{"id":2936,"date":"2024-02-09T20:28:26","date_gmt":"2024-02-09T20:28:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2936"},"modified":"2026-03-24T17:38:23","modified_gmt":"2026-03-24T17:38:23","slug":"9-6-1-greatest-common-factor","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/9-6-1-greatest-common-factor\/","title":{"raw":"9.6.1: Greatest Common Factor","rendered":"9.6.1: Greatest Common Factor"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Determine the greatest common factor of multiple numbers<\/li>\r\n \t<li>Determine the greatest common factor of monomials<\/li>\r\n \t<li>Factor a polynomial using the GCF<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Factor<\/strong>: a divisor<\/li>\r\n \t<li><strong>Common factor<\/strong>: a term that is a factor of two or more other terms<\/li>\r\n \t<li><strong>Greatest common factor<\/strong>: the common factor that has the highest degree<\/li>\r\n \t<li><strong>Factoring<\/strong>: writing a product or a sum as the multiplication of factors<\/li>\r\n \t<li><strong>Prime number<\/strong>: a whole number greater than or equal to 2 that has exactly 2 factors; 1 and itself<\/li>\r\n \t<li><strong>Prime factor<\/strong>: a factor that is also a prime number<\/li>\r\n \t<li><strong>Prime factorization<\/strong>: the process of factoring a number into prime factors<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Greatest Common Factor<\/h2>\r\nIn chapter 1, we defined a <em><strong>factor<\/strong><\/em> as a number that divides exactly into another number. We multiplied factors together to get a product.\u00a0\u00a0Factors are the building blocks of multiplication. They are the numbers that we can multiply together to produce another number: [latex]2[\/latex] and [latex]10[\/latex] are factors of [latex]20[\/latex], as are \u00a0[latex]4[\/latex] and [latex]5[\/latex] and [latex]1[\/latex] and [latex]20[\/latex]. To factor a number is to rewrite it as a product. For example, [latex]20=4\\cdot5[\/latex]. In algebra, we use the word factor as both a noun - something being multiplied - and as a verb - rewriting a sum or difference as a product.\r\n\r\nFinding a product is multiplying two or more terms together. Each term is a\u00a0<em><strong>factor<\/strong><\/em> of the product. Splitting a product into factors is called <em><strong>factoring<\/strong><\/em>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224609\/CNX_BMath_Figure_10_06_001_img.png\" alt=\"Example showing 8 and 7 as factors of 56. Also 2x and x plus 3 as factors of 2x squared plus 6x.\" width=\"442\" height=\"255\" \/>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine the greatest common factor of 18 and 24.\r\n<h4>Solution<\/h4>\r\nThe factors of 18 are <strong>1<\/strong>, <strong>2<\/strong>, <strong>3<\/strong>, <strong>6<\/strong>, 9, and 18\r\n\r\nThe factors of 24 are <strong>1<\/strong>, <strong>2<\/strong>,<strong> 3<\/strong>, 4, <strong>6<\/strong>, 8, 12, and 24\r\n\r\nThe common factors are the numbers that appear in both lists: 1, 2, 3, and <strong>6<\/strong>.\r\n\r\nThe greatest common factor is the largest of the common factors: 6.\r\n<h4>Answer<\/h4>\r\nGCF(18, 24) = 6\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine the greatest common factor of:\r\n<ol>\r\n \t<li>28 and 42<\/li>\r\n \t<li>10 and 55<\/li>\r\n \t<li>64, 16, and 40<\/li>\r\n \t<li>7, 12, 15<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm157\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm157\"]\r\n<ol>\r\n \t<li>GCF(28, 42) = 14<\/li>\r\n \t<li>GCF(10, 55) = 5<\/li>\r\n \t<li>GCF(64, 16, 40) = 8<\/li>\r\n \t<li>GCF(7, 12, 15) = 1<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA <em><strong>prime factor<\/strong><\/em> is a <em><strong>prime number<\/strong><\/em>\u2014it has only itself and 1 as factors\u2014that is a factor. The process of breaking a number down into its prime factors is called <em><strong>prime factorization<\/strong><\/em>. Prime factorization is unique. Each number has only one set of prime factors. For example, [latex]10=2\\cdot 5[\/latex] is the only way to write [latex]10[\/latex] as a product of primes.\u00a0To find the GCF, we can factor each number into its prime factors, identify the prime factors the numbers have in common, and then multiply those prime factors together.