{"id":10892,"date":"2017-06-05T21:40:55","date_gmt":"2017-06-05T21:40:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10892"},"modified":"2024-04-30T23:19:24","modified_gmt":"2024-04-30T23:19:24","slug":"finding-the-greatest-common-factor-of-a-polynomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/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 monomial out of a polynomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Factor a Polynomial<\/span>\r\n\r\n[caption id=\"attachment_4784\" align=\"alignleft\" width=\"300\"]<img class=\"size-medium wp-image-4784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/10184115\/Screen-Shot-2016-06-10-at-11.40.44-AM-300x190.png\" alt=\"Beetles pinned to a surface as a collection with a mini volkswagen beetle car in the mix.\" width=\"300\" height=\"190\" \/> One of these things is not like the others.[\/caption]\r\n\r\nBefore we solve polynomial\u00a0equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nTo help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Polynomial<\/th>\r\n<th>Terms<\/th>\r\n<th>Common Factors<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]6x+9[\/latex]<\/td>\r\n<td>[latex]6x[\/latex] and [latex]9[\/latex]<\/td>\r\n<td>[latex]3[\/latex] is a factor of [latex]6x[\/latex] and \u00a0[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a^{2}\u20132a[\/latex]<\/td>\r\n<td>[latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\r\n<td><i>a<\/i> is a factor of [latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4c^{3}+4c[\/latex]<\/td>\r\n<td>[latex]4c^{3}[\/latex] and [latex]4c[\/latex]<\/td>\r\n<td>[latex]4[\/latex] and <i>c<\/i> are factors of [latex]4c^{3}[\/latex] and \u00a0[latex]4c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember 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\r\nTo factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the <strong>distributive property of multiplication over addition<\/strong> states that a product of a number and a sum is the same as the sum of the products.\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<h4>Distributive Property Forward and Backward<\/h4>\r\nForward: Product of a number and a sum: [latex]a\\left(b+c\\right)=a\\cdot{b}+a\\cdot{c}[\/latex]. You can say that \u201c[latex]a[\/latex] is being distributed over [latex]b+c[\/latex].\u201d\r\n\r\nBackward: Sum of the products: [latex]a\\cdot{b}+a\\cdot{c}=a\\left(b+c\\right)[\/latex]. Here you can say that \u201c<em>a<\/em> is being factored out.\u201d\r\n\r\n<\/div>\r\nWe first learned that we could distribute a factor over a sum or difference, now we are learning that we can \"undo\" the distributive property with factoring.\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 class=\"alignnone\" 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=\"The image shows the prime factorization of 2 x written as the equation 2 x equals 2 times x. Below this equation is another showing the prime factorization of 14 written as the equation 14 equals 2 times 7. The two equations line up vertically at the equal symbol. The 2's in both of the prime factorizations align with each other, and the pair is circled. The x from the prime factorization of 2 x and the 7 from the prime factorization of 14 do not align with any factors from the prime factorizations. A horizontal line is drawn under the prime factorization of 14. Below this line is the equation GCF equals 2.\" width=\"127\" height=\"77\" \/><\/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\nNotice in the next example how, when we factor 3 out of the expression, we are left with a factor of 1.\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 class=\"alignnone\" 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=\"Find the GCF of 3a and 3. Next to this statement is the prime factorization of 3 a written as 3 a equals 3 times a. Below this equation is another for the prime factorization of 3 written as 3 equals 3. The two equations line up vertically at the equal symbol. The 3's from the prime factorizations align vertically, and the pair is circled. The a from the prime factorization of 3 a does not align with any factors from the prime factorization of 3. A horizontal line is drawn under the prime factorization of 3. Below this line is the equation GCF equals 3.\" width=\"514\" height=\"78\" \/><\/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 prime 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 class=\"alignnone\" 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=\"Find the GCF of 12 x and 60. Next to this statement is the prime factorization of 12 x written as 12 x equals 2 times 2 times 3 times x. Below this equation is another for the prime factorization of 60 written as 60 equals 2 times 2 times 3 times 5. The two equations line up vertically at the equal symbol. The 2's and 3's from both of the prime factorizations align vertically, and the vertical pairs are circled. The x from the prime factorization of 12 x and the 5 from the prime factorization of 60 do not align with any other factors from the prime factorizations. A horizontal line is drawn under the prime factorization of 60. Below this line is the equation GCF equals 2 times 2 times 3, which simplifies to GCF equals 12.