&#43;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836608328\">\n<p id=\"fs-id1167836608330\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd117887ca53592f2b6a4a1645cd01de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836533023\">When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836287935\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836287937\">\n<div data-type=\"problem\" id=\"fs-id1167836287939\">\n<p>Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-886f1777228058e3e4b09e1a9878d5dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#55;&#120;&#121;&#43;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825949241\">\n<table class=\"unnumbered unstyled\" summary=\"The trinomial 18 x squared minus 37xy plus 15y squared is already in descending order. Factoring the first term, we get 1x times 18x, 2x times 9x and 3x times 6x.To find the factors of the last term, consider the signs. Since 15 is positive and the coefficient of the middle term is negative, we use the negative factors minus 1, minus 5 and minus 5, minus 1.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The trinomial is already in descending order.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836800465\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the factors of the first term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833197132\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the factors of the last term. Consider the signs.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since 15 is positive and the coefficient of the middle<\/p>\n<div data-type=\"newline\"><\/div>\n<p>term is negative, we use the negative factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836554867\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_009c_img_Errata.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829789946\">Consider all the combinations of factors.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836521754\" data-alt=\"This table shows the possible factors and corresponding products of the trinomial 18 x squared minus 37xy plus 15 y squared. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: open parentheses x minus 1y close parentheses open parentheses 18x minus 15y close parentheses, highlighted. Factor, open parentheses x minus 15y close parentheses open parentheses 18x minus 1y close parentheses; product: 18 x squared minus 271xy plus 15 y squared. Factor open parentheses x minus 3y close parentheses open parentheses 18x minus 5 y close parentheses; product: 18 x squared minus 59xy plus 15 y squared. Factor: open parentheses x minus 5y close parentheses open parentheses 18x minus 3y close parentheses highlighted. Factor: open parentheses 2x minus 1y close parentheses open parentheses 9x minus 15y close parentheses highlighted. Factor: open parentheses 2x minus 15y close parentheses open parentheses 9x minus 1y close parentheses; product 18 x squared minus 137 xy plus 15y squared. Factor: open parentheses 2x minus 3y close parentheses open parentheses 9x minus 5y close parentheses; product: 18 x squared minus 37xy plus 15 y squared, which is the original trinomial. Factor: open parentheses 2x minus 57 close parentheses open parentheses 9x minus 3y close parentheses highlighted. Factor: open parentheses 3x minus 1y close parentheses open parentheses 6x minus 15y close parentheses highlighted. Factor: open parentheses 3x minus 15y close parentheses highlighted open parentheses 6x minus 1y close parentheses. Factor: open parentheses 3x minus 3y close parentheses highlighted open parentheses 6x minus 5y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table shows the possible factors and corresponding products of the trinomial 18 x squared minus 37xy plus 15 y squared. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: open parentheses x minus 1y close parentheses open parentheses 18x minus 15y close parentheses, highlighted. Factor, open parentheses x minus 15y close parentheses open parentheses 18x minus 1y close parentheses; product: 18 x squared minus 271xy plus 15 y squared. Factor open parentheses x minus 3y close parentheses open parentheses 18x minus 5 y close parentheses; product: 18 x squared minus 59xy plus 15 y squared. Factor: open parentheses x minus 5y close parentheses open parentheses 18x minus 3y close parentheses highlighted. Factor: open parentheses 2x minus 1y close parentheses open parentheses 9x minus 15y close parentheses highlighted. Factor: open parentheses 2x minus 15y close parentheses open parentheses 9x minus 1y close parentheses; product 18 x squared minus 137 xy plus 15y squared. Factor: open parentheses 2x minus 3y close parentheses open parentheses 9x minus 5y close parentheses; product: 18 x squared minus 37xy plus 15 y squared, which is the original trinomial. Factor: open parentheses 2x minus 57 close parentheses open parentheses 9x minus 3y close parentheses highlighted. Factor: open parentheses 3x minus 1y close parentheses open parentheses 6x minus 15y close parentheses highlighted. Factor: open parentheses 3x minus 15y close parentheses highlighted open parentheses 6x minus 1y close parentheses. Factor: open parentheses 3x minus 3y close parentheses highlighted open parentheses 6x minus 5y.\" \/><\/span><\/p>\n<p id=\"fs-id1167829627829\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d6280a22728ed98720d771f4b5c0846_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;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#111;&#114;&#114;&#101;&#99;&#116;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#111;&#115;&#101;&#32;&#119;&#104;&#111;&#115;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#111;&#114;&#105;&#103;&#105;&#110;&#97;&#108;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#121;&#45;&#50;&#55;&#120;&#121;&#43;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#55;&#120;&#121;&#43;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"171\" width=\"630\" style=\"vertical-align: -81px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829747027\">\n<div data-type=\"problem\" id=\"fs-id1167836512666\">\n<p>Factor completely using trial and error <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2f33999a069ef19cd25ef2e09bd02ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#121;&#45;&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833086736\">\n<p id=\"fs-id1167836551348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-171931faccc01d4c98d57ec25bad28e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824755152\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833379398\">\n<div data-type=\"problem\" id=\"fs-id1167833379400\">\n<p id=\"fs-id1167833379402\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca6b473a49e529fcfcd14d9dcda1eb05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#51;&#120;&#121;&#45;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829790284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e61d81ebc7286c9eac31e23fff02b567_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#120;&#45;&#50;&#49;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167824765027\">Don\u2019t forget to look for a GCF first and remember if the leading coefficient is negative, so is the GCF.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836614266\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836614268\">\n<div data-type=\"problem\" id=\"fs-id1167833339780\">\n<p id=\"fs-id1167833339782\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa76b00fede05959b082f17a4c287076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"164\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825660234\">\n<table id=\"fs-id1167829621071\" class=\"unnumbered unstyled\" summary=\"The trinomial is minus 10 y to the power 4 minus 55 y cubed minus 60 y squared. Factoring the GCF, we get minus 5 y squared open parentheses 2 y squared plus 11y plus 12 close parentheses. The factors of the first term of the trinomial in the parentheses are y and 2y. The factor pairs of the last term are 1 and 12, 2 and 6, 3 and 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830123177\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Notice the greatest common factor, so factor it first.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025652\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833019180\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836534006\">Consider all the combinations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836534010\" data-alt=\"This table shows the possible factors and product of the trinomial 2 y squared plus 11y plus 12. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: y plus 1, 2y plus 12 highlighted. Factor: y plus 12, 2y plus 1; product: 2 y squared plus 25y plus 12. Factor: y plus 2, 2y plus 6 highlighted. Factor: y plus 6, 2y plus 2 highlighted. Factor: y plus 3, 2y plus 4 highlighted. Factor: y plus 4, 2y plus 3; product: 2 y squared plus 11y plus 12. This is the original trinomial.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table shows the possible factors and product of the trinomial 2 y squared plus 11y plus 12. In some pairs of factors, when one factor contains two terms with a common factor, that factor is highlighted. In such cases, product is not an option because if trinomial has no common factors, then neither factor can contain a common factor. Factor: y plus 1, 2y plus 12 highlighted. Factor: y plus 12, 2y plus 1; product: 2 y squared plus 25y plus 12. Factor: y plus 2, 2y plus 6 highlighted. Factor: y plus 6, 2y plus 2 highlighted. Factor: y plus 3, 2y plus 4 highlighted. Factor: y plus 4, 2y plus 3; product: 2 y squared plus 11y plus 12. This is the original trinomial.