\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<h4>Solution<\/h4>\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<th style=\"width: 194px;\" colspan=\"3\"><strong>Finding the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex]<\/strong><strong>\r\n<\/strong><\/th>\r\n<\/tr>\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 class=\"alignnone\" 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=\"Using a factor tree to find the factors of 24 to be 2 cubed and 3. Using a factor tree to find the factors of 36 to be 2 squared times 3 squared.\" width=\"460\" height=\"193\" \/><\/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 class=\"alignnone\" 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=\"Listing all the factors of 24 and 36, lining up common factors in a column.\" width=\"459\" height=\"65\" \/><\/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 class=\"alignnone\" 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=\"Circling the common factors of 24 and 36. They are 2 squared times 3, with a resulting greatest common factor of 12.\" width=\"459\" height=\"141\" \/><\/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\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=42[\/latex]\r\n\r\n<\/div>\r\n<div>Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and [latex]168[\/latex] by [latex]42[\/latex]\u2014they are both exactly divisible by this number!)<\/div>\r\nThe video that follows shows another example of finding the greatest common factor of two whole numbers.\r\n\r\nhttps:\/\/youtu.be\/KbBJcdDY_VE\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-1.docx\">Transcript-9.6.1-1<\/a>\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\n<h2>Greatest Common Factor of Monomials<\/h2>\r\nDetermining the GCF of monomials works the same way as numbers: [latex]4x[\/latex]\u00a0is the GCF of [latex]16x[\/latex]\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest term that divides exactly into both [latex]16x[\/latex] and [latex]20x^2[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Greatest Common Factor of monomials<\/h3>\r\nThe greatest common factor (GCF) of two or more monomials is the largest term that is a factor of all the monomials.\r\n\r\n<\/div>\r\nFinding the greatest common factor in a set of monomials is the same as finding the GCF of two whole numbers. The only difference is that there will be variables involved. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.\r\n<div class=\"textbox shaded\">\r\n<h3>determining 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 exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]5x\\text{ and }15[\/latex].\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td><\/td>\r\n<\/tr>\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 class=\"alignnone\" 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=\"Finding the greatest common factor between 5x (5 times x) and 15 (3 times 5). The answer is 5.\" width=\"232\" height=\"75\" \/><\/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<\/div>\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 greatest common factor 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<h4>Solution<\/h4>\r\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 774.244px;\" colspan=\"2\">Find the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 334.425px;\">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 style=\"width: 428.656px;\"><img class=\"alignnone\" 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=\"Finding the greatest common factor between 12x squared and 18x cubed. Answer is 6x squared.\" width=\"249\" height=\"113\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 334.425px;\"><\/td>\r\n<td style=\"width: 428.656px;\">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<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}&amp; 25b^{3}&amp;=5\\cdot5\\cdot{b}\\cdot{b}\\cdot{b}\\\\10b^{2}&amp;=5\\cdot2\\cdot{b}\\cdot{b}\\\\ \\text{GCF} &amp;=5\\cdot{b}\\cdot{b}\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nThe monomials have the factors [latex]5[\/latex], <i>b<\/i>, and <i>b<\/i> in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}[\/latex], or simply [latex]5b^{2}[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=5b^{2}[\/latex]\r\n\r\n<\/div>\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\n&nbsp;\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<h4>Solution<\/h4>\r\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says, \">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\">Finding the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex]<\/th>\r\n<\/tr>\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 class=\"alignnone\" 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=\"Finding the greatest common factors between 14x cubed, 8x squared, and 10x. The answer is 2x.