\" width=\"427\" height=\"101\" \/><\/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 class=\"alignnone\" 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=\"Find the GCF of 3 y squared, 6 y and 9. Next to this statement is the prime factorization of 3 y squared written as 3 y squared equals 3 times y times y. Below this equation is another for the prime factorization of 6 y written as 6 y equals 2 times 3 times y. Below this equation is a third for the prime factorization of 9 written as 9 equals 3 times 3. The three equations line up vertically at the equal symbol. The 2 from the prime factorization of 6 y does not vertically align with any factors from the other prime factorizations. The 3's from the prime factorizations of 3 y squared and 6 y vertically align with each other and the first 3 from the prime factorization of 9, and the trio is circled. The second 3 from the prime factorization of 9 does not vertically align with any factors from the other prime factorizations. The first y in the prime factorization of 3 y squared aligns with the y from the prime factorization of 6 y but there is no factor from the prime factorization of 9 which aligns with these y's. The second y from the prime factorization of 3 y squared does not align with any factors from the other prime factorizations. A horizontal line is drawn under the prime factorization of 9. Below this line is the equation GCF equals 3, which aligns with the trio of threes from above.\" width=\"485\" height=\"111\" \/><\/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 class=\"alignnone\" 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=\"The prime factorization of 6 x squared written as 2 times 3 times x times x. Below this equation is the prime factorization of 5 x written as 5 times x. Neither the 2 nor the 3 from the prime factorization of 6 x squared vertically align with factors from the prime factorization of 5 x. The same can be said for the 5 from the prime factorization of 5 x. The first x from the prime factorization of 6 x squared vertically aligns with the x in the prime factorization of 5 x, and the pair is circled. The second x in the prime factorization of 6 x squared does not align with any factors from the prime factorization of 6 x squared. A horizontal line is drawn under the prime factorization of 5 x. Below this line is the equation GCF equals x.\" width=\"186\" height=\"84\" \/><\/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 class=\"alignnone\" 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=\"Find the GCF of 4 x cubed and 20 x squared. Next to this statement is the prime factorization of 4 x cubed written as 4 x cubed equals 2 times 2 times x times x times x. Below this equation is another for the prime factorization of 20 x squared written as 20 x squared equals 2 times 2 times 5 times x times x. The two equations line up vertically at the equal symbol. Both of the two 2's in the prime factorizations vertically align, and each vertical pair is circled. The 5 in the prime factorization of 20 x sqaured does not align with any factors from the prime factorization of 4 x cubed. The first 2 x's in the prime factorization of 4 x cubed align with the 2 x's in the prime factorization of 20 x squared, and each vertical pair is circled. The third x in the prime factorization of 4 x cubed does not align with any factors from the prime factorization of 20 x squared. A horizontal line is drawn under the prime factorization of 20 x sqaured. Below this line is the equation GCF equals 2 times 2 times x times x. Each term in the GCF equation aligns with the respective circled pair from above. The GCF equation simplifies to GCF equals 4 x squared.\" width=\"505\" height=\"108\" \/><\/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<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]25b^{3}+10b^{2}[\/latex].\r\n\r\n[reveal-answer q=\"716902\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716902\"]Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] to be [latex]5b^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,25b^{3}=5\\cdot5\\cdot{b}\\cdot{b}\\cdot{b}\\\\\\,\\,10b^{2}=5\\cdot2\\cdot{b}\\cdot{b}\\\\\\text{GCF}=5\\cdot{b}\\cdot{b}=5b^{2}\\end{array}[\/latex]<\/p>\r\nRewrite each term with the GCF as one factor.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}25b^{3} = 5b^{2}\\cdot5b\\\\10b^{2}=5b^{2}\\cdot2\\end{array}[\/latex]<\/p>\r\nRewrite the polynomial using the factored terms in place of the original terms.\r\n<p style=\"text-align: center;\">[latex]5b^{2}\\left(5b\\right)+5b^{2}\\left(2\\right)[\/latex]<\/p>\r\nFactor out the [latex]5b^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5b^{2}\\left(5b+2\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe factored form of the polynomial [latex]25b^{3}+10b^{2}[\/latex] is [latex]5b^{2}\\left(5b+2\\right)[\/latex]. You can check this by doing the multiplication. [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].\r\n\r\nNote that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.\r\n\r\nFor example:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}25b^{3}+10b^{2}=5\\left(5b^{3}+2b^{2}\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }5.\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5b^{2}\\left(5b+2\\right) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }b^{2}.