{"id":4414,"date":"2020-04-13T12:49:02","date_gmt":"2020-04-13T12:49:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4414"},"modified":"2021-02-06T00:05:09","modified_gmt":"2021-02-06T00:05:09","slug":"finding-the-greatest-common-factor-of-a-polynomial","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/finding-the-greatest-common-factor-of-a-polynomial\/","title":{"raw":"Finding the Greatest Common Factor of a Polynomial","rendered":"Finding the Greatest Common Factor of a Polynomial"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Factor the greatest common factor from a polynomial<\/li>\r\n<\/ul>\r\n<\/div>\r\nJust like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, [latex]12[\/latex] as [latex]2\\cdot 6\\text{ or }3\\cdot 4[\/latex]) in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 2\\left(x + 7\\right)&amp;\\text{factors}\\hfill \\\\ \\hfill 2\\cdot x + 2\\cdot 7\\hfill \\\\ \\hfill 2x + 14&amp;\\text{product}\\hfill \\end{array}[\/latex]<\/p>\r\nHere, we will start with a product, like [latex]2x+14[\/latex], and end with its factors, [latex]2\\left(x+7\\right)[\/latex]. To do this we apply the Distributive Property \"in reverse\".\r\n<div class=\"textbox shaded\">\r\n<h3>Distributive Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are real numbers, then\r\n\r\n[latex]a\\left(b+c\\right)=ab+ac\\text{ and }ab+ac=a\\left(b+c\\right)[\/latex]\r\n\r\n<\/div>\r\nThe form on the left is used to multiply. The form on the right is used to factor.\r\n\r\nSo how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]2x+14[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469670620\" class=\"unnumbered unstyled\" summary=\"Three columns are shown. The top row of the first column says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1:<\/strong> Find the GCF of all the terms of the polynomial.<\/td>\r\n<td>Find the GCF of [latex]2x[\/latex] and [latex]14[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224621\/CNX_BMath_Figure_10_06_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2:<\/strong> Rewrite each term as a product using the GCF.<\/td>\r\n<td>Rewrite [latex]2x[\/latex] and [latex]14[\/latex] as products of their GCF, [latex]2[\/latex].\r\n\r\n[latex]2x=2\\cdot x[\/latex]\r\n\r\n[latex]14=2\\cdot 7[\/latex]<\/td>\r\n<td>[latex]2x+14[\/latex]\r\n\r\n[latex]\\color{red}{2}\\cdot x+\\color{red}{2}\\cdot7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3:<\/strong> Use the Distributive Property 'in reverse' to factor the expression.<\/td>\r\n<td><\/td>\r\n<td>[latex]2\\left(x+7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4:<\/strong> Check by multiplying the factors.<\/td>\r\n<td><\/td>\r\n<td>Check:\r\n\r\n[latex]2(x+7)[\/latex]\r\n\r\n[latex]2\\cdot{x}+2\\cdot{7}[\/latex]\r\n\r\n[latex]2x+14\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146330[\/ohm_question]\r\n\r\n<\/div>\r\nNotice that in the example, we used the word <em>factor<\/em> as both a noun and a verb:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\text{Noun}\\hfill &amp; &amp; &amp; 7\\text{ is a factor of }14\\hfill \\\\ \\text{Verb}\\hfill &amp; &amp; &amp; \\text{factor }2\\text{ from }2x+14\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Factor the greatest common factor from a polynomial<\/h3>\r\n<ol id=\"eip-id1168469803720\" class=\"stepwise\">\r\n \t<li>Find the GCF of all the terms of the polynomial.<\/li>\r\n \t<li>Rewrite each term as a product using the GCF.<\/li>\r\n \t<li>Use the Distributive Property \u2018in reverse\u2019 to factor the expression.<\/li>\r\n \t<li>Check by multiplying the factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]3a+3[\/latex]\r\n[reveal-answer q=\"486634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486634\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468725870\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224623\/CNX_BMath_Figure_10_06_029_img-01.png\" alt=\".\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3a+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term as a product using the GCF.<\/td>\r\n<td>[latex]\\color{red}{3}\\cdot a+\\color{red}{3}\\cdot1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Distributive Property 'in reverse' to factor the GCF.<\/td>\r\n<td>[latex]3(a+1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying the factors to get the original polynomial.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3(a+1)[\/latex]\r\n\r\n[latex]3\\cdot{a}+3\\cdot{1}[\/latex]\r\n\r\n[latex]3a+3\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146331[\/ohm_question]\r\n\r\n<\/div>\r\nThe expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]12x - 60[\/latex]\r\n[reveal-answer q=\"176014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176014\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466179717\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224628\/CNX_BMath_Figure_10_06_030_img-01.png\" alt=\".