\" \/><\/span><\/p>\n<p id=\"fs-id1167824720957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ee18593f16ecbb30e8707df4c145936_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;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#111;&#114;&#114;&#101;&#99;&#116;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#111;&#115;&#101;&#32;&#119;&#104;&#111;&#115;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#114;&#105;&#103;&#105;&#110;&#97;&#108;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#32;&#82;&#101;&#109;&#101;&#109;&#98;&#101;&#114;&#32;&#116;&#111;&#32;&#105;&#110;&#99;&#108;&#117;&#100;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#102;&#97;&#99;&#116;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#43;&#51;&#121;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"215\" width=\"363\" style=\"vertical-align: -103px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836550866\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836550870\">\n<div data-type=\"problem\" id=\"fs-id1167833061507\">\n<p id=\"fs-id1167833061509\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-220587fd9e653a390ba696e074870fe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#48;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836391301\">\n<p id=\"fs-id1167836391303\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0847b7d9a9970f4aff5c5a7aa57a5aab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#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>\n<div data-type=\"note\" id=\"fs-id1167829830348\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829830352\">\n<div data-type=\"problem\" id=\"fs-id1167836557643\">\n<p id=\"fs-id1167836557645\">Factor completely using trial and error: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59ddadfc923d065b44a2f4b990844d29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#54;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#50;&#48;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#54;&#113;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833369061\">\n<p id=\"fs-id1167833369063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afae38aa98d032d24f91a0498b054f63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#113;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#113;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833135490\">\n<h3 data-type=\"title\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u201cac\u201d Method<\/h3>\n<p id=\"fs-id1167836487108\">Another way to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> is the \u201cac\u201d method. (The \u201cac\u201d method is sometimes called the grouping method.) The \u201cac\u201d method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!<\/p>\n<div data-type=\"example\" id=\"fs-id1167829812485\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor Trinomials using the \u201cac\u201d Method<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829812487\">\n<div data-type=\"problem\" id=\"fs-id1167829812489\">\n<p id=\"fs-id1167833018423\">Factor using the <em data-effect=\"italics\">\u2018ac\u2019<\/em> method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7a00d3fc5e41f4cbb0e9986860537d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836553592\"><span data-type=\"media\" id=\"fs-id1167836553594\" data-alt=\"Step 1 is to factor the GCF. There is none in 6 x squared plus 7x plus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to factor the GCF. There is none in 6 x squared plus 7x plus 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167824763512\" data-alt=\"Step 2 is to find the product of a and c. The product of 6 and 2 is 12.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the product of a and c. The product of 6 and 2 is 12.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167824732580\" data-alt=\"Step 3 is to find 2 numbers m and n such that mn is ac and m plus n is b. So we need to numbers that multiply to 12 and add to 7. Both factors must be positive. 3 times 4 is 12 and 3 plus 4 is 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find 2 numbers m and n such that mn is ac and m plus n is b. So we need to numbers that multiply to 12 and add to 7. Both factors must be positive. 3 times 4 is 12 and 3 plus 4 is 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833022512\" data-alt=\"Step 4 is to split the middle term using m and n. So we rewrite 7 x as 3x plus 4x. It would give the same result if we used 4x plus 3x. Rewriting, we get 6 x squared plus 3x plus 4x plus 2. Notice that this is the same as the original polynomial. We just split the middle term to get a more useful form\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to split the middle term using m and n. So we rewrite 7 x as 3x plus 4x. It would give the same result if we used 4x plus 3x. Rewriting, we get 6 x squared plus 3x plus 4x plus 2. Notice that this is the same as the original polynomial. We just split the middle term to get a more useful form\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829713653\" data-alt=\"Step 5 is to factor by grouping. So, we get, 3x open parentheses 2x plus 1 close parentheses plus 2 open parentheses 2x plus 1 close parentheses. This is equal to 2x plus 1, 3x plus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to factor by grouping. So, we get, 3x open parentheses 2x plus 1 close parentheses plus 2 open parentheses 2x plus 1 close parentheses. This is equal to 2x plus 1, 3x plus 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836392737\" data-alt=\"Step 6 is to check by multiplying the factors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_011f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check by multiplying the factors.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833050735\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833050739\">\n<div data-type=\"problem\" id=\"fs-id1167833050741\">\n<p id=\"fs-id1167833018138\">Factor using the \u2018ac\u2019 method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afcd3ed97adf79b0715ad436a402b85f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836548815\">\n<p id=\"fs-id1167836548817\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b02e438b42c86860a6cabce38146cd25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#49;&#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>\n<div data-type=\"note\" id=\"fs-id1167836387643\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836387646\">\n<div data-type=\"problem\" id=\"fs-id1167836387648\">\n<p id=\"fs-id1167836728730\">Factor using the \u2018ac\u2019 method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e03e343b72f524b3bf60bad826bb4a86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836576087\">\n<p id=\"fs-id1167836576089\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b9e229016a63a2f98d72cfd55e8e79d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836629022\">The \u201cac\u201d method is summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833082444\" class=\"howto\">\n<div data-type=\"title\">Factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u201cac\u201d method.<\/div>\n<ol id=\"fs-id1167824735184\" type=\"1\" class=\"stepwise\">\n<li>Factor any GCF.<\/li>\n<li>Find the product <em data-effect=\"italics\">ac<\/em>.<\/li>\n<li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that:\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-fb5a53899394494c51b3c3d99b8b5555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#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;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#109;&middot;&#110;&#61;&#97;&middot;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#116;&#111;&#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;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#109;&#43;&#110;&#61;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"287\" style=\"vertical-align: -24px;\" \/><\/li>\n<li>Split the middle term using <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-073fe987c134a5cb3fc153eea173e386_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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#120;&#43;&#110;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/li>\n<li>Factor by grouping.<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836614469\">Don\u2019t forget to look for a common factor!<\/p>\n<div data-type=\"example\" id=\"fs-id1167832926004\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832926007\">\n<div data-type=\"problem\" id=\"fs-id1167832926009\">\n<p id=\"fs-id1167829810980\">Factor using the <em data-effect=\"italics\">\u2018ac\u2019<\/em> method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15100a34d5be608a1c0a46f3a2051cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#53;&#121;&#43;&#55;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829749936\">\n<table id=\"fs-id1167833257770\" class=\"unnumbered unstyled\" summary=\"The GCF in 10 y squared minus 55y plus 70 is 5. Factoring this, we get 5 open parentheses 2 y squared minus 11 y plus 14. The trinomial inside the parentheses has a leading coefficient that is not 1. So we find the product ac, which is 28. Now we find two numbers that multiply to ac and add to b. Minus 4 times minus 7 is 28 and minus 4 minus 7 is minus 11. Splitting the middle term of the trinomial, we get, 5 open parentheses 2 y squared minus 7y minus 4 y plus 14 close parentheses. We factor by grouping to get 5 open parentheses y minus 2 close parentheses open parentheses 2y minus 7 close parentheses. Now we check by multiplying all three factors to get the original polynomial.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is there a greatest common factor?<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Yes. The GCF is 5.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586669\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor it.