\\end{array}[\/latex]<\/p>\r\nNotice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.\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 class=\"alignnone\" 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=\"The prime factorization of 21 y squared is written as 21 y squared equals 3 times 7 times y times y. Below this equation is another for the prime factorization of 35 y written as 35 y equals 5 times 7 times y. The two equations line up vertically at the equal symbol. The 3 from the prime factorization of 21 y squared does not vertically align with any factors from the prime factorization of 35 y. The same can be said for the 5 from the prime factorization of 35 y aligning with no factors from the prime factorization of 21 y squared. The 7's from both prime factorizations align vertically, and the pair is circled. The same can be said for the first y in the prime factorization of 21 y squared and the y in the prime factorization of 35 y. The second y in the prime factorization of 21 y squared does not align with any factors from the prime factorization of 35 y. A horizontal line is drawn under the prime factorization of 35 y. Below this line is the equation GCF equals 7 times y. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 7 y.\" width=\"196\" height=\"113\" \/><\/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 class=\"alignnone\" 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=\"The prime factorization of 9 y written as 9 y equals 3 times 3 times y. Below this equation is another for the prime factorization of 27 written as 27 equals 3 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 3's from both prime factorizations vertically align with each other, and each vertical pair is circled. The third 3 from the prime factorization of 27 does not align with any factors from the prime factorization of 9 y. Similarly, the y from the prime factorization of 9 y does not align with any factors. A horizontal line is drawn under the prime factorization of 27. Below this line is the equation GCF equals 3 times 3. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 9.\" width=\"162\" height=\"104\" \/><\/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 class=\"alignnone\" 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=\"The prime factorization of 4 a squared written as 4 a squared equals 2 times 2 times a times a. Below this equation is another for the prime factorization of 16 a written as 16 as equals 2 times 2 times 2 times 2 times a. The two equations line up vertically at the equal symbol. The first two 2's from both prime factorizations vertically align, and each vertical pair is circled. The remaining 2's in the prime factorization of 16 a do not align vertically with any factors from the prime factorization of 4 a squared. The first a from both prime factorizations align with each other, and the vertical pair is circled. The remaining a from the prime factorization of 4 a squared does not vertically align with any other factors. A horizontal line is drawn below the prime factorization of 16 a. Below this line is the equation GCF equals 2 times 2 times a. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 4 a.\" width=\"218\" height=\"108\" \/><\/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>\r\nThis next example shows factoring a binomial when there are two different variables in the expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]81c^{3}d+45c^{2}d^{2}[\/latex].\r\n\r\n[reveal-answer q=\"809701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"809701\"]Factor [latex]81c^{3}d[\/latex].\r\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot9\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}[\/latex]<\/p>\r\nFactor [latex]45c^{2}d^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}[\/latex]<\/p>\r\nFind the GCF.\r\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}=9c^{2}d[\/latex]<\/p>\r\nRewrite each term as the product of the GCF and the remaining terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,81c^{3}d=9c^{2}d\\left(9c\\right)\\\\45c^{2}d^{2}=9c^{2}d\\left(5d\\right)\\end{array}[\/latex]<\/p>\r\nRewrite the polynomial expression using the factored terms in place of the original terms.\r\n<p style=\"text-align: center;\">[latex]9c^{2}d\\left(9c\\right)+9c^{2}d\\left(5d\\right)[\/latex]<\/p>\r\nFactor out [latex]9c^{2}d[\/latex]<i>.<\/i>\r\n<p style=\"text-align: center;\">[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides two more examples of finding the greatest common factor of a binomial\r\n\r\nhttps:\/\/youtu.be\/25_f_mVab_4\r\n\r\nThis last example shows finding the greatest common factors of trinomials.\r\n\r\nhttps:\/\/youtu.be\/3f1RFTIw2Ng\r\n<h2>Summary<\/h2>\r\nA whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\\cdot{b}+a\\cdot{c}[\/latex] form has been rewritten as [latex]a\\left(b+c\\right)[\/latex], where <i>a<\/i> is the GCF, the polynomial is in factored form.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Factor the greatest common monomial out of a polynomial<\/li>\n<\/ul>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Factor a Polynomial<\/span><\/p>\n<div id=\"attachment_4784\" style=\"width: 310px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4784\" class=\"size-medium wp-image-4784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/10184115\/Screen-Shot-2016-06-10-at-11.40.44-AM-300x190.png\" alt=\"Beetles pinned to a surface as a collection with a mini volkswagen beetle car in the mix.