\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]12x-60[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term as a product using the GCF.<\/td>\r\n<td>[latex]\\color{red}{12}\\cdot x-\\color{red}{12}\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the GCF.<\/td>\r\n<td>[latex]12(x-5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying the factors.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]12(x-5)[\/latex]\r\n\r\n[latex]12\\cdot{x}-12\\cdot{5}[\/latex]\r\n\r\n[latex]12x-60\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146332[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples of factoring the GCF from a binomial.\r\n\r\nhttps:\/\/youtu.be\/68M_AJNpAu4\r\n\r\nNow we\u2019ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]3{y}^{2}+6y+9[\/latex]\r\n[reveal-answer q=\"151582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151582\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468743634\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224634\/CNX_BMath_Figure_10_06_031_img-01.png\" alt=\".\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3y^2+6y+9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term as a product using the GCF.<\/td>\r\n<td>[latex]\\color{red}{3}\\cdot{y}^{2}+\\color{red}{3}\\cdot 2y+\\color{red}{3}\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the GCF.<\/td>\r\n<td>[latex]3(y^{2}+2y+3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3(y^{2}+2y+3)[\/latex]\r\n\r\n[latex]3\\cdot{y^2}+3\\cdot{2y}+3\\cdot{3}[\/latex]\r\n\r\n[latex]3y^{2}+6y+9\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146333[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example, we factor a variable from a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]6{x}^{2}+5x[\/latex]\r\n[reveal-answer q=\"694506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"694506\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466314102\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6{x}^{2}+5x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the GCF of [latex]6{x}^{2}[\/latex] and [latex]5x[\/latex] and the math that goes with it.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224639\/CNX_BMath_Figure_10_06_013_img-1.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term as a product.<\/td>\r\n<td>[latex]\\color{red}{x}\\cdot{6x}+\\color{red}{x}\\cdot{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the GCF.<\/td>\r\n<td>[latex]x\\left(6x+5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x\\left(6x+5\\right)[\/latex]\r\n\r\n[latex]x\\cdot 6x+x\\cdot 5[\/latex]\r\n\r\n[latex]6{x}^{2}+5x\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146335[\/ohm_question]\r\n\r\n<\/div>\r\nWhen there are several common factors, as we\u2019ll see in the next two examples, good organization and neat work helps!\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]4{x}^{3}-20{x}^{2}[\/latex]\r\n[reveal-answer q=\"834508\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834508\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467249818\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224641\/CNX_BMath_Figure_10_06_033_img-01.png\" alt=\".\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]4x^3-20x^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term.<\/td>\r\n<td><\/td>\r\n<td>[latex]\\color{red}{4{x}^{2}}\\cdot x - \\color{red}{4{x}^{2}}\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the GCF.<\/td>\r\n<td><\/td>\r\n<td>[latex]4x^2(x-5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.<\/td>\r\n<td>[latex]4x^2(x-5)[\/latex]\r\n\r\n[latex]4x^2\\cdot{x}-4x^2\\cdot{5}[\/latex]\r\n\r\n[latex]4x^3-20x^2\\quad\\checkmark[\/latex]\r\n\r\n&nbsp;<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146337[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]21{y}^{2}+35y[\/latex]\r\n[reveal-answer q=\"771418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"771418\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468229244\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr style=\"height: 115px\">\r\n<td style=\"height: 115px\">Find the GCF of [latex]21{y}^{2}[\/latex] and [latex]35y[\/latex]<\/td>\r\n<td style=\"height: 115px\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224646\/CNX_BMath_Figure_10_06_034_img-01.png\" alt=\".\" \/><\/td>\r\n<td style=\"height: 115px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 23.4619px\">\r\n<td style=\"height: 23.4619px\"><\/td>\r\n<td style=\"height: 23.4619px\"><\/td>\r\n<td style=\"height: 23.4619px\">[latex]21y^2+35y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px\">\r\n<td style=\"height: 23px\">Rewrite each term.<\/td>\r\n<td style=\"height: 23px\"><\/td>\r\n<td style=\"height: 23px\">[latex]\\color{red}{7y}\\cdot 3y + \\color{red}{7y}\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Factor the GCF.