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833057789\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The trinomial inside the parentheses has a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>leading coefficient that is not 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829861944\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the product <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2064bb97f14cb3682bab0335f710f1c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-509279a1db901479253a437b01dbb4b3_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;&#51;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#99;&#61;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find two numbers that multiply to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fe68776986c55a6eacdb12c5a99552a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a66c36c14c449f024267bc0964cc673_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;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">and add to <em data-effect=\"italics\">b<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ef884485d48df3275e165ff354d43de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Split the middle term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477566\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/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_06_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial by grouping.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693285\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836747903\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_012j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check by multiplying all three factors.<\/p>\n<div data-type=\"newline\"><\/div>\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-cb6337a05b5f56574cef2731435a5149_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#121;&#45;&#52;&#121;&#43;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#121;&#43;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#53;&#121;&#43;&#55;&#48;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"171\" style=\"vertical-align: -37px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836356159\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836356163\">\n<div data-type=\"problem\" id=\"fs-id1167836356165\">\n<p id=\"fs-id1167829720098\">Factor using the \u2018ac\u2019 method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e2151306cb3d5da02bb99c57fcec21b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&#120;&#43;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"128\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833048819\">\n<p id=\"fs-id1167833048821\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aeb07fcef2c1f39f5453711b50d96282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#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;&#45;&#49;&#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>\n<div data-type=\"note\" id=\"fs-id1167836539503\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836539508\">\n<div data-type=\"problem\" id=\"fs-id1167836539510\">\n<p id=\"fs-id1167829621078\">Factor using the \u2018ac\u2019 method: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a48f4c966e6b4052b2908183d4e6b67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#57;&#119;&#43;&#49;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833274653\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a54dae888426f01f9d7b983fd30716c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#119;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#119;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836440705\">\n<h3 data-type=\"title\">Factor Using Substitution<\/h3>\n<p id=\"fs-id1167836440710\">Sometimes a trinomial does not appear to be in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> form. However, we can often make a thoughtful substitution that will allow us to make it fit the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> form. This is called <span data-type=\"term\" class=\"no-emphasis\">factoring by substitution<\/span>. It is standard to use <em data-effect=\"italics\">u<\/em> for the substitution.<\/p>\n<p id=\"fs-id1167833051242\">In the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d344f942e04348123541737872f536d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"100\" style=\"vertical-align: -4px;\" \/> the middle term has a variable, <em data-effect=\"italics\">x<\/em>, and its square, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f82905c002b530c14921e8d459fe64b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> is the variable part of the first term. Look for this relationship as you try to find a substitution.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829908071\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829908073\">\n<div data-type=\"problem\" id=\"fs-id1167836743430\">\n<p id=\"fs-id1167836743432\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53347368d8bebc0acef6b5ef0e4b0b9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836649904\">\n<p id=\"fs-id1167836649906\">The variable part of the middle term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> and its square, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c634c0ca31404b640ada24b1d6ac36e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> is the variable part of the first term. (We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-349242239d764a85e3839450b59b7e7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"85\" style=\"vertical-align: -7px;\" \/> If we let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d128926844ed50bdbab9542af5b5cd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -4px;\" \/> we can put our trinomial in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> form we need to factor it.<\/p>\n<table id=\"fs-id1167836615497\" class=\"unnumbered unstyled\" summary=\"In the polynomial x to the power 4 minus 4 x squared minus 5, substitute x squared with u. We get the trinomial u squared minus 4u minus 5. We factor this to get u plus 1, u minus 5. Replacing u with x squared, we get x squared plus 1, x squared minus 5. We check by multiplying the factors to get the original polynomial.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833271796\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832971395\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506ad41ae004b491ff353f08fd6eed0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833053520\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833024758\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecbdeaa97c968725a27882437601678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756401\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\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-0a4142315f19e8c6d3d41350ced5bf57_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"136\" style=\"vertical-align: -23px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824735528\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824735532\">\n<div data-type=\"problem\" id=\"fs-id1167824735534\">\n<p id=\"fs-id1167832971372\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6902eb5b6ab6f5d7a5bb12b2fe5b749e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#104;&#125;&#94;&#123;&#52;&#125;&#43;&#52;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829789387\">\n<p id=\"fs-id1167829789389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08d8a7eccb5b83c272f0a198ccce7267_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#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>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829597653\">\n<div data-type=\"problem\" id=\"fs-id1167829597655\">\n<p id=\"fs-id1167829741295\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af902c9066e39f3d60ab4ab4de067ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836508040\">\n<p id=\"fs-id1167836508042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7ea1352bea880d455293d0708d64264_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;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#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>\n<\/div>\n<p id=\"fs-id1171791549871\">Sometimes the expression to be substituted is not a monomial.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836320647\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836320649\">\n<div data-type=\"problem\" id=\"fs-id1167836320651\">\n<p id=\"fs-id1167836320653\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b8645061e4b7c57dcc4dbacf3dbf35a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836319286\">\n<p id=\"fs-id1167836319288\">The binomial in the middle term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d857c9fe62db023c7fd751b0bf09300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -4px;\" \/> is squared in the first term. If we let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad9c7f486afd000c57fe590e0b1820c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\" \/> and substitute, our trinomial will be in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> form.<\/p>\n<table id=\"fs-id1167836398883\" class=\"unnumbered unstyled\" summary=\"The polynomial is open parentheses x minus 2 close parentheses squared plus 7 open parentheses x minus 2 close parentheses plus 12. Substituting x minus 2 with u, we get u squared plus 7u plus 12. We factor this to get u plus 3, u plus 4. Replacing u with x minus 2, we get open parentheses x minus 2 plus 3 close parentheses open parentheses x minus 2 plus 4 close parentheses. Simplifying, we get x plus 1, x plus 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748361\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691720\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad9c7f486afd000c57fe590e0b1820c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829579793\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836699186\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfdc25fb64ffc9ad0325b336b33b7ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829712623\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify inside the parentheses.