{"id":18273,"date":"2022-04-14T20:59:50","date_gmt":"2022-04-14T20:59:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18273"},"modified":"2022-04-28T23:15:48","modified_gmt":"2022-04-28T23:15:48","slug":"cr-10-factors-the-greatest-common-factor","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-10-factors-the-greatest-common-factor\/","title":{"raw":"CR.10: Factors: The Greatest Common Factor","rendered":"CR.10: Factors: The Greatest Common Factor"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the greatest common factor of a list of expressions<\/li>\r\n \t<li>Find the greatest common factor of a polynomial<\/li>\r\n \t<li>Factor a four term polynomial by grouping terms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot{5}[\/latex] or [latex]20=1\\cdot{20}[\/latex]. In algebra, we use the word factor as both a noun \u2013 something being multiplied \u2013 and as a verb \u2013 the action of rewriting a sum or difference as a product.\u00a0<strong>Factoring<\/strong> is very helpful in simplifying expressions and solving equations involving\u00a0polynomials.\r\n\r\nThe <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers: [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of two or more algebraic expressions works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest algebraic expression that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].\r\n<h3>Find the GCF of a list of algebraic expressions<\/h3>\r\nWe begin by finding the GCF of a list of numbers, then we'll extend the technique to monomial expressions containing variables.\r\n\r\nA good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].\r\n\r\n[reveal-answer q=\"863750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"863750\"]\r\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\r\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors--matching common factors in a column.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\r\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\r\n<td style=\"width: 426.45px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\r\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\r\n<td style=\"width: 426.45px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere's a summary of the technique.\r\n<div class=\"textbox shaded\">\r\n<h3>Find the greatest common factor<\/h3>\r\n<ol id=\"eip-id1168468531103\" class=\"stepwise\">\r\n \t<li>Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/li>\r\n \t<li>List all factors\u2014matching common factors in a column. In each column, circle the common factors.<\/li>\r\n \t<li>Bring down the common factors that all expressions share.<\/li>\r\n \t<li>Multiply the factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146326[\/ohm_question]\r\n\r\n<\/div>\r\nIn the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.\r\n\r\nTo find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here's an example using the matching method from the example above.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]5x\\text{ and }15[\/latex].\r\n[reveal-answer q=\"470279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"470279\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each number into primes.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224617\/CNX_BMath_Figure_10_06_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]5x[\/latex] and [latex]15[\/latex] is [latex]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]146327[\/ohm_question]\r\n\r\n<\/div>\r\nIn the examples so far, the GCF was a constant. In the next two examples we will get variables in the greatest common factor.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex].\r\n[reveal-answer q=\"35972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35972\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each coefficient into primes and write\r\n\r\nthe variables with exponents in expanded form.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.\r\n\r\nMultiply the factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224619\/CNX_BMath_Figure_10_06_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex] is [latex]6{x}^{2}[\/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]146328[\/ohm_question]\r\n\r\n<\/div>\r\nHere are some examples of finding the GCF of a list of more than two expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex].\r\n[reveal-answer q=\"215868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"215868\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>Factor each coefficient into primes and write\r\n\r\nthe variables with exponents in expanded form.\r\n\r\nCircle the common factors in each column.\r\n\r\nBring down the common factors.\r\n\r\nMultiply the factors.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224620\/CNX_BMath_Figure_10_06_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The GCF of [latex]14{x}^{3}[\/latex] and [latex]8{x}^{2}[\/latex] and [latex]10x[\/latex] is [latex]2x[\/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]146329[\/ohm_question]\r\n\r\n<\/div>\r\nThe common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term.\r\n\r\nIn fact,\u00a0the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.