<\/td>\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]7y(3y+5)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146338[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]14{x}^{3}+8{x}^{2}-10x[\/latex]\r\n[reveal-answer q=\"421054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421054\"]\r\n\r\nSolution\r\nPreviously, we found the GCF of [latex]14{x}^{3},8{x}^{2},\\text{and}10x[\/latex] to be [latex]2x[\/latex].\r\n<table id=\"eip-id1168468533932\" class=\"unnumbered unstyled\" summary=\"The top line shows 14 x cubed plus 8 x squared minus 10x. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]14{x}^{3}+8{x}^{2}-10x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Rewrite each term using the GCF, 2x.<\/td>\r\n<td style=\"height: 15px\">[latex]\\color{red}{2x}\\cdot 7{x}^{2}+\\color{red}{2x}\\cdot4x-\\color{red}{2x}\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Factor the GCF.<\/td>\r\n<td style=\"height: 15px\">[latex]2x\\left(7{x}^{2}+4x - 5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 82px\">\r\n<td style=\"height: 82px\">[latex]2x(7x^2+4x-5)[\/latex]\r\n\r\n[latex]2x\\cdot{7x^2}+2x\\cdot{4x}-2x\\cdot{5}[\/latex]\r\n\r\n[latex]14x^3+8x^2-10x\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"height: 82px\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146339[\/ohm_question]\r\n\r\n<\/div>\r\nWhen the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]-9y - 27[\/latex]\r\n[reveal-answer q=\"949641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"949641\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469451209\" class=\"unnumbered unstyled\" summary=\"The text says, \">\r\n<tbody>\r\n<tr>\r\n<td>When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of [latex]9y[\/latex] and [latex]27[\/latex] is [latex]9[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224653\/CNX_BMath_Figure_10_06_036_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Since the expression [latex]\u22129y\u221227[\/latex] has a negative leading coefficient, we use [latex]\u22129[\/latex] as the GCF.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9y - 27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite each term using the GCF.<\/td>\r\n<td>[latex]\\color{red}{-9}\\cdot y + \\color{red}{(-9)}\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the GCF.<\/td>\r\n<td>[latex]-9\\left(y+3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.\r\n\r\n[latex]-9(y+3)[\/latex]\r\n\r\n[latex]-9\\cdot{y}+(-9)\\cdot{3}[\/latex]\r\n\r\n[latex]-9y-27\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146340[\/ohm_question]\r\n\r\n<\/div>\r\nPay close attention to the signs of the terms in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFactor: [latex]-4{a}^{2}+16a[\/latex]\r\n[reveal-answer q=\"756063\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"756063\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466034961\" class=\"unnumbered unstyled\" summary=\"The text says, \">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\" colspan=\"2\">The leading coefficient is negative, so the GCF will be negative.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 111.5px\">\r\n<td style=\"height: 111.5px\"><\/td>\r\n<td style=\"height: 111.5px\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224656\/CNX_BMath_Figure_10_06_037_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\" colspan=\"2\">Since the leading coefficient is negative, the GCF is negative, [latex]\u22124a[\/latex].<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]-4{a}^{2}+16a[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Rewrite each term.<\/td>\r\n<td style=\"height: 15px\">[latex]\\color{red}{-4a}\\cdot{a}-\\color{red}{(-4a)}\\cdot{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Factor the GCF.<\/td>\r\n<td style=\"height: 15px\">[latex]-4a\\left(a - 4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Check on your own by multiplying.<\/td>\r\n<td style=\"height: 15px\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question]146341[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Factor the greatest common factor from a polynomial<\/li>\n<\/ul>\n<\/div>\n<p>Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, [latex]12[\/latex] as [latex]2\\cdot 6\\text{ or }3\\cdot 4[\/latex]) in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 2\\left(x + 7\\right)&\\text{factors}\\hfill \\\\ \\hfill 2\\cdot x + 2\\cdot 7\\hfill \\\\ \\hfill 2x + 14&\\text{product}\\hfill \\end{array}[\/latex]<\/p>\n<p>Here, we will start with a product, like [latex]2x+14[\/latex], and end with its factors, [latex]2\\left(x+7\\right)[\/latex]. To do this we apply the Distributive Property &#8220;in reverse&#8221;.<\/p>\n<div class=\"textbox shaded\">\n<h3>Distributive Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p>[latex]a\\left(b+c\\right)=ab+ac\\text{ and }ab+ac=a\\left(b+c\\right)[\/latex]<\/p>\n<\/div>\n<p>The form on the left is used to multiply. The form on the right is used to factor.