\" width=\"296\" height=\"118\" \/><\/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<\/div>\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\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\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-2.docx\">Transcript-9.6.1-2<\/a>\r\n<h2>Factoring a Polynomial<\/h2>\r\nA polynomial is made up of the sum (or difference) of monomial terms. If all of these monomial terms have a greatest common factor, we can factor the polynomial.\u00a0<em><strong>Factoring<\/strong><\/em>\u00a0a polynomial is very helpful in simplifying and solving polynomial equations.\r\n\r\nConsider the polynomial [latex]10x^4+30x^3-45x^2[\/latex]. The monomials that make up this polynomial are\u00a0[latex]10x^4\\text{, }30x^3[\/latex] and\u00a0[latex]-45x^2[\/latex]. The GCF of these monomials is\u00a0[latex]5x^2[\/latex]. This means that\u00a0[latex]5x^2[\/latex] divides exactly into each of the monomials.\u00a0 If we were to divide the polynomial by the GCF we would get:\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}&amp; \\;\\;\\;\\;\\frac{10x^4+30x^3-45x^2}{5x^2} \\\\ &amp;=\\frac{10x^4}{5x^2}+\\frac{30x^3}{5x^2}-\\frac{45x^2}{5x^2} \\\\ &amp;=2x^2+6x-9\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nThis means that\u00a0[latex]5x^2[\/latex] and [latex]2x^2+6x-9[\/latex] are factors of [latex]10x^4+30x^3-45x^2[\/latex].\r\n\r\nIn other words,\u00a0[latex]10x^4+30x^3-45x^2=5x^2\\left (2x^2+6x-9\\right )[\/latex].\u00a0 We have factored the polynomial\u00a0[latex]10x^4+30x^3-45x^2[\/latex] using the greatest common factor.\r\n\r\nNote that we can always check our work by using the distributive property to multiply [latex]5x^2\\left (2x^2+6x-9\\right )[\/latex] to get back to the original polynomial\u00a0[latex]10x^4+30x^3-45x^2[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFactor the polynomial [latex]16x^7+24x^5-56x^3[\/latex] using the GCF.\r\n<h4>Solution<\/h4>\r\nFind the GCF of the monomials that make up the polynomial:\r\n\r\n[latex]16x^7=2\\cdot 2\\cdot 2\\cdot 2 \\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]24x^5=2\\cdot 2\\cdot 2\\cdot 3\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]56x^3=2\\cdot 2\\cdot 2\\cdot 7 \\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\nGCF = [latex]2\\cdot 2\\cdot 2\\cdot x\\cdot x\\cdot x=8x^3[\/latex]\r\n\r\n&nbsp;\r\n\r\nDivide the polynomial by the GCF:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\;\\frac{16x^7+24x^5-56x^3}{8x^3}\\\\&amp;=\\frac{16x^7}{8x^3}+\\frac{24x^5}{8x^3}-\\frac{56x^3}{8x^3}\\\\&amp;=2x^4+3x^2-7\\end{aligned}\\end{equation}[\/latex]\r\n\r\n&nbsp;\r\n\r\nWrite the polynomial as the product of factors:\r\n\r\n[latex]16x^7+24x^5-56x^3=8x^3\\left (2x^4+3x^2-7\\right )[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFactor the polynomial [latex]14x^6+21x^5-63x^4-77x^3[\/latex] using the GCF.\r\n<h4>Solution<\/h4>\r\nFind the GCF of the monomials that make up the polynomial:\r\n\r\n[latex]14x^6=2\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]21x^5=3\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]63x^4=3\\cdot 3\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]77x^3=7\\cdot 11\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\nGCF=[latex]7x^3[\/latex]\r\n\r\n&nbsp;\r\n\r\nDivide the polynomial by the GCF:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\;\\frac{14x^6+21x^5-63x^4-77x^3}{7x^3}\\\\&amp;=\\frac{14x^6}{7x^3}+\\frac{21x^5}{7x^3}-\\frac{63x^4}{7x^3}-\\frac{77x^3}{7x^3}\\\\&amp;=2x^3+3x^2-9x-11\\end{aligned}\\end{equation}[\/latex]\r\n\r\n&nbsp;\r\n\r\nWrite the polynomial as the product of factors:\r\n\r\n[latex]14x^6+21x^5-63x^4-77x^3=7x^3\\left (2x^3+3x^2-9x-11\\right )[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFactor the polynomial [latex]12x^4-21x^3+15x^2[\/latex] using the GCF.