\r\n\r\nWatch the following video to see another example of how to find the GCF of two monomials that have one variable.\r\n\r\nhttps:\/\/youtu.be\/EhkVBXRBC2s\r\n\r\nSometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example we find the GCF of two terms which both contain two variables.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].\r\n[reveal-answer q=\"930504\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930504\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^{3}d=3\\cdot3\\cdot3\\cdot3\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}\\\\45c^{2}d^{2}=3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=9c^{2}d[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.\r\n\r\nhttps:\/\/youtu.be\/GfJvoIO3gKQ\r\n<h3>Find the GCF of a polynomial<\/h3>\r\nNow that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring\u00a0the GCF out of a polynomial. That is, doing the distributive property \"backwards\" to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial.\r\n\r\nRecall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions.\r\n\r\nNo matter how large the polynomial, you can use the same technique described below to identify its GCF.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial expression, find the greatest common factor.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the GCF of the coefficients.<\/li>\r\n \t<li>Identify the GCF of the variables.<\/li>\r\n \t<li>Combine to find the GCF of the expression.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the GCF of [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n[reveal-answer q=\"572595\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572595\"]\r\n\r\nThe GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex].\r\nThe GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex].\r\nAnd the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex].\r\nCombine these to find the GCF of the polynomial, [latex]3xy[\/latex].\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]14137[\/ohm_question]\r\n\r\n<\/div>\r\nRecall that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of polynomials works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].\r\n\r\nFinding and factoring out a GCF from a polynomial is the first skill involved in factoring polynomials.\r\n<h2>Factoring a GCF out of a polynomial<\/h2>\r\nWhen factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Greatest Common Factor<\/h3>\r\nThe <strong>greatest common factor<\/strong> (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.\r\n\r\n<\/div>\r\nTo factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property \"backwards\" 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<h4>Distributive Property Forward and Backward<\/h4>\r\n<p style=\"text-align: left;\">Forward:\u00a0<em>We distribute [latex]a[\/latex] over [latex]b+c[\/latex]<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex].<\/p>\r\nBackward:\u00a0<em>We factor [latex]a[\/latex] out of [latex]ab+ac[\/latex].<\/em>\r\n<p style=\"text-align: center;\">[latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/p>\r\nWe have seen that we can distribute a factor over a sum or difference. Now we see that we can \"undo\" the distributive property with factoring.\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.\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<h4>Analysis<\/h4>\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\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we see two more examples of how to find and factor the GCF from binomials.\r\n\r\nhttps:\/\/youtu.be\/25_f_mVab_4\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial expression, factor out the greatest common factor<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the GCF of the coefficients.<\/li>\r\n \t<li>Identify the GCF of the variables.<\/li>\r\n \t<li>Combine to find the GCF of the expression.<\/li>\r\n \t<li>Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\r\n \t<li>Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Factoring the Greatest Common Factor<\/h3>\r\nFactor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n[reveal-answer q=\"113189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113189\"]\r\n\r\nFirst find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions of the form [latex]{x}^{n}[\/latex] will always be the lowest exponent.) The GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].\r\n\r\nNext, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3}, 3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].\r\n\r\nFinally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.\r\n<div style=\"text-align: center;\">[latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\r\n<div>\r\n<h4>Analysis of the Solution<\/h4>\r\nAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\nThe GCF may not always be a monomial. Here is an example of a GCF that is a binomial.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFactor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.\r\n\r\n[reveal-answer q=\"94532\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"94532\"]\r\n\r\n[latex]\\left({b}^{2}-a\\right)\\left(x+6\\right)[\/latex][\/hidden-answer]\r\n[ohm_question]7888[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video to see more examples of how to factor the GCF from a trinomial.