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408611\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833054731\">This could also be factored by first multiplying out the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f391bb87b8937e0d7c7ea83c311b8495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#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;\" \/> and the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45bc494d3e14c4ca1a6675a97f3b5720_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> and then combining like terms and then factoring. Most students prefer the substitution method.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829849379\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829589752\">\n<div data-type=\"problem\" id=\"fs-id1167829589754\">\n<p id=\"fs-id1167829589756\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8337ab1020e67dc2ae0e797582fe75fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826025462\">\n<p id=\"fs-id1167826025464\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7186b37c61b54a8ae2f877862254c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#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>\n<div data-type=\"note\" id=\"fs-id1167829614310\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829614315\">\n<div data-type=\"problem\" id=\"fs-id1167829979385\">\n<p id=\"fs-id1167829979387\">Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e29d4e524672e1d7409abbfe0746748d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836386289\">\n<p id=\"fs-id1167836440365\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0e0e4396db4db070534e9addb0e4c01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#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=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832977093\" class=\"media-2\">\n<p id=\"fs-id1167832977097\">Access this online resource for additional instruction and practice with factoring.<\/p>\n<ul id=\"fs-id1167832977100\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37ACmethod\">Factor a trinomial using the AC method<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167824737671\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836545407\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/strong>\n<ol id=\"fs-id1167836521132\" type=\"1\" class=\"stepwise\">\n<li>Write the factors as two binomials with first terms <em data-effect=\"italics\">x<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ae3800cab695eda122b1e0b585266f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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=\"41\" width=\"91\" style=\"vertical-align: -15px;\" \/><\/li>\n<li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that\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-cd95839cf3336d94c86d09f4a22f86a0_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;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#99;&#44;&#109;&middot;&#110;&#61;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#100;&#100;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#98;&#44;&#109;&#43;&#110;&#61;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"214\" style=\"vertical-align: -15px;\" \/><\/li>\n<li>Use <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> as the last terms of the factors. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-873e3ab318ba825e28f98fb27fb426ea_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;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Strategy for Factoring Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/>:<\/strong> When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.\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-d881a69ae528ef374b903e21dd189710_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#101;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#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;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#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;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#44;&#110;&#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;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;&#44;&#110;&#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;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#101;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#118;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#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=\"261\" width=\"494\" style=\"vertical-align: -124px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Notice that, in the case when <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> have opposite signs, the sign of the one with the larger absolute value matches the sign of <em data-effect=\"italics\">b<\/em>.<\/li>\n<li><strong data-effect=\"bold\">How to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using trial and error.<\/strong>\n<ol id=\"fs-id1167833086362\" type=\"1\" class=\"stepwise\">\n<li>Write the trinomial in descending order of degrees as needed.<\/li>\n<li>Factor any GCF.<\/li>\n<li>Find all the factor pairs of the first term.<\/li>\n<li>Find all the factor pairs of the third term.<\/li>\n<li>Test all the possible combinations of the factors until the correct product is found.<\/li>\n<li>Check by multiplying.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to factor trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u201cac\u201d method.<\/strong>\n<ol id=\"fs-id1167829696444\" type=\"1\" class=\"stepwise\">\n<li>Factor any GCF.<\/li>\n<li>Find the product <em data-effect=\"italics\">ac<\/em>.<\/li>\n<li>Find two numbers <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em> that:\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-bfdcf03dbdd497f0d587636332c5fcd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#111;&#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;&#97;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#109;&middot;&#110;&#61;&#97;&middot;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#116;&#111;&#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;&#98;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#109;&#43;&#110;&#61;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"291\" style=\"vertical-align: -24px;\" \/><\/li>\n<li>Split the middle term using <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7acf9b39ba0330c71a387527f88038b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#120;&#43;&#110;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/><\/li>\n<li>Factor by grouping.<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836487152\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829785803\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167829579829\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/strong><\/p>\n<p id=\"fs-id1167836409781\">In the following exercises, factor each trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836504047\">\n<div data-type=\"problem\" id=\"fs-id1167836504049\">\n<p id=\"fs-id1167836504051\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbd6b7ba78fd9952a3f4867444b46cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#112;&#43;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833356059\">\n<p id=\"fs-id1167836625804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb68d89e2f4eaf5380d8d2c4ecf90299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#54;&#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-id1167829931154\">\n<div data-type=\"problem\" id=\"fs-id1167829931156\">\n<p id=\"fs-id1167829931158\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27740b5b1c8a9b763dabe52cbde71255_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#50;&#49;\" 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-id1167836712400\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836409429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e85c231f51abca640162a4b38e2d619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#57;&#110;&#43;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"108\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833380083\">\n<p id=\"fs-id1167833380085\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e24e1e322ddf1d8621b24366f603c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624541\">\n<div data-type=\"problem\" id=\"fs-id1167829624543\">\n<p id=\"fs-id1167829624545\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9832a29c700c61ed4bf62e44db5f3d54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#98;&#43;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824754942\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167824735504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e7418992e639667de4b2871446d54c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#97;&#43;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833256132\">\n<p id=\"fs-id1167833256134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f71be17e4892e848d7a4110fad73a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836667054\">\n<div data-type=\"problem\" id=\"fs-id1167836667056\">\n<p id=\"fs-id1167836667058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d13d40ee8356fdb1eb27e01033dcd50f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#49;&#117;&#43;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833082098\">\n<div data-type=\"problem\" id=\"fs-id1167833082100\">\n<p id=\"fs-id1167833082102\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82b40c9869282c51ca0c7530ca8cac95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"97\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719705\">\n<p id=\"fs-id1167829719708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0df50359022c5bb7946f87e19b54eaf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#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-id1167836578775\">\n<div data-type=\"problem\" id=\"fs-id1167836578778\">\n<p id=\"fs-id1167836578780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2821b9c86cebe3ccc7aaba5b3eed1a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#113;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836579148\">\n<div