\" width=\"300\" height=\"190\" \/><\/p>\n<p id=\"caption-attachment-4784\" class=\"wp-caption-text\">One of these things is not like the others.<\/p>\n<\/div>\n<p>Before we solve polynomial\u00a0equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>To help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Polynomial<\/th>\n<th>Terms<\/th>\n<th>Common Factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]6x+9[\/latex]<\/td>\n<td>[latex]6x[\/latex] and [latex]9[\/latex]<\/td>\n<td>[latex]3[\/latex] is a factor of [latex]6x[\/latex] and \u00a0[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a^{2}\u20132a[\/latex]<\/td>\n<td>[latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\n<td><i>a<\/i> is a factor of [latex]a^{2}[\/latex] and [latex]\u22122a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4c^{3}+4c[\/latex]<\/td>\n<td>[latex]4c^{3}[\/latex] and [latex]4c[\/latex]<\/td>\n<td>[latex]4[\/latex] and <i>c<\/i> are factors of [latex]4c^{3}[\/latex] and \u00a0[latex]4c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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<p>To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the <strong>distributive property of multiplication over addition<\/strong> states that a product of a number and a sum is the same as the sum of the products.<\/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<h4>Distributive Property Forward and Backward<\/h4>\n<p>Forward: Product of a number and a sum: [latex]a\\left(b+c\\right)=a\\cdot{b}+a\\cdot{c}[\/latex]. You can say that \u201c[latex]a[\/latex] is being distributed over [latex]b+c[\/latex].\u201d<\/p>\n<p>Backward: Sum of the products: [latex]a\\cdot{b}+a\\cdot{c}=a\\left(b+c\\right)[\/latex]. Here you can say that \u201c<em>a<\/em> is being factored out.\u201d<\/p>\n<\/div>\n<p>We first learned that we could distribute a factor over a sum or difference, now we are learning that we can &#8220;undo&#8221; the distributive property with factoring.<\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The image shows the prime factorization of 2 x written as the equation 2 x equals 2 times x. Below this equation is another showing the prime factorization of 14 written as the equation 14 equals 2 times 7. The two equations line up vertically at the equal symbol. The 2's in both of the prime factorizations align with each other, and the pair is circled. The x from the prime factorization of 2 x and the 7 from the prime factorization of 14 do not align with any factors from the prime factorizations. A horizontal line is drawn under the prime factorization of 14. Below this line is the equation GCF equals 2.\" width=\"127\" height=\"77\" \/><\/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>Notice in the next example how, when we factor 3 out of the expression, we are left with a factor of 1.<\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Find the GCF of 3a and 3. Next to this statement is the prime factorization of 3 a written as 3 a equals 3 times a. Below this equation is another for the prime factorization of 3 written as 3 equals 3. The two equations line up vertically at the equal symbol. The 3's from the prime factorizations align vertically, and the pair is circled. The a from the prime factorization of 3 a does not align with any factors from the prime factorization of 3. A horizontal line is drawn under the prime factorization of 3. Below this line is the equation GCF equals 3.\" width=\"514\" height=\"78\" \/><\/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 prime 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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Find the GCF of 12 x and 60. Next to this statement is the prime factorization of 12 x written as 12 x equals 2 times 2 times 3 times x. Below this equation is another for the prime factorization of 60 written as 60 equals 2 times 2 times 3 times 5. The two equations line up vertically at the equal symbol. The 2's and 3's from both of the prime factorizations align vertically, and the vertical pairs are circled. The x from the prime factorization of 12 x and the 5 from the prime factorization of 60 do not align with any other factors from the prime factorizations. A horizontal line is drawn under the prime factorization of 60. Below this line is the equation GCF equals 2 times 2 times 3, which simplifies to GCF equals 12.\" width=\"427\" height=\"101\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Find the GCF of 3 y squared, 6 y and 9. Next to this statement is the prime factorization of 3 y squared written as 3 y squared equals 3 times y times y. Below this equation is another for the prime factorization of 6 y written as 6 y equals 2 times 3 times y. Below this equation is a third for the prime factorization of 9 written as 9 equals 3 times 3. The three equations line up vertically at the equal symbol. The 2 from the prime factorization of 6 y does not vertically align with any factors from the other prime factorizations. The 3's from the prime factorizations of 3 y squared and 6 y vertically align with each other and the first 3 from the prime factorization of 9, and the trio is circled. The second 3 from the prime factorization of 9 does not vertically align with any factors from the other prime factorizations. The first y in the prime factorization of 3 y squared aligns with the y from the prime factorization of 6 y but there is no factor from the prime factorization of 9 which aligns with these y's. The second y from the prime factorization of 3 y squared does not align with any factors from the other prime factorizations. A horizontal line is drawn under the prime factorization of 9. Below this line is the equation GCF equals 3, which aligns with the trio of threes from above.