\r\n\r\nhttps:\/\/youtu.be\/3f1RFTIw2Ng\r\n<h2>Factor a Four Term Polynomial by Grouping Terms<\/h2>\r\nWhen we learned to multiply two binomials, we found that the result, before combining like terms, was a four term polynomial, as in this example: [latex]\\left(x+4\\right)\\left(x+2\\right)=x^{2}+2x+4x+8[\/latex].\r\n\r\nWe can apply what we have learned about factoring out a common monomial to return a four term polynomial to the product of two binomials. Why would we even want to do this?\r\n<div id=\"attachment_4825\" class=\"wp-caption aligncenter\" style=\"width: 401px;\">\r\n\r\n<img class=\" wp-image-4825\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/12195953\/Screen-Shot-2016-06-12-at-12.59.11-PM-300x138.png\" alt=\"Thought bubble with the words .....and i should care why?\" width=\"391\" height=\"180\" \/>\r\n<p class=\"wp-caption-text\">Why Should I Care?<\/p>\r\n\r\n<\/div>\r\nBecause it is an important step in learning techniques for factoring trinomials, such as the one you get when you simplify the product of the two binomials from above:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(x+4\\right)\\left(x+2\\right)\\\\=x^{2}+2x+4x+8\\\\=x^2+6x+8\\end{array}[\/latex]<\/p>\r\nAdditionally, factoring by grouping is a technique that allows us to factor a polynomial whose terms don\u2019t all share a GCF. In the following example, we will introduce you to the technique. Remember, one of the main reasons to factor is because it will help solve polynomial equations.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]a^2+3a+5a+15[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q437455\">Show Solution<\/span>\r\n<div id=\"q437455\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThere isn\u2019t a common factor between all four terms, so we will group the terms into pairs that will enable us to find a GCF for them. For example, we wouldn\u2019t want to group [latex]a^2\\text{ and }15[\/latex] because they don\u2019t share a common factor.\r\n<p style=\"text-align: center;\">[latex]\\left(a^2+3a\\right)+\\left(5a+15\\right)[\/latex]<\/p>\r\nFind the GCF of the first pair of terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,a^2=a\\cdot{a}\\\\\\,\\,\\,\\,3a=3\\cdot{a}\\\\\\text{GCF}=a\\end{array}[\/latex]<\/p>\r\nFactor the GCF, <i>a<\/i>, out of the first group.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left(a\\cdot{a}+a\\cdot{3}\\right)+\\left(5a+15\\right)\\\\a\\left(a+3\\right)+\\left(5a+15\\right)\\end{array}[\/latex]<\/p>\r\nFind the GCF of the second pair of terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5a=5\\cdot{a}\\\\15=5\\cdot3\\\\\\text{GCF}=5\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nFactor [latex]5[\/latex] out of the second group.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a\\left(a+3\\right)+\\left(5\\cdot{a}+5\\cdot3\\right)\\\\a\\left(a+3\\right)+5\\left(a+3\\right)\\end{array}[\/latex]<\/p>\r\nNotice that the two terms have a common factor [latex]\\left(a+3\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]a\\left(a+3\\right)+5\\left(a+3\\right)[\/latex]<\/p>\r\nFactor out the common factor [latex]\\left(a+3\\right)[\/latex] from the two terms.\r\n<p style=\"text-align: center;\">[latex]\\left(a+3\\right)\\left(a+5\\right)[\/latex]<\/p>\r\nNote how the a and 5 become a binomial sum, and the other factor. This is probably the most confusing part of factoring by grouping.\r\n<h4>Answer<\/h4>\r\n[latex]a^2+3a+5a+15=\\left(a+3\\right)\\left(a+5\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nNotice that when you factor a two term polynomial, the result is a monomial times a polynomial. But the factored form of a four-term polynomial is the product of two binomials. As we noted before, this is an important middle step in learning how to factor a three term polynomial.\r\n\r\nThis process is called the <i>grouping technique<\/i>. Broken down into individual steps, here\u2019s how to do it (you can also follow this process in the example below).\r\n<ul>\r\n \t<li>Group the terms of the polynomial into pairs that share a GCF.<\/li>\r\n \t<li>Find the greatest common factor and then use the distributive property to pull out the GCF<\/li>\r\n \t<li>Look for the common binomial between the factored terms<\/li>\r\n \t<li>Factor the common binomial out of the groups, the other factors will make the other binomial<\/li>\r\n<\/ul>\r\nLet\u2019s try factoring a few more four-term polynomials. Note how there is a now a constant in front of the [latex]x^2[\/latex] term. We will just consider this another factor when we are finding the GCF.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]2x^{2}+4x+5x+10[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q313122\">Show Solution<\/span>\r\n<div id=\"q313122\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nGroup terms of the polynomial into pairs.\r\n<p style=\"text-align: center;\">[latex]\\left(2x^{2}+4x\\right)+\\left(5x+10\\right)[\/latex]<\/p>\r\nFactor out the like factor, [latex]2x[\/latex], from the first group.