\r\n\r\n[reveal-answer q=\"hjm749\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm749\"][latex]12x^4-21x^3+15x^2=3x^2\\left (4x^2-7x+5\\right )[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFactor the polynomial [latex]18x^8+54x^5-27x^4+45x^3[\/latex] using the GCF.\r\n\r\n[reveal-answer q=\"hjm992\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm992\"][latex]18x^8+54x^5-27x^4+45x^3=9x^3\\left (2x^5+6x^2-3x+5\\right )[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that every time, the exponent on the variable in the GCF is always the lowest exponent in the polynomial. This will always be the case.\r\n\r\nWhen the leading coefficient of a polynomial is negative, we include the negative sign as part of the GCF.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFactor the polynomial [latex]-5x^6+15x^5-35x^4+45x^3[\/latex] using the GCF.\r\n<h4>Solution<\/h4>\r\nFind the GCF of the monomials that make up the polynomial:\r\n\r\n[latex]5x^6=5\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x [\/latex]\r\n\r\n[latex]15x^5=3\\cdot 5\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]35x^4=5\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\n[latex]45x^3=3\\cdot 3\\cdot 5\\cdot x\\cdot x\\cdot x[\/latex]\r\n\r\nGCF = [latex]5x^3[\/latex]\r\n\r\n&nbsp;\r\n\r\nDivide the polynomial by the GCF and <strong>include the negative sign of the leading coefficient<\/strong>:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\;\\frac{-5x^6+15x^5-35x^4+45x^3}{-5x^3}\\\\&amp;=\\frac{-5x^6}{-5x^3}+\\frac{15x^5}{-5x^3}-\\frac{35x^4}{-5x^3}+\\frac{45x^3}{-5x^3}\\\\&amp;=x^3-3x^2+7x-9\\end{aligned}\\end{equation}[\/latex]\r\n\r\n&nbsp;\r\n\r\nWrite the polynomial as the product of factors:\r\n\r\n[latex]-5x^6+15x^5-35x^4+45x^3=-5x^3\\left (x^3-3x^2+7x-9\\right )[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFactor the polynomial [latex]-6x^6+18x^5-24x^4[\/latex] using the GCF.\r\n\r\n[reveal-answer q=\"hjm781\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm781\"][latex]-6x^6+18x^5-24x^4=-6x^4\\left (x^2-3x+4\\right )[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nAs you gain confidence with factoring you will find that you can skip the division step and simply use the distributive property to factor.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFactor the polynomial [latex]18x^3-12x^2[\/latex] using the GCF.\r\n<h4>Solution<\/h4>\r\nThe GCF of the monomials is 2.\r\n\r\nSo,\u00a0[latex]18x^3-12x^4=2(\\text{a binomial})[\/latex] and we need to find that trinomial.\r\n\r\nWe ask ourselves, [latex]2[\/latex] times what equals [latex]18x^3[\/latex]?\u00a0 [latex]2(9x^3)=18x^3[\/latex] so the first term of the binomial is [latex]9x^3[\/latex].\r\n\r\n[latex]18x^3-12x^2=2(9x^3+...)[\/latex]\r\n\r\nNow we ask, what times[latex]2[\/latex] equals [latex]-12x^2[\/latex]?\u00a0 Well,\u00a0[latex]2(-6x^2)=-12x^2[\/latex], so the second term of the binomial is [latex]-6x^2[\/latex].\r\n\r\n[latex]18x^3-12x^2=2(9x^3-6x^2)[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]18x^3-12x^2=2(9x^3-6x^2)[\/latex]TRY\u00a0IT\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146341[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n[ohm_question]146339[\/ohm_question]\r\n\r\n<\/div>\r\nThe following videos provide more examples of factoring a polynomial using the distributive property.\r\n\r\nhttps:\/\/youtu.be\/25_f_mVab_4\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-3.docx\">Transcript-9.6.1-3<\/a>\r\n\r\nhttps:\/\/youtu.be\/3f1RFTIw2Ng\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-4.docx\">Transcript-9.6.1-4<\/a>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFactor the polynomial [latex]14x^4-21x^3-7x^2[\/latex] using the GCF.\r\n\r\n[reveal-answer q=\"hjm268\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm268\"][latex]14x^4-21x^3-7x^2=7x^2\\left (2x^2-3x-1\\right )[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFactor the polynomial [latex]-6x^4+21x^3-18x^2[\/latex] using the GCF.