<\/p>\n<p>So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]2x+14[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469670620\" class=\"unnumbered unstyled\" summary=\"Three columns are shown. The top row of the first column says,\">\n<tbody>\n<tr>\n<td><strong>Step 1:<\/strong> Find the GCF of all the terms of the polynomial.<\/td>\n<td>Find the GCF of [latex]2x[\/latex] and [latex]14[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224621\/CNX_BMath_Figure_10_06_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2:<\/strong> Rewrite each term as a product using the GCF.<\/td>\n<td>Rewrite [latex]2x[\/latex] and [latex]14[\/latex] as products of their GCF, [latex]2[\/latex].<\/p>\n<p>[latex]2x=2\\cdot x[\/latex]<\/p>\n<p>[latex]14=2\\cdot 7[\/latex]<\/td>\n<td>[latex]2x+14[\/latex]<\/p>\n<p>[latex]\\color{red}{2}\\cdot x+\\color{red}{2}\\cdot7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3:<\/strong> Use the Distributive Property &#8216;in reverse&#8217; to factor the expression.<\/td>\n<td><\/td>\n<td>[latex]2\\left(x+7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4:<\/strong> Check by multiplying the factors.<\/td>\n<td><\/td>\n<td>Check:<\/p>\n<p>[latex]2(x+7)[\/latex]<\/p>\n<p>[latex]2\\cdot{x}+2\\cdot{7}[\/latex]<\/p>\n<p>[latex]2x+14\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146330\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146330&theme=oea&iframe_resize_id=ohm146330&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Notice that in the example, we used the word <em>factor<\/em> as both a noun and a verb:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\text{Noun}\\hfill & & & 7\\text{ is a factor of }14\\hfill \\\\ \\text{Verb}\\hfill & & & \\text{factor }2\\text{ from }2x+14\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Factor the greatest common factor from a polynomial<\/h3>\n<ol id=\"eip-id1168469803720\" class=\"stepwise\">\n<li>Find the GCF of all the terms of the polynomial.<\/li>\n<li>Rewrite each term as a product using the GCF.<\/li>\n<li>Use the Distributive Property \u2018in reverse\u2019 to factor the expression.<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]3a+3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q486634\">Show Solution<\/span><\/p>\n<div id=\"q486634\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468725870\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224623\/CNX_BMath_Figure_10_06_029_img-01.png\" alt=\".\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3a+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term as a product using the GCF.<\/td>\n<td>[latex]\\color{red}{3}\\cdot a+\\color{red}{3}\\cdot1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Distributive Property &#8216;in reverse&#8217; to factor the GCF.<\/td>\n<td>[latex]3(a+1)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying the factors to get the original polynomial.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]3(a+1)[\/latex]<\/p>\n<p>[latex]3\\cdot{a}+3\\cdot{1}[\/latex]<\/p>\n<p>[latex]3a+3\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146331\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146331&theme=oea&iframe_resize_id=ohm146331&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]12x - 60[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q176014\">Show Solution<\/span><\/p>\n<div id=\"q176014\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466179717\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224628\/CNX_BMath_Figure_10_06_030_img-01.png\" alt=\".\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]12x-60[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term as a product using the GCF.<\/td>\n<td>[latex]\\color{red}{12}\\cdot x-\\color{red}{12}\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the GCF.<\/td>\n<td>[latex]12(x-5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying the factors.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]12(x-5)[\/latex]<\/p>\n<p>[latex]12\\cdot{x}-12\\cdot{5}[\/latex]<\/p>\n<p>[latex]12x-60\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146332\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146332&theme=oea&iframe_resize_id=ohm146332&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples of factoring the GCF from a binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Factor a Binomial - Greatest Common Factor (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/68M_AJNpAu4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now we\u2019ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]3{y}^{2}+6y+9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q151582\">Show Solution<\/span><\/p>\n<div id=\"q151582\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468743634\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224634\/CNX_BMath_Figure_10_06_031_img-01.png\" alt=\".