\" width=\"485\" height=\"111\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The prime factorization of 6 x squared written as 2 times 3 times x times x. Below this equation is the prime factorization of 5 x written as 5 times x. Neither the 2 nor the 3 from the prime factorization of 6 x squared vertically align with factors from the prime factorization of 5 x. The same can be said for the 5 from the prime factorization of 5 x. The first x from the prime factorization of 6 x squared vertically aligns with the x in the prime factorization of 5 x, and the pair is circled. The second x in the prime factorization of 6 x squared does not align with any factors from the prime factorization of 6 x squared. A horizontal line is drawn under the prime factorization of 5 x. Below this line is the equation GCF equals x.\" width=\"186\" height=\"84\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Find the GCF of 4 x cubed and 20 x squared. Next to this statement is the prime factorization of 4 x cubed written as 4 x cubed equals 2 times 2 times x times x times x. Below this equation is another for the prime factorization of 20 x squared written as 20 x squared equals 2 times 2 times 5 times x times x. The two equations line up vertically at the equal symbol. Both of the two 2's in the prime factorizations vertically align, and each vertical pair is circled. The 5 in the prime factorization of 20 x sqaured does not align with any factors from the prime factorization of 4 x cubed. The first 2 x's in the prime factorization of 4 x cubed align with the 2 x's in the prime factorization of 20 x squared, and each vertical pair is circled. The third x in the prime factorization of 4 x cubed does not align with any factors from the prime factorization of 20 x squared. A horizontal line is drawn under the prime factorization of 20 x sqaured. Below this line is the equation GCF equals 2 times 2 times x times x. Each term in the GCF equation aligns with the respective circled pair from above. The GCF equation simplifies to GCF equals 4 x squared.\" width=\"505\" height=\"108\" \/><\/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<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]25b^{3}+10b^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716902\">Show Solution<\/span><\/p>\n<div id=\"q716902\" class=\"hidden-answer\" style=\"display: none\">Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex] to be [latex]5b^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,25b^{3}=5\\cdot5\\cdot{b}\\cdot{b}\\cdot{b}\\\\\\,\\,10b^{2}=5\\cdot2\\cdot{b}\\cdot{b}\\\\\\text{GCF}=5\\cdot{b}\\cdot{b}=5b^{2}\\end{array}[\/latex]<\/p>\n<p>Rewrite each term with the GCF as one factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}25b^{3} = 5b^{2}\\cdot5b\\\\10b^{2}=5b^{2}\\cdot2\\end{array}[\/latex]<\/p>\n<p>Rewrite the polynomial using the factored terms in place of the original terms.<\/p>\n<p style=\"text-align: center;\">[latex]5b^{2}\\left(5b\\right)+5b^{2}\\left(2\\right)[\/latex]<\/p>\n<p>Factor out the [latex]5b^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5b^{2}\\left(5b+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The factored form of the polynomial [latex]25b^{3}+10b^{2}[\/latex] is [latex]5b^{2}\\left(5b+2\\right)[\/latex]. You can check this by doing the multiplication. [latex]5b^{2}\\left(5b+2\\right)=25b^{3}+10b^{2}[\/latex].<\/p>\n<p>Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.<\/p>\n<p>For example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}25b^{3}+10b^{2}=5\\left(5b^{3}+2b^{2}\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }5.\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=5b^{2}\\left(5b+2\\right) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Factor out }b^{2}.\\end{array}[\/latex]<\/p>\n<p>Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.<\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The prime factorization of 21 y squared is written as 21 y squared equals 3 times 7 times y times y. Below this equation is another for the prime factorization of 35 y written as 35 y equals 5 times 7 times y. The two equations line up vertically at the equal symbol. The 3 from the prime factorization of 21 y squared does not vertically align with any factors from the prime factorization of 35 y. The same can be said for the 5 from the prime factorization of 35 y aligning with no factors from the prime factorization of 21 y squared. The 7's from both prime factorizations align vertically, and the pair is circled. The same can be said for the first y in the prime factorization of 21 y squared and the y in the prime factorization of 35 y. The second y in the prime factorization of 21 y squared does not align with any factors from the prime factorization of 35 y. A horizontal line is drawn under the prime factorization of 35 y. Below this line is the equation GCF equals 7 times y. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 7 y.\" width=\"196\" height=\"113\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The prime factorization of 9 y written as 9 y equals 3 times 3 times y. Below this equation is another for the prime factorization of 27 written as 27 equals 3 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 3's from both prime factorizations vertically align with each other, and each vertical pair is circled. The third 3 from the prime factorization of 27 does not align with any factors from the prime factorization of 9 y. Similarly, the y from the prime factorization of 9 y does not align with any factors. A horizontal line is drawn under the prime factorization of 27. Below this line is the equation GCF equals 3 times 3. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 9.\" width=\"162\" height=\"104\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The prime factorization of 4 a squared written as 4 a squared equals 2 times 2 times a times a. Below this equation is another for the prime factorization of 16 a written as 16 as equals 2 times 2 times 2 times 2 times a. The two equations line up vertically at the equal symbol. The first two 2's from both prime factorizations vertically align, and each vertical pair is circled. The remaining 2's in the prime factorization of 16 a do not align vertically with any factors from the prime factorization of 4 a squared. The first a from both prime factorizations align with each other, and the vertical pair is circled. The remaining a from the prime factorization of 4 a squared does not vertically align with any other factors. A horizontal line is drawn below the prime factorization of 16 a. Below this line is the equation GCF equals 2 times 2 times a. Each term in the GCF equation aligns with the respective circled pair from above. The equation simplifies to GCF equals 4 a.\" width=\"218\" height=\"108\" \/><\/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<p>This next example shows factoring a binomial when there are two different variables in the expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]81c^{3}d+45c^{2}d^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q809701\">Show Solution<\/span><\/p>\n<div id=\"q809701\" class=\"hidden-answer\" style=\"display: none\">Factor [latex]81c^{3}d[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot9\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}[\/latex]<\/p>\n<p>Factor [latex]45c^{2}d^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}[\/latex]<\/p>\n<p>Find the GCF.<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}=9c^{2}d[\/latex]<\/p>\n<p>Rewrite each term as the product of the GCF and the remaining terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,81c^{3}d=9c^{2}d\\left(9c\\right)\\\\45c^{2}d^{2}=9c^{2}d\\left(5d\\right)\\end{array}[\/latex]<\/p>\n<p>Rewrite the polynomial expression using the factored terms in place of the original terms.<\/p>\n<p style=\"text-align: center;\">[latex]9c^{2}d\\left(9c\\right)+9c^{2}d\\left(5d\\right)[\/latex]<\/p>\n<p>Factor out [latex]9c^{2}d[\/latex]<i>.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]9c^{2}d\\left(9c+5d\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides two more examples of finding the greatest common factor of a binomial<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Identify GCF and Factor a Binomial\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/25_f_mVab_4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This last example shows finding the greatest common factors of trinomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2:  Identify GCF and Factor a Trinomial\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3f1RFTIw2Ng?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\\cdot{b}+a\\cdot{c}[\/latex] form has been rewritten as [latex]a\\left(b+c\\right)[\/latex], where <i>a<\/i> is the GCF, the polynomial is in factored form.<\/p>\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-10892\">\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":21046,"menu_order":5,"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":"0ba3102927a746d3905c5e25ebe7b6f2, 7018cc6d7e2a4c65a48950ca9c6b7ecc","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10892","chapter","type-chapter","status-publish","hentry"],"part":16188,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10892","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/21046"}],"version-history":[{"count":43,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10892\/revisions"}],"predecessor-version":[{"id":20486,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10892\/revisions\/20486"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16188"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10892\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10892"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10892"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10892"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=10892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}