\r\n\r\n[reveal-answer q=\"hjm494\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm494\"][latex]-6x^4+21x^3-18x^2=-3x^2\\left (2x^2-7x+9\\right )[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSometimes the GCF of the monomials in a polynomial is [latex]1[\/latex]. For example, the polynomial [latex]6x^2-7x+2[\/latex] has a greatest common factor of [latex]1[\/latex] across the monomials making up the polynomial. Factoring the polynomial as [latex]6x^2-7x+2=1\\left (6x^2-7x+2\\right )[\/latex] isn't of much use. However, we will see in the next sections that there are other ways to factor a polynomial. Indeed,\u00a0[latex]6x^2-7x+2[\/latex] factors into binomial factors [latex](3x-2)(2x-1)[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Determine the greatest common factor of multiple numbers<\/li>\n<li>Determine the greatest common factor of monomials<\/li>\n<li>Factor a polynomial using the GCF<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Factor<\/strong>: a divisor<\/li>\n<li><strong>Common factor<\/strong>: a term that is a factor of two or more other terms<\/li>\n<li><strong>Greatest common factor<\/strong>: the common factor that has the highest degree<\/li>\n<li><strong>Factoring<\/strong>: writing a product or a sum as the multiplication of factors<\/li>\n<li><strong>Prime number<\/strong>: a whole number greater than or equal to 2 that has exactly 2 factors; 1 and itself<\/li>\n<li><strong>Prime factor<\/strong>: a factor that is also a prime number<\/li>\n<li><strong>Prime factorization<\/strong>: the process of factoring a number into prime factors<\/li>\n<\/ul>\n<\/div>\n<h2>Greatest Common Factor<\/h2>\n<p>In chapter 1, we defined a <em><strong>factor<\/strong><\/em> as a number that divides exactly into another number. We multiplied factors together to get a product.\u00a0\u00a0Factors are the building blocks of multiplication. They are the numbers that we can multiply together to produce another number: [latex]2[\/latex] and [latex]10[\/latex] are factors of [latex]20[\/latex], as are \u00a0[latex]4[\/latex] and [latex]5[\/latex] and [latex]1[\/latex] and [latex]20[\/latex]. To factor a number is to rewrite it as a product. For example, [latex]20=4\\cdot5[\/latex]. In algebra, we use the word factor as both a noun &#8211; something being multiplied &#8211; and as a verb &#8211; rewriting a sum or difference as a product.<\/p>\n<p>Finding a product is multiplying two or more terms together. Each term is a\u00a0<em><strong>factor<\/strong><\/em> of the product. Splitting a product into factors is called <em><strong>factoring<\/strong><\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224609\/CNX_BMath_Figure_10_06_001_img.png\" alt=\"Example showing 8 and 7 as factors of 56. Also 2x and x plus 3 as factors of 2x squared plus 6x.\" width=\"442\" height=\"255\" \/><\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine the greatest common factor of 18 and 24.<\/p>\n<h4>Solution<\/h4>\n<p>The factors of 18 are <strong>1<\/strong>, <strong>2<\/strong>, <strong>3<\/strong>, <strong>6<\/strong>, 9, and 18<\/p>\n<p>The factors of 24 are <strong>1<\/strong>, <strong>2<\/strong>,<strong> 3<\/strong>, 4, <strong>6<\/strong>, 8, 12, and 24<\/p>\n<p>The common factors are the numbers that appear in both lists: 1, 2, 3, and <strong>6<\/strong>.<\/p>\n<p>The greatest common factor is the largest of the common factors: 6.<\/p>\n<h4>Answer<\/h4>\n<p>GCF(18, 24) = 6<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine the greatest common factor of:<\/p>\n<ol>\n<li>28 and 42<\/li>\n<li>10 and 55<\/li>\n<li>64, 16, and 40<\/li>\n<li>7, 12, 15<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm157\">Show Answer<\/span><\/p>\n<div id=\"qhjm157\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>GCF(28, 42) = 14<\/li>\n<li>GCF(10, 55) = 5<\/li>\n<li>GCF(64, 16, 40) = 8<\/li>\n<li>GCF(7, 12, 15) = 1<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>A <em><strong>prime factor<\/strong><\/em> is a <em><strong>prime number<\/strong><\/em>\u2014it has only itself and 1 as factors\u2014that is a factor. The process of breaking a number down into its prime factors is called <em><strong>prime factorization<\/strong><\/em>. Prime factorization is unique. Each number has only one set of prime factors. For example, [latex]10=2\\cdot 5[\/latex] is the only way to write [latex]10[\/latex] as a product of primes.