\r\n<p style=\"text-align: center;\">[latex]2x\\left(x+2\\right)+\\left(5x+10\\right)[\/latex]<\/p>\r\nFactor out the like factor, [latex]5[\/latex], from the second group.\r\n<p style=\"text-align: center;\">[latex]2x\\left(x+2\\right)+5\\left(x+2\\right)[\/latex]<\/p>\r\nLook for common factors between the factored forms of the paired terms. Here, the common factor is [latex](x+2)[\/latex].\r\n\r\nFactor out the common factor, [latex]\\left(x+2\\right)[\/latex], from both terms.\r\n<p style=\"text-align: center;\">[latex]\\left(x+2\\right)\\left(2x+5\\right)[\/latex]<\/p>\r\nThe polynomial is now factored.\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+2\\right)\\left(2x+5\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nAnother example follows that contains subtraction. Note how we choose a positive GCF from each group of terms, and the negative signs stay.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]2x^{2}\u20133x+8x\u201312[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q715080\">Show Solution<\/span>\r\n<div id=\"q715080\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nGroup terms into pairs.\r\n<p style=\"text-align: center;\">[latex](2x^{2}\u20133x)+(8x\u201312)[\/latex]<\/p>\r\nFactor the common factor, [latex]x[\/latex], out of the first group and the common factor, [latex]4[\/latex], out of the second group.\r\n<p style=\"text-align: center;\">[latex]x\\left(2x\u20133\\right)+4\\left(2x\u20133\\right)[\/latex]<\/p>\r\nFactor out the common factor, [latex]\\left(2x\u20133\\right)[\/latex], from both terms.\r\n<p style=\"text-align: center;\">[latex]\\left(x+4\\right)\\left(2x\u20133\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+4\\right)\\left(2x-3\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe video that follows provides another example of factoring by grouping.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/RR5nj7RFSiU?feature=oembed&amp;rel=0\" width=\"500\" height=\"375\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nIn the next example, we will have a GCF that is negative. It is important to pay attention to what happens to the resulting binomial when the GCF is negative.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]3x^{2}+3x\u20132x\u20132[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q744005\">Show Solution<\/span>\r\n<div id=\"q744005\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nGroup terms into pairs.\r\n<p style=\"text-align: center;\">[latex]\\left(3x^{2}+3x\\right)+\\left(-2x-2\\right)[\/latex]<\/p>\r\nFactor the common factor [latex]3x[\/latex] out of first group.\r\n<p style=\"text-align: center;\">[latex]3x\\left(x+1\\right)+\\left(-2x-2\\right)[\/latex]<\/p>\r\nFactor the common factor [latex]\u22122[\/latex] out of the second group. Notice what happens to the signs within the parentheses once [latex]\u22122[\/latex] is factored out.\r\n<p style=\"text-align: center;\">[latex]3x\\left(x+1\\right)-2\\left(x+1\\right)[\/latex]<\/p>\r\nFactor out the common factor, [latex]\\left(x+1\\right)[\/latex], from both terms.\r\n<p style=\"text-align: center;\">[latex]\\left(x+1\\right)\\left(3x-2\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+1\\right)\\left(3x-2\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video we present another example of factoring by grouping when one of the GCF is negative.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/0dvGmDGVC5U?feature=oembed&amp;rel=0\" width=\"500\" height=\"375\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nSometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFactor [latex]7x^{2}\u201321x+5x\u20135[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q262926\">Show Solution<\/span>\r\n<div id=\"q262926\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nGroup terms into pairs.\r\n<p style=\"text-align: center;\">[latex]\\left(7x^{2}\u201321x\\right)+\\left(5x\u20135\\right)[\/latex]<\/p>\r\nFactor the common factor [latex]7x[\/latex] out of the first group.\r\n<p style=\"text-align: center;\">[latex]7x\\left(x-3\\right)+\\left(5x-5\\right)[\/latex]<\/p>\r\nFactor the common factor [latex]5[\/latex] out of the second group.\r\n<p style=\"text-align: center;\">[latex]7x\\left(x-3\\right)+5\\left(x-1\\right)[\/latex]<\/p>\r\nThe two groups [latex]7x\\left(x\u20133\\right)[\/latex] and [latex]5\\left(x\u20131\\right)[\/latex] do not have any common factors, so this polynomial cannot be factored any further.\r\n<p style=\"text-align: center;\">[latex]7x\\left(x\u20133\\right)+5\\left(x\u20131\\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nCannot be factored\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the example above, each pair can be factored, but then there is no common factor between the pairs! This means that it cannot be factored.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the greatest common factor of a list of expressions<\/li>\n<li>Find the greatest common factor of a polynomial<\/li>\n<li>Factor a four term polynomial by grouping terms<\/li>\n<\/ul>\n<\/div>\n<p><strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[\/latex] and\u00a0[latex]10[\/latex] are factors of\u00a0[latex]20[\/latex], as are\u00a0[latex]4, 5, 1, 20[\/latex]. To factor a number is to rewrite it as a product. [latex]20=4\\cdot{5}[\/latex] or [latex]20=1\\cdot{20}[\/latex]. In algebra, we use the word factor as both a noun \u2013 something being multiplied \u2013 and as a verb \u2013 the action of rewriting a sum or difference as a product.