data-type=\"problem\" id=\"fs-id1167833008962\">\n<p id=\"fs-id1167833008964\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5bc5a539355c726d7584b814971cc2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#121;&#43;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829877863\">\n<p id=\"fs-id1167829877865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-534a38338261fc0032d992224f04c575_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;&#121;&#45;&#49;&#53;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167836662669\">\n<div data-type=\"problem\" id=\"fs-id1167836707254\">\n<p id=\"fs-id1167836707256\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29eb5ebb4d2301375321dda70c5568c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#109;&#43;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836600208\">\n<div data-type=\"problem\" id=\"fs-id1167829651488\">\n<p id=\"fs-id1167829651490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5afe4f2772ccf956980f8497093a321a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836485691\">\n<p id=\"fs-id1167836485694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b105ab4fe4a05b269e83cde642371315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#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-id1167836619465\">\n<div data-type=\"problem\" id=\"fs-id1167836409749\">\n<p id=\"fs-id1167836409751\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffaad88ce25a5da4f0fbd7e1f0eece4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#121;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833142068\">\n<div data-type=\"problem\" id=\"fs-id1167836578916\">\n<p id=\"fs-id1167836578918\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed918d07ad84fabededb4374c4f9fbb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#112;&#45;&#54;&#43;&#123;&#112;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832926693\">\n<p id=\"fs-id1167832926695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d90f6cc209a7c285bd7da3f94951f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#54;&#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-id1167836447414\">\n<div data-type=\"problem\" id=\"fs-id1167836546028\">\n<p id=\"fs-id1167836546030\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69657a52a5fad7c6c4fad59eb4961396_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#110;&#45;&#55;&#43;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836597968\">\n<div data-type=\"problem\" id=\"fs-id1167836691283\">\n<p id=\"fs-id1167836691285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85484b1df24265c6f86ae96c0fc6b6bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#54;&#120;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836650093\">\n<p id=\"fs-id1167836650096\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71c546f918f08ea1b2d15bd988882e6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#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-id1167829686747\">\n<div data-type=\"problem\" id=\"fs-id1167836743445\">\n<p id=\"fs-id1167836743448\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8e2be4340c8de3b3e8bc962358287d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836704362\">\n<div data-type=\"problem\" id=\"fs-id1167829833455\">\n<p id=\"fs-id1167829833457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70ccb0dbcdeb4f66d2f65de51b9c000a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#45;&#49;&#49;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836731142\">\n<p id=\"fs-id1167836731144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d7df8c46c7938adc4a70005aaaf0fe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#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-id1167836662752\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829714524\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bacfd79c92ae9937d13748a5a0ebb4d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#45;&#49;&#48;&#120;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836701267\">In the following exercises, factor each trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-601fada7dc0d9f3ff1ea382efa1a3a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#121;&#43;&#99;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836409536\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e9dabd39d1677fbd9bff5b831054f38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#121;&#45;&#56;&#48;&#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 data-type=\"solution\" id=\"fs-id1167836792241\">\n<p id=\"fs-id1167836792243\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d74d49c669a27edfd790deb6cf7437e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824720795\">\n<div data-type=\"problem\" id=\"fs-id1167824720798\">\n<p id=\"fs-id1167824720800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e1624a0a67861a2a7f6b558d059af3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#112;&#113;&#45;&#54;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829791592\">\n<div data-type=\"problem\" id=\"fs-id1167829791595\">\n<p id=\"fs-id1167829791597\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8915e7407dee793ab654af1f6656a4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#109;&#110;&#45;&#54;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"147\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826025369\">\n<p id=\"fs-id1167826025371\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87221a3b707f1fd2f9af4ded4405b4c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#54;&#53;&#110;&#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-id1167830123538\">\n<div data-type=\"problem\" id=\"fs-id1167830123540\">\n<p id=\"fs-id1167830123542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a46345427b417f7b9144f8b3c79f1510_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#112;&#113;&#45;&#51;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833366578\">\n<div data-type=\"problem\" id=\"fs-id1167833366580\">\n<p id=\"fs-id1167833366582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e95c6cf2e6326170ed4d7e80a314247f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#97;&#98;&#45;&#50;&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744468\">\n<p id=\"fs-id1167829744470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15322acc9adc219f4a56d83786ed991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#56;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#51;&#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-id1167833082568\">\n<div data-type=\"problem\" id=\"fs-id1167833082570\">\n<p id=\"fs-id1167833082572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8f815647c4de8458399eb0456a4122e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#114;&#115;&#45;&#50;&#56;&#123;&#115;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829620568\">\n<div data-type=\"problem\" id=\"fs-id1167826170218\">\n<p id=\"fs-id1167826170220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8ad236b7639a8a8587461894cb8d2f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#121;&#45;&#49;&#52;&#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 data-type=\"solution\" id=\"fs-id1167829690069\">\n<p id=\"fs-id1167829690071\">Prime<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829789140\">\n<p id=\"fs-id1167829789142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fce73092ce06edfb304d2ad07e93766b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#117;&#118;&#45;&#50;&#52;&#123;&#118;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"123\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833339029\">\n<div data-type=\"problem\" id=\"fs-id1167833339031\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b0c7adbc105b6cad71e461d82e7b9f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#109;&#110;&#43;&#51;&#48;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836667115\">\n<p id=\"fs-id1167836667117\">Prime<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836738050\">\n<div data-type=\"problem\" id=\"fs-id1167836738052\">\n<p id=\"fs-id1167836738054\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c5d65a937c9bc16b0c09b8d4a97b23e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#99;&#100;&#43;&#49;&#56;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836691590\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> Using Trial and Error<\/strong><\/p>\n<p id=\"fs-id1167824617569\">In the following exercises, factor completely using trial and error.