\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3y^2+6y+9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term as a product using the GCF.<\/td>\n<td>[latex]\\color{red}{3}\\cdot{y}^{2}+\\color{red}{3}\\cdot 2y+\\color{red}{3}\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the GCF.<\/td>\n<td>[latex]3(y^{2}+2y+3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]3(y^{2}+2y+3)[\/latex]<\/p>\n<p>[latex]3\\cdot{y^2}+3\\cdot{2y}+3\\cdot{3}[\/latex]<\/p>\n<p>[latex]3y^{2}+6y+9\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146333\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146333&theme=oea&iframe_resize_id=ohm146333&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example, we factor a variable from a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]6{x}^{2}+5x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q694506\">Show Solution<\/span><\/p>\n<div id=\"q694506\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466314102\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]6{x}^{2}+5x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Find the GCF of [latex]6{x}^{2}[\/latex] and [latex]5x[\/latex] and the math that goes with it.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224639\/CNX_BMath_Figure_10_06_013_img-1.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term as a product.<\/td>\n<td>[latex]\\color{red}{x}\\cdot{6x}+\\color{red}{x}\\cdot{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the GCF.<\/td>\n<td>[latex]x\\left(6x+5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]x\\left(6x+5\\right)[\/latex]<\/p>\n<p>[latex]x\\cdot 6x+x\\cdot 5[\/latex]<\/p>\n<p>[latex]6{x}^{2}+5x\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146335\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146335&theme=oea&iframe_resize_id=ohm146335&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When there are several common factors, as we\u2019ll see in the next two examples, good organization and neat work helps!<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]4{x}^{3}-20{x}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834508\">Show Solution<\/span><\/p>\n<div id=\"q834508\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467249818\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224641\/CNX_BMath_Figure_10_06_033_img-01.png\" alt=\".\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>[latex]4x^3-20x^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term.<\/td>\n<td><\/td>\n<td>[latex]\\color{red}{4{x}^{2}}\\cdot x - \\color{red}{4{x}^{2}}\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the GCF.<\/td>\n<td><\/td>\n<td>[latex]4x^2(x-5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/td>\n<td>[latex]4x^2(x-5)[\/latex]<\/p>\n<p>[latex]4x^2\\cdot{x}-4x^2\\cdot{5}[\/latex]<\/p>\n<p>[latex]4x^3-20x^2\\quad\\checkmark[\/latex]<\/p>\n<p>&nbsp;<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146337\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146337&theme=oea&iframe_resize_id=ohm146337&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]21{y}^{2}+35y[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q771418\">Show Solution<\/span><\/p>\n<div id=\"q771418\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468229244\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr style=\"height: 115px\">\n<td style=\"height: 115px\">Find the GCF of [latex]21{y}^{2}[\/latex] and [latex]35y[\/latex]<\/td>\n<td style=\"height: 115px\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224646\/CNX_BMath_Figure_10_06_034_img-01.png\" alt=\".\" \/><\/td>\n<td style=\"height: 115px\"><\/td>\n<\/tr>\n<tr style=\"height: 23.4619px\">\n<td style=\"height: 23.4619px\"><\/td>\n<td style=\"height: 23.4619px\"><\/td>\n<td style=\"height: 23.4619px\">[latex]21y^2+35y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 23px\">\n<td style=\"height: 23px\">Rewrite each term.<\/td>\n<td style=\"height: 23px\"><\/td>\n<td style=\"height: 23px\">[latex]\\color{red}{7y}\\cdot 3y + \\color{red}{7y}\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Factor the GCF.<\/td>\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]7y(3y+5)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146338\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146338&theme=oea&iframe_resize_id=ohm146338&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]14{x}^{3}+8{x}^{2}-10x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421054\">Show Solution<\/span><\/p>\n<div id=\"q421054\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nPreviously, we found the GCF of [latex]14{x}^{3},8{x}^{2},\\text{and}10x[\/latex] to be [latex]2x[\/latex].