\u00a0<strong>Factoring<\/strong> is very helpful in simplifying expressions and solving equations involving\u00a0polynomials.<\/p>\n<p>The <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers: [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of two or more algebraic expressions works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest algebraic expression that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/p>\n<h3>Find the GCF of a list of algebraic expressions<\/h3>\n<p>We begin by finding the GCF of a list of numbers, then we&#8217;ll extend the technique to monomial expressions containing variables.<\/p>\n<p>A good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q863750\">Show Solution<\/span><\/p>\n<div id=\"q863750\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says,\">\n<tbody>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors&#8211;matching common factors in a column.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\n<td style=\"width: 426.45px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\n<td style=\"width: 426.45px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here&#8217;s a summary of the technique.<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the greatest common factor<\/h3>\n<ol id=\"eip-id1168468531103\" class=\"stepwise\">\n<li>Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/li>\n<li>List all factors\u2014matching common factors in a column. In each column, circle the common factors.<\/li>\n<li>Bring down the common factors that all expressions share.<\/li>\n<li>Multiply the factors.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146326&theme=oea&iframe_resize_id=ohm146326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients.<\/p>\n<p>To find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here&#8217;s an example using the matching method from the example above.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]5x\\text{ and }15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q470279\">Show Solution<\/span><\/p>\n<div id=\"q470279\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each number into primes.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224617\/CNX_BMath_Figure_10_06_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]5x[\/latex] and [latex]15[\/latex] is [latex]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=\"ohm146327\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146327&theme=oea&iframe_resize_id=ohm146327&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the examples so far, the GCF was a constant. In the next two examples we will get variables in the greatest common factor.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35972\">Show Solution<\/span><\/p>\n<div id=\"q35972\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each coefficient into primes and write<\/p>\n<p>the variables with exponents in expanded form.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/p>\n<p>Multiply the factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224619\/CNX_BMath_Figure_10_06_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex] is [latex]6{x}^{2}[\/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=\"ohm146328\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146328&theme=oea&iframe_resize_id=ohm146328&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Here are some examples of finding the GCF of a list of more than two expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [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=\"q215868\">Show Solution<\/span><\/p>\n<div id=\"q215868\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>Factor each coefficient into primes and write<\/p>\n<p>the variables with exponents in expanded form.<\/p>\n<p>Circle the common factors in each column.<\/p>\n<p>Bring down the common factors.<\/p>\n<p>Multiply the factors.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224620\/CNX_BMath_Figure_10_06_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The GCF of [latex]14{x}^{3}[\/latex] and [latex]8{x}^{2}[\/latex] and [latex]10x[\/latex] is [latex]2x[\/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=\"ohm146329\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146329&theme=oea&iframe_resize_id=ohm146329&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term.<\/p>\n<p>In fact,\u00a0the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.<\/p>\n<p>Watch the following video to see another example of how to find the GCF of two monomials that have one variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine the GCF of Two Monomials (One Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EhkVBXRBC2s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example we find the GCF of two terms which both contain two variables.