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167824617573\">\n<div data-type=\"problem\" id=\"fs-id1167824617575\">\n<p id=\"fs-id1167824617577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f17027ba6e14e30dfcd28b7dfd0c9a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025378\">\n<p id=\"fs-id1167836705241\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49d779eb84e68155843057f483ba62a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836635464\">\n<div data-type=\"problem\" id=\"fs-id1167836635466\">\n<p id=\"fs-id1167836635468\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba15e5b1b5b6c3e46bdba40876e616a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#45;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833051603\">\n<div data-type=\"problem\" id=\"fs-id1167833051605\">\n<p id=\"fs-id1167833051607\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a9e2f5748ae6b1a36f07559448b5516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#49;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"149\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836790669\">\n<p id=\"fs-id1167836790671\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2386291f89100538b3886674c9ae427_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696647\">\n<div data-type=\"problem\" id=\"fs-id1167836487008\">\n<p id=\"fs-id1167836487011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d3a42b203943adbebffd5aa5d706371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#53;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#52;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824734155\">\n<div data-type=\"problem\" id=\"fs-id1167836513313\">\n<p id=\"fs-id1167836513315\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df23dccc5bda26658bbce0a16d22f2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829905215\">\n<p id=\"fs-id1167829905218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f85fccfd7af52abc30e0d4ecc361ce7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#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=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836621263\">\n<p id=\"fs-id1167836621266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57cb27a47bc1dfa91cbdbc400a52d3d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#43;&#49;&#50;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836515214\">\n<div data-type=\"problem\" id=\"fs-id1167836515216\">\n<p id=\"fs-id1167836515218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33227ba7cf7e139762f69a8c5133b07c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#116;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833328966\">\n<p id=\"fs-id1167833328968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41b1388fdff3638f7ee432c1cb90d042_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#116;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836511666\">\n<div data-type=\"problem\" id=\"fs-id1167836511668\">\n<p id=\"fs-id1167829731913\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fae916de984e9498c07c3de874a05992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#121;&#43;&#49;&#49;\" 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-id1167833114742\">\n<div data-type=\"problem\" id=\"fs-id1167833114744\">\n<p id=\"fs-id1167833114746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eef182114928b35beb7d9fa6cba344ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"115\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734715\">\n<p id=\"fs-id1167824734718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-250c5146101e049dac3233d8e9c406dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#120;&#43;&#49;&#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=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836623750\">\n<div data-type=\"problem\" id=\"fs-id1167836623752\">\n<p id=\"fs-id1167836623754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9837e33e6a8853ea38b2754780f3a82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#48;&#98;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836627631\">\n<div data-type=\"problem\" id=\"fs-id1167836627633\">\n<p id=\"fs-id1167836627635\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da6f8e727f5eb978277d95e75e5eef49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#119;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829749988\">\n<p id=\"fs-id1167829749990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd5ed64763e7e69dce34320e6ab17e3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#119;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#49;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167836635483\">\n<div data-type=\"problem\" id=\"fs-id1167836635485\">\n<p id=\"fs-id1167836635487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dafbb544e010e77d0c37ae98a4b8674_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#55;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836532737\">\n<div data-type=\"problem\" id=\"fs-id1167836532740\">\n<p id=\"fs-id1167836532742\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25b95e43b6869849996e04a2e6b24afc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#113;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836599429\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b22ff8596c6187ed88258353aaeb710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#113;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#50;&#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-id1167833023066\">\n<div data-type=\"problem\" id=\"fs-id1167833023068\">\n<p id=\"fs-id1167833023070\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c1fe0859f95e07d147593f474f0d41a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#51;&#121;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381421\">\n<div data-type=\"problem\" id=\"fs-id1167833381423\">\n<p id=\"fs-id1167833381425\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6fe62b5a4e2bb2b44c5c2ddcb795ede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#112;&#113;&#43;&#49;&#48;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836349251\">\n<p id=\"fs-id1167836349253\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34995200e30792eb45665f2d39de1013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#112;&#45;&#53;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#45;&#50;&#113;&#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-id1167836524333\">\n<div data-type=\"problem\" id=\"fs-id1167829930941\">\n<p id=\"fs-id1167829930943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33ff0830e9a4e9054a8d1e15c85b2c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#57;&#109;&#110;&#43;&#49;&#48;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"164\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833224665\">\n<div data-type=\"problem\" id=\"fs-id1167833224667\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a031701a657671de61e2eea697e62485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#97;&#98;&#45;&#49;&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836356298\">\n<p id=\"fs-id1167836356300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37c6a7994a5f65b3ade734fee5e45ba7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#45;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#98;&#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>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f93c9fd2ae59dd65ebddd3f35304b721_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#117;&#118;&#45;&#49;&#52;&#123;&#118;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624130\">\n<div data-type=\"problem\" id=\"fs-id1167829624132\">\n<p id=\"fs-id1167829624134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ac4ea7674dd37f69f360e71421d7141_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&#120;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"138\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734903\">\n<p id=\"fs-id1167824734905\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3aff50a1f08ba12cbbda5aa6efbf991d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#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-id1167833412554\">\n<div data-type=\"problem\" id=\"fs-id1167833412556\">\n<p id=\"fs-id1167833412558\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26a517b9f4baae594ead76a247a8a9e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#49;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#51;&#97;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829619774\">\n<div data-type=\"problem\" id=\"fs-id1167829619776\">\n<p id=\"fs-id1167829619779\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7830d0d4b556d34e626d2daf5deab9ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#52;&#48;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#48;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752404\">\n<p id=\"fs-id1167829752406\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b652a25fc9a524efdbcafde81e4f82b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#113;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#113;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#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-id1167829908109\">\n<div data-type=\"problem\" id=\"fs-id1167832940635\">\n<p id=\"fs-id1167832940637\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f60dc8dfcf4a392eda7c1cc21b6b8c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#53;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833036714\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u2018ac\u2019 Method<\/strong><\/p>\n<p id=\"fs-id1167833023204\">In the following exercises, factor using the \u2018ac\u2019 method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829752649\">\n<div data-type=\"problem\" id=\"fs-id1167829752651\">\n<p id=\"fs-id1167829752653\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd676ee55fd7727883f6010628e395e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#110;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"108\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829624525\">\n<p id=\"fs-id1167829624527\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46fa15866f443b2369a660760442104b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719186\">\n<div data-type=\"problem\" id=\"fs-id1167829719188\">\n<p id=\"fs-id1167824733201\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ffae8e9435050922316d218f4eb550d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#119;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829784219\">\n<div data-type=\"problem\" id=\"fs-id1167829784221\">\n<p id=\"fs-id1167829784223\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dec6e8848805b752dcd13bbd402aa07b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#107;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836790645\">\n<p id=\"fs-id1167836790647\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8670324056ee1ea1634d78df1026c007_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#107;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#107;&#45;&#53;&#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-id1167836429494\">\n<div data-type=\"problem\" id=\"fs-id1167836429496\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46e1c755031f14990c2f95bfb439664a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#115;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829789158\">\n<div