\u00a0To find the GCF, we can factor each number into its prime factors, identify the prime factors the numbers have in common, and then multiply those prime factors together.<\/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<h4>Solution<\/h4>\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<th style=\"width: 194px;\" colspan=\"3\"><strong>Finding the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex]<\/strong><strong><br \/>\n<\/strong><\/th>\n<\/tr>\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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Using a factor tree to find the factors of 24 to be 2 cubed and 3. Using a factor tree to find the factors of 36 to be 2 squared times 3 squared.\" width=\"460\" height=\"193\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Listing all the factors of 24 and 36, lining up common factors in a column.\" width=\"459\" height=\"65\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Circling the common factors of 24 and 36. They are 2 squared times 3, with a resulting greatest common factor of 12.\" width=\"459\" height=\"141\" \/><\/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 class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=42[\/latex]<\/p>\n<\/div>\n<div>Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and [latex]168[\/latex] by [latex]42[\/latex]\u2014they are both exactly divisible by this number!)<\/div>\n<p>The video that follows shows another example of finding the greatest common factor of two whole numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Determining the Greatest Common Factor\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KbBJcdDY_VE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-1.docx\">Transcript-9.6.1-1<\/a><\/p>\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<h2>Greatest Common Factor of Monomials<\/h2>\n<p>Determining the GCF of monomials works the same way as numbers: [latex]4x[\/latex]\u00a0is the GCF of [latex]16x[\/latex]\u00a0and [latex]20x^2[\/latex]\u00a0because it is the largest term that divides exactly into both [latex]16x[\/latex] and [latex]20x^2[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Greatest Common Factor of monomials<\/h3>\n<p>The greatest common factor (GCF) of two or more monomials is the largest term that is a factor of all the monomials.<\/p>\n<\/div>\n<p>Finding the greatest common factor in a set of monomials is the same as finding the GCF of two whole numbers. The only difference is that there will be variables involved. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.<\/p>\n<div class=\"textbox shaded\">\n<h3>determining 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 exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]5x\\text{ and }15[\/latex].<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168466996785\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td><\/td>\n<\/tr>\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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Finding the greatest common factor between 5x (5 times x) and 15 (3 times 5). The answer is 5.\" width=\"232\" height=\"75\" \/><\/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 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 greatest common factor 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<h4>Solution<\/h4>\n<table id=\"eip-id1168469763176\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<th style=\"width: 774.244px;\" colspan=\"2\">Find the greatest common factor of [latex]12{x}^{2}[\/latex] and [latex]18{x}^{3}[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 334.425px;\">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 style=\"width: 428.656px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Finding the greatest common factor between 12x squared and 18x cubed. Answer is 6x squared.