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]81c^{3}d[\/latex] and [latex]45c^{2}d^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930504\">Show Solution<\/span><\/p>\n<div id=\"q930504\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,81c^{3}d=3\\cdot3\\cdot3\\cdot3\\cdot{c}\\cdot{c}\\cdot{c}\\cdot{d}\\\\45c^{2}d^{2}=3\\cdot3\\cdot5\\cdot{c}\\cdot{c}\\cdot{d}\\cdot{d}\\\\\\,\\,\\,\\,\\text{GCF}=3\\cdot3\\cdot{c}\\cdot{c}\\cdot{d}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=9c^{2}d[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Determine the GCF of Two Monomials (Two Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GfJvoIO3gKQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Find the GCF of a polynomial<\/h3>\n<p>Now that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring\u00a0the GCF out of a polynomial. That is, doing the distributive property &#8220;backwards&#8221; to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial.<\/p>\n<p>Recall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions.<\/p>\n<p>No matter how large the polynomial, you can use the same technique described below to identify its GCF.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial expression, find the greatest common factor.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the GCF of the coefficients.<\/li>\n<li>Identify the GCF of the variables.<\/li>\n<li>Combine to find the GCF of the expression.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the GCF of [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q572595\">Show Solution<\/span><\/p>\n<div id=\"q572595\" class=\"hidden-answer\" style=\"display: none\">\n<p>The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex].<br \/>\nThe GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex].<br \/>\nAnd the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex].<br \/>\nCombine these to find the GCF of the polynomial, [latex]3xy[\/latex].\n<\/p><\/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=\"ohm14137\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14137&theme=oea&iframe_resize_id=ohm14137&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Recall that the <strong>greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex]. The GCF of polynomials works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/p>\n<p>Finding and factoring out a GCF from a polynomial is the first skill involved in factoring polynomials.<\/p>\n<h2>Factoring a GCF out of a polynomial<\/h2>\n<p>When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Greatest Common Factor<\/h3>\n<p>The <strong>greatest common factor<\/strong> (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.<\/p>\n<\/div>\n<p>To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property &#8220;backwards&#8221; 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<h4>Distributive Property Forward and Backward<\/h4>\n<p style=\"text-align: left;\">Forward:\u00a0<em>We distribute [latex]a[\/latex] over [latex]b+c[\/latex]<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex].<\/p>\n<p>Backward:\u00a0<em>We factor [latex]a[\/latex] out of [latex]ab+ac[\/latex].<\/em><\/p>\n<p style=\"text-align: center;\">[latex]ab+ac=a\\left(b+c\\right)[\/latex].<\/p>\n<p>We have seen that we can distribute a factor over a sum or difference. Now we see that we can &#8220;undo&#8221; the distributive property with factoring.<\/p>\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.<\/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<h4>Analysis<\/h4>\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>\n<\/div>\n<\/div>\n<p>In the following video we see two more examples of how to find and factor the GCF from binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<div class=\"textbox\">\n<h3>How To: Given a polynomial expression, factor out the greatest common factor<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the GCF of the coefficients.<\/li>\n<li>Identify the GCF of the variables.<\/li>\n<li>Combine to find the GCF of the expression.<\/li>\n<li>Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\n<li>Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Factoring the Greatest Common Factor<\/h3>\n<p>Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113189\">Show Solution<\/span><\/p>\n<div id=\"q113189\" class=\"hidden-answer\" style=\"display: none\">\n<p>First find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions of the form [latex]{x}^{n}[\/latex] will always be the lowest exponent.) The GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].<\/p>\n<p>Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3}, 3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].<\/p>\n<p>Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\n<div>\n<h4>Analysis of the Solution<\/h4>\n<p>After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Factor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q94532\">Show Solution<\/span><\/p>\n<div id=\"q94532\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left({b}^{2}-a\\right)\\left(x+6\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm7888\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7888&theme=oea&iframe_resize_id=ohm7888&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of how to factor the GCF from a trinomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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>Factor a Four Term Polynomial by Grouping Terms<\/h2>\n<p>When we learned to multiply two binomials, we found that the result, before combining like terms, was a four term polynomial, as in this example: [latex]\\left(x+4\\right)\\left(x+2\\right)=x^{2}+2x+4x+8[\/latex].