data-type=\"problem\" id=\"fs-id1167829789160\">\n<p id=\"fs-id1167833239776\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-468e93567126068de619c481f4ed1e92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829614527\">\n<p id=\"fs-id1167829614529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dd0f3960be7fea235c550b22949d666_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167824734630\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d15e4c29ee313e7cca3bc96b9c760f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#112;&#45;&#50;&#50;\" 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-id1167833085475\">\n<div data-type=\"problem\" id=\"fs-id1167833085477\">\n<p id=\"fs-id1167833024802\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fe8f1705df3d52303da0634caafdd13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#55;&#110;&#45;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747883\">\n<p id=\"fs-id1167829747885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8b33e2ad54f86c60e20aa2a26e5c080_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#53;&#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-id1167836398908\">\n<div data-type=\"problem\" id=\"fs-id1167836398910\">\n<p id=\"fs-id1167836398912\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c52e50b18ce9284c37161a1d4a6a2632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#49;&#122;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"121\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836731922\">\n<div data-type=\"problem\" id=\"fs-id1167836731924\">\n<p id=\"fs-id1167836731926\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4edfa5e0a8d7272d50cdcd69e4738a63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#57;&#48;&#121;&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836519248\">\n<p id=\"fs-id1167836333653\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8567e978415cb0c8ff935d3c82953a79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836532781\">\n<div data-type=\"problem\" id=\"fs-id1167836532783\">\n<p id=\"fs-id1167836508660\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5d9dc65824ca6478f50201033dba1da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#54;&#117;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833087032\">\n<div data-type=\"problem\" id=\"fs-id1167833087034\">\n<p id=\"fs-id1167833087036\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5454f9e92512d924e6a5e10e6eb13ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#122;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#48;&#50;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#53;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"148\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836537843\">\n<p id=\"fs-id1167836537845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b76c7792bfd9321361c187dec4ac241_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#122;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#122;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#122;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829908836\">\n<div data-type=\"problem\" id=\"fs-id1167829908838\">\n<p id=\"fs-id1167836697363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7f4641c071727093df7f0f48fb5fe7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#49;&#54;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"153\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829783678\">\n<div data-type=\"problem\" id=\"fs-id1167829783680\">\n<p id=\"fs-id1167829783682\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f25546aef872e204503ba4a6ad37f7dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#48;&#115;&#43;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836791403\">\n<p id=\"fs-id1167836791405\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5522934384e8b7cad05a334a56138ff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#115;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#115;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833023438\">\n<div data-type=\"problem\" id=\"fs-id1167833023440\">\n<p id=\"fs-id1167833023442\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de6637de383cd2481a04e8927944ea4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#48;&#112;&#43;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829712919\">\n<div data-type=\"problem\" id=\"fs-id1167829712921\">\n<p id=\"fs-id1167829712923\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb8f02373a70afc732b141ee826ce8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829791297\">\n<p id=\"fs-id1167836485977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d27e51d5482407bc725e194d54ced57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#45;&#51;&#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=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836585285\">\n<div data-type=\"problem\" id=\"fs-id1167836585287\">\n<p id=\"fs-id1167836585289\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acc8e9c24775daf57a227390b815dbd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#53;&#120;&#45;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832925672\"><strong data-effect=\"bold\">Factor Using Substitution<\/strong><\/p>\n<p id=\"fs-id1167832925678\">In the following exercises, factor using substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836705253\">\n<div data-type=\"problem\" id=\"fs-id1167836705255\">\n<p id=\"fs-id1167836705257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba9dcfeb59946bac858951b2a5616e85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833136734\">\n<p id=\"fs-id1167833136736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d580fc326f5d888a5926d1dfea39bb03_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;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#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-id1167836535867\">\n<div data-type=\"problem\" id=\"fs-id1167836535869\">\n<p id=\"fs-id1167836535872\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a74f26fe94e632cb33c154e57faa47f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"97\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833224803\">\n<div data-type=\"problem\" id=\"fs-id1167833224805\">\n<p id=\"fs-id1167833224807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-726fa6a687b8a4657891b51e1ca552eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"106\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824735216\">\n<p id=\"fs-id1167824735218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df3b75b2773d2d43e5c236ce6a77a84e_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;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#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-id1167829894287\">\n<div data-type=\"problem\" id=\"fs-id1167829894289\">\n<p id=\"fs-id1167829894292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8471ef59e0eff9c69673dee06fc317c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"115\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833050525\">\n<div data-type=\"problem\" id=\"fs-id1167833050527\">\n<p id=\"fs-id1167833050529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-717c8844b9e06bfcb6c79740df6727c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833386411\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d7df8c46c7938adc4a70005aaaf0fe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#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-id1167829752883\">\n<div data-type=\"problem\" id=\"fs-id1167829752885\">\n<p id=\"fs-id1167833396915\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7426dec44d9f5e8abb225a13a76f70b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833041786\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836727764\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-934c7f265518428e142de288944ceef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"183\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167826025137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dd40613b22e456bc58bc37b51a140bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832937141\">\n<div data-type=\"problem\" id=\"fs-id1167832937143\">\n<p id=\"fs-id1167832937145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d57f515ea7b8b515e12154da401afcdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"205\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833339369\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p id=\"fs-id1167833339375\">In the following exercises, factor each expression using any method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833339378\">\n<div data-type=\"problem\" id=\"fs-id1167833082156\">\n<p id=\"fs-id1167833082158\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cfa768ce54f16a047d30ddf88a707a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#117;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829942496\">\n<p id=\"fs-id1167829942498\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-053e1c26f08866b6e8a1e8e2a9588746_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#54;&#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-id1167836652948\">\n<div data-type=\"problem\" id=\"fs-id1167836652950\">\n<p id=\"fs-id1167836652952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2da61c5acc94736edf320e635200c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#45;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836732422\">\n<div data-type=\"problem\" id=\"fs-id1167836732424\">\n<p id=\"fs-id1167836732426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa873a6882192b27fef3822d23b3130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#114;&#115;&#43;&#54;&#52;&#123;&#115;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833082178\">\n<p