<\/p>\n<table id=\"eip-id1168468533932\" class=\"unnumbered unstyled\" summary=\"The top line shows 14 x cubed plus 8 x squared minus 10x. The next line says,\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]14{x}^{3}+8{x}^{2}-10x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Rewrite each term using the GCF, 2x.<\/td>\n<td style=\"height: 15px\">[latex]\\color{red}{2x}\\cdot 7{x}^{2}+\\color{red}{2x}\\cdot4x-\\color{red}{2x}\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Factor the GCF.<\/td>\n<td style=\"height: 15px\">[latex]2x\\left(7{x}^{2}+4x - 5\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 82px\">\n<td style=\"height: 82px\">[latex]2x(7x^2+4x-5)[\/latex]<\/p>\n<p>[latex]2x\\cdot{7x^2}+2x\\cdot{4x}-2x\\cdot{5}[\/latex]<\/p>\n<p>[latex]14x^3+8x^2-10x\\quad\\checkmark[\/latex]<\/td>\n<td style=\"height: 82px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146339\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146339&theme=oea&iframe_resize_id=ohm146339&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]-9y - 27[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q949641\">Show Solution<\/span><\/p>\n<div id=\"q949641\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469451209\" class=\"unnumbered unstyled\" summary=\"The text says,\">\n<tbody>\n<tr>\n<td>When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of [latex]9y[\/latex] and [latex]27[\/latex] is [latex]9[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224653\/CNX_BMath_Figure_10_06_036_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Since the expression [latex]\u22129y\u221227[\/latex] has a negative leading coefficient, we use [latex]\u22129[\/latex] as the GCF.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9y - 27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite each term using the GCF.<\/td>\n<td>[latex]\\color{red}{-9}\\cdot y + \\color{red}{(-9)}\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the GCF.<\/td>\n<td>[latex]-9\\left(y+3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/p>\n<p>[latex]-9(y+3)[\/latex]<\/p>\n<p>[latex]-9\\cdot{y}+(-9)\\cdot{3}[\/latex]<\/p>\n<p>[latex]-9y-27\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146340\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146340&theme=oea&iframe_resize_id=ohm146340&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Pay close attention to the signs of the terms in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Factor: [latex]-4{a}^{2}+16a[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q756063\">Show Solution<\/span><\/p>\n<div id=\"q756063\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466034961\" class=\"unnumbered unstyled\" summary=\"The text says,\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\" colspan=\"2\">The leading coefficient is negative, so the GCF will be negative.<\/td>\n<\/tr>\n<tr style=\"height: 111.5px\">\n<td style=\"height: 111.5px\"><\/td>\n<td style=\"height: 111.5px\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224656\/CNX_BMath_Figure_10_06_037_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\" colspan=\"2\">Since the leading coefficient is negative, the GCF is negative, [latex]\u22124a[\/latex].<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]-4{a}^{2}+16a[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Rewrite each term.<\/td>\n<td style=\"height: 15px\">[latex]\\color{red}{-4a}\\cdot{a}-\\color{red}{(-4a)}\\cdot{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Factor the GCF.<\/td>\n<td style=\"height: 15px\">[latex]-4a\\left(a - 4\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Check on your own by multiplying.<\/td>\n<td style=\"height: 15px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146341\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146341&theme=oea&iframe_resize_id=ohm146341&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4414\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Factor a Binomial - Greatest Common Factor (Basic). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/68M_AJNpAu4\">https:\/\/youtu.be\/68M_AJNpAu4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Factor a Binomial - Greatest Common Factor (Basic)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/68M_AJNpAu4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4414","chapter","type-chapter","status-web-only","hentry"],"part":4412,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4414","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4414\/revisions"}],"predecessor-version":[{"id":4415,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4414\/revisions\/4415"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/4412"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4414\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4414"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4414"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4414"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}