<\/p>\n<p>We can apply what we have learned about factoring out a common monomial to return a four term polynomial to the product of two binomials. Why would we even want to do this?<\/p>\n<div id=\"attachment_4825\" class=\"wp-caption aligncenter\" style=\"width: 401px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4825\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/12195953\/Screen-Shot-2016-06-12-at-12.59.11-PM-300x138.png\" alt=\"Thought bubble with the words .....and i should care why?\" width=\"391\" height=\"180\" \/><\/p>\n<p class=\"wp-caption-text\">Why Should I Care?<\/p>\n<\/div>\n<p>Because it is an important step in learning techniques for factoring trinomials, such as the one you get when you simplify the product of the two binomials from above:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(x+4\\right)\\left(x+2\\right)\\\\=x^{2}+2x+4x+8\\\\=x^2+6x+8\\end{array}[\/latex]<\/p>\n<p>Additionally, factoring by grouping is a technique that allows us to factor a polynomial whose terms don\u2019t all share a GCF. In the following example, we will introduce you to the technique. Remember, one of the main reasons to factor is because it will help solve polynomial equations.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]a^2+3a+5a+15[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q437455\">Show Solution<\/span><\/p>\n<div id=\"q437455\" class=\"hidden-answer\" style=\"display: none;\">\n<p>There isn\u2019t a common factor between all four terms, so we will group the terms into pairs that will enable us to find a GCF for them. For example, we wouldn\u2019t want to group [latex]a^2\\text{ and }15[\/latex] because they don\u2019t share a common factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^2+3a\\right)+\\left(5a+15\\right)[\/latex]<\/p>\n<p>Find the GCF of the first pair of terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,a^2=a\\cdot{a}\\\\\\,\\,\\,\\,3a=3\\cdot{a}\\\\\\text{GCF}=a\\end{array}[\/latex]<\/p>\n<p>Factor the GCF, <i>a<\/i>, out of the first group.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left(a\\cdot{a}+a\\cdot{3}\\right)+\\left(5a+15\\right)\\\\a\\left(a+3\\right)+\\left(5a+15\\right)\\end{array}[\/latex]<\/p>\n<p>Find the GCF of the second pair of terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5a=5\\cdot{a}\\\\15=5\\cdot3\\\\\\text{GCF}=5\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Factor [latex]5[\/latex] out of the second group.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a\\left(a+3\\right)+\\left(5\\cdot{a}+5\\cdot3\\right)\\\\a\\left(a+3\\right)+5\\left(a+3\\right)\\end{array}[\/latex]<\/p>\n<p>Notice that the two terms have a common factor [latex]\\left(a+3\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]a\\left(a+3\\right)+5\\left(a+3\\right)[\/latex]<\/p>\n<p>Factor out the common factor [latex]\\left(a+3\\right)[\/latex] from the two terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+3\\right)\\left(a+5\\right)[\/latex]<\/p>\n<p>Note how the a and 5 become a binomial sum, and the other factor. This is probably the most confusing part of factoring by grouping.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]a^2+3a+5a+15=\\left(a+3\\right)\\left(a+5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that when you factor a two term polynomial, the result is a monomial times a polynomial. But the factored form of a four-term polynomial is the product of two binomials. As we noted before, this is an important middle step in learning how to factor a three term polynomial.<\/p>\n<p>This process is called the <i>grouping technique<\/i>. Broken down into individual steps, here\u2019s how to do it (you can also follow this process in the example below).<\/p>\n<ul>\n<li>Group the terms of the polynomial into pairs that share a GCF.<\/li>\n<li>Find the greatest common factor and then use the distributive property to pull out the GCF<\/li>\n<li>Look for the common binomial between the factored terms<\/li>\n<li>Factor the common binomial out of the groups, the other factors will make the other binomial<\/li>\n<\/ul>\n<p>Let\u2019s try factoring a few more four-term polynomials. Note how there is a now a constant in front of the [latex]x^2[\/latex] term. We will just consider this another factor when we are finding the GCF.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]2x^{2}+4x+5x+10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q313122\">Show Solution<\/span><\/p>\n<div id=\"q313122\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Group terms of the polynomial into pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2x^{2}+4x\\right)+\\left(5x+10\\right)[\/latex]<\/p>\n<p>Factor out the like factor, [latex]2x[\/latex], from the first group.<\/p>\n<p style=\"text-align: center;\">[latex]2x\\left(x+2\\right)+\\left(5x+10\\right)[\/latex]<\/p>\n<p>Factor out the like factor, [latex]5[\/latex], from the second group.