id=\"fs-id1167833082180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3fa10e33eb2b2c7a0a45d3fb10e6445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#52;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#49;&#54;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829691171\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829691175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caf61a877008e44a11df159de5dd7371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#57;&#113;&#114;&#45;&#57;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836664187\">\n<div data-type=\"problem\" id=\"fs-id1167836664189\">\n<p id=\"fs-id1167836664191\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039ebde601737780f83fdedcee96579e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#57;&#121;&#43;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833086526\">\n<p id=\"fs-id1167833086528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c5f860b3548a6813399618cbdce4873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832936294\">\n<div data-type=\"problem\" id=\"fs-id1167832936296\">\n<p id=\"fs-id1167832936298\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c84978af756a9fb85f209855fdaedbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#121;&#45;&#50;&#52;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836614302\">\n<div data-type=\"problem\" id=\"fs-id1167829696487\">\n<p id=\"fs-id1167829696489\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8be7ae333f2582ee9fe431070dea9f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#110;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"99\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829851524\">\n<p id=\"fs-id1167829851526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60133057c403fcdce8f3af05c13f4102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#43;&#52;&#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-id1167829619062\">\n<div data-type=\"problem\" id=\"fs-id1167829619064\">\n<p id=\"fs-id1167829619067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14552d2ab03bcee7624de9322f840c66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836363546\">\n<div data-type=\"problem\" id=\"fs-id1167836363548\">\n<p id=\"fs-id1167836363550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cb54002198120c36f8a5d5c7b365706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#57;&#122;&#45;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829924882\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-187b7c9f22d3a5048a215e287c7fc356_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#122;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836662701\">\n<div data-type=\"problem\" id=\"fs-id1167836662703\">\n<p id=\"fs-id1167836662705\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9a4da1ee90eba974493d767a5f94d55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#114;&#43;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836598121\">\n<div data-type=\"problem\" id=\"fs-id1167829713020\">\n<p id=\"fs-id1167829713022\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-838fb498ff183d42e2c836cdf00590e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/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-db6eea9b7a8790bd600d300bc11cecd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#112;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829594556\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829594561\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b163e8752d10371e75323c71b65cef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#49;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833053777\">\n<div data-type=\"problem\" id=\"fs-id1167833053780\">\n<p id=\"fs-id1167833053782\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb60612dfc9e75c8f18d177884d88df0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#114;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836614430\">\n<p id=\"fs-id1167836614432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a9902001ae0a4a21a579343c603393e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836578943\">\n<div data-type=\"problem\" id=\"fs-id1167836578945\">\n<p id=\"fs-id1167836578947\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5239e31cafcf55ac91a32afc1fab6f44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#109;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836625666\">\n<div data-type=\"problem\" id=\"fs-id1167836789914\">\n<p id=\"fs-id1167836789916\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c98c8132476fc00179d1c3dcaa4335a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#110;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"117\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750344\">\n<p id=\"fs-id1167829750346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77742698e5be3708d1926512bf7b903d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#43;&#49;&#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-id1167836409387\">\n<div data-type=\"problem\" id=\"fs-id1167836409389\">\n<p id=\"fs-id1167836409392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2da40da0cfdeb54aec3944bdaca8531_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#97;&#43;&#50;\" 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-id1167833004862\">\n<div data-type=\"problem\" id=\"fs-id1167833004864\">\n<p id=\"fs-id1167833004866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76f16f26f9718f6c1a50b27019f4c412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836574849\">\n<p id=\"fs-id1167836574851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06edc1a7ed69ad4b9475cef274eea1b8_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167832980660\">\n<div data-type=\"problem\" id=\"fs-id1167832980662\">\n<p id=\"fs-id1167832980664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc1b4507ea4cb0b8fccc41c4e8c098ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"97\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829716233\">\n<div data-type=\"problem\" id=\"fs-id1167829716235\">\n<p id=\"fs-id1167829716237\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3abe8def72e1792bee35d284c690462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829686722\">\n<p id=\"fs-id1167829686724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aeb616ce1654468a6ccfedf2c055aa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#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-id1167832936260\">\n<div data-type=\"problem\" id=\"fs-id1167832936262\">\n<p id=\"fs-id1167832936264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-401397a7d84dcd63bc53afbc74a2a6ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"198\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167824763176\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829752897\">\n<div data-type=\"problem\" id=\"fs-id1167829752899\">\n<p id=\"fs-id1167829752901\">Many trinomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/> factor into the product of two binomials <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35a2e7b6e9b9ce600db1f1052e8b0548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/> Explain how you find the values of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829761405\">\n<p id=\"fs-id1167829761407\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836790885\">\n<div data-type=\"problem\" id=\"fs-id1167836790887\">\n<p id=\"fs-id1167836790889\">Tommy factored <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66d931a636e5d477f7869c9407fde46c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a71f5d404cc00c4468d3172e6de9f12_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;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/> Sara factored it as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-324c7f3fee9b1516772af590e8ea902f_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;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/> Ernesto factored it as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fccc2bd0e3f91527c1700a31375d321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/> Who is correct? Explain why the other two are wrong.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836305370\">\n<div data-type=\"problem\" id=\"fs-id1167836305373\">\n<p id=\"fs-id1167836305375\">List, in order, all the steps you take when using the \u201cac\u201d method to factor a trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13fc8234342051cf7b36d52613e9e1f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829785037\">\n<p id=\"fs-id1167836391259\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836391265\">\n<div data-type=\"problem\" id=\"fs-id1167836391267\">\n<p id=\"fs-id1167836391269\">How is the \u201cac\u201d method similar to the \u201cundo FOIL\u201d method? How is it different?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836399159\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836399164\"><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-id1167836399173\" data-alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the \u201cac\u201d method, factor using substitution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the \u201cac\u201d method, factor using substitution.\" \/><\/span><\/p>\n<p id=\"fs-id1167833380224\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3047","chapter","type-chapter","status-publish","hentry"],"part":2962,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3047","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3047\/revisions"}],"predecessor-version":[{"id":15243,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3047\/revisions\/15243"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2962"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3047\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3047"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3047"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3047"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3047"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}