<\/p>\n<p style=\"text-align: center;\">[latex]2x\\left(x+2\\right)+5\\left(x+2\\right)[\/latex]<\/p>\n<p>Look for common factors between the factored forms of the paired terms. Here, the common factor is [latex](x+2)[\/latex].<\/p>\n<p>Factor out the common factor, [latex]\\left(x+2\\right)[\/latex], from both terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+2\\right)\\left(2x+5\\right)[\/latex]<\/p>\n<p>The polynomial is now factored.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+2\\right)\\left(2x+5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another example follows that contains subtraction. Note how we choose a positive GCF from each group of terms, and the negative signs stay.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]2x^{2}\u20133x+8x\u201312[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q715080\">Show Solution<\/span><\/p>\n<div id=\"q715080\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Group terms into pairs.<\/p>\n<p style=\"text-align: center;\">[latex](2x^{2}\u20133x)+(8x\u201312)[\/latex]<\/p>\n<p>Factor the common factor, [latex]x[\/latex], out of the first group and the common factor, [latex]4[\/latex], out of the second group.<\/p>\n<p style=\"text-align: center;\">[latex]x\\left(2x\u20133\\right)+4\\left(2x\u20133\\right)[\/latex]<\/p>\n<p>Factor out the common factor, [latex]\\left(2x\u20133\\right)[\/latex], from both terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+4\\right)\\left(2x\u20133\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+4\\right)\\left(2x-3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows provides another example of factoring by grouping.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/RR5nj7RFSiU?feature=oembed&amp;rel=0\" width=\"500\" height=\"375\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we will have a GCF that is negative. It is important to pay attention to what happens to the resulting binomial when the GCF is negative.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]3x^{2}+3x\u20132x\u20132[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q744005\">Show Solution<\/span><\/p>\n<div id=\"q744005\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Group terms into pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3x^{2}+3x\\right)+\\left(-2x-2\\right)[\/latex]<\/p>\n<p>Factor the common factor [latex]3x[\/latex] out of first group.<\/p>\n<p style=\"text-align: center;\">[latex]3x\\left(x+1\\right)+\\left(-2x-2\\right)[\/latex]<\/p>\n<p>Factor the common factor [latex]\u22122[\/latex] out of the second group. Notice what happens to the signs within the parentheses once [latex]\u22122[\/latex] is factored out.<\/p>\n<p style=\"text-align: center;\">[latex]3x\\left(x+1\\right)-2\\left(x+1\\right)[\/latex]<\/p>\n<p>Factor out the common factor, [latex]\\left(x+1\\right)[\/latex], from both terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+1\\right)\\left(3x-2\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+1\\right)\\left(3x-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present another example of factoring by grouping when one of the GCF is negative.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/0dvGmDGVC5U?feature=oembed&amp;rel=0\" width=\"500\" height=\"375\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Factor [latex]7x^{2}\u201321x+5x\u20135[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q262926\">Show Solution<\/span><\/p>\n<div id=\"q262926\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Group terms into pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(7x^{2}\u201321x\\right)+\\left(5x\u20135\\right)[\/latex]<\/p>\n<p>Factor the common factor [latex]7x[\/latex] out of the first group.<\/p>\n<p style=\"text-align: center;\">[latex]7x\\left(x-3\\right)+\\left(5x-5\\right)[\/latex]<\/p>\n<p>Factor the common factor [latex]5[\/latex] out of the second group.<\/p>\n<p style=\"text-align: center;\">[latex]7x\\left(x-3\\right)+5\\left(x-1\\right)[\/latex]<\/p>\n<p>The two groups [latex]7x\\left(x\u20133\\right)[\/latex] and [latex]5\\left(x\u20131\\right)[\/latex] do not have any common factors, so this polynomial cannot be factored any further.<\/p>\n<p style=\"text-align: center;\">[latex]7x\\left(x\u20133\\right)+5\\left(x\u20131\\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Cannot be factored<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example above, each pair can be factored, but then there is no common factor between the pairs! This means that it cannot be factored.<\/p>\n","protected":false},"author":264444,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18273","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18273","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18273\/revisions"}],"predecessor-version":[{"id":18705,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18273\/revisions\/18705"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18273\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18273"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18273"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18273"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18273"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}