\" width=\"249\" height=\"113\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 334.425px;\"><\/td>\n<td style=\"width: 428.656px;\">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 class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]25b^{3}[\/latex] and [latex]10b^{2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}& 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}\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>The monomials have the factors [latex]5[\/latex], <i>b<\/i>, and <i>b<\/i> in common, which means their greatest common factor is [latex]5\\cdot{b}\\cdot{b}[\/latex], or simply [latex]5b^{2}[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=5b^{2}[\/latex]<\/p>\n<\/div>\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>&nbsp;<\/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<h4>Solution<\/h4>\n<table id=\"eip-id1168469756907\" class=\"unnumbered unstyled\" summary=\"The left side says,\">\n<tbody>\n<tr>\n<th colspan=\"2\">Finding the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex]<\/th>\n<\/tr>\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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Finding the greatest common factors between 14x cubed, 8x squared, and 10x. The answer is 2x.\" width=\"296\" height=\"118\" \/><\/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 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>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-2\" 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><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-2.docx\">Transcript-9.6.1-2<\/a><\/p>\n<h2>Factoring a Polynomial<\/h2>\n<p>A polynomial is made up of the sum (or difference) of monomial terms. If all of these monomial terms have a greatest common factor, we can factor the polynomial.\u00a0<em><strong>Factoring<\/strong><\/em>\u00a0a polynomial is very helpful in simplifying and solving polynomial equations.<\/p>\n<p>Consider the polynomial [latex]10x^4+30x^3-45x^2[\/latex]. The monomials that make up this polynomial are\u00a0[latex]10x^4\\text{, }30x^3[\/latex] and\u00a0[latex]-45x^2[\/latex]. The GCF of these monomials is\u00a0[latex]5x^2[\/latex]. This means that\u00a0[latex]5x^2[\/latex] divides exactly into each of the monomials.\u00a0 If we were to divide the polynomial by the GCF we would get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}& \\;\\;\\;\\;\\frac{10x^4+30x^3-45x^2}{5x^2} \\\\ &=\\frac{10x^4}{5x^2}+\\frac{30x^3}{5x^2}-\\frac{45x^2}{5x^2} \\\\ &=2x^2+6x-9\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>This means that\u00a0[latex]5x^2[\/latex] and [latex]2x^2+6x-9[\/latex] are factors of [latex]10x^4+30x^3-45x^2[\/latex].<\/p>\n<p>In other words,\u00a0[latex]10x^4+30x^3-45x^2=5x^2\\left (2x^2+6x-9\\right )[\/latex].\u00a0 We have factored the polynomial\u00a0[latex]10x^4+30x^3-45x^2[\/latex] using the greatest common factor.<\/p>\n<p>Note that we can always check our work by using the distributive property to multiply [latex]5x^2\\left (2x^2+6x-9\\right )[\/latex] to get back to the original polynomial\u00a0[latex]10x^4+30x^3-45x^2[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Factor the polynomial [latex]16x^7+24x^5-56x^3[\/latex] using the GCF.<\/p>\n<h4>Solution<\/h4>\n<p>Find the GCF of the monomials that make up the polynomial:<\/p>\n<p>[latex]16x^7=2\\cdot 2\\cdot 2\\cdot 2 \\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]24x^5=2\\cdot 2\\cdot 2\\cdot 3\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]56x^3=2\\cdot 2\\cdot 2\\cdot 7 \\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>GCF = [latex]2\\cdot 2\\cdot 2\\cdot x\\cdot x\\cdot x=8x^3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Divide the polynomial by the GCF:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\;\\frac{16x^7+24x^5-56x^3}{8x^3}\\\\&=\\frac{16x^7}{8x^3}+\\frac{24x^5}{8x^3}-\\frac{56x^3}{8x^3}\\\\&=2x^4+3x^2-7\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Write the polynomial as the product of factors:<\/p>\n<p>[latex]16x^7+24x^5-56x^3=8x^3\\left (2x^4+3x^2-7\\right )[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Factor the polynomial [latex]14x^6+21x^5-63x^4-77x^3[\/latex] using the GCF.<\/p>\n<h4>Solution<\/h4>\n<p>Find the GCF of the monomials that make up the polynomial:<\/p>\n<p>[latex]14x^6=2\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]21x^5=3\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]63x^4=3\\cdot 3\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]77x^3=7\\cdot 11\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>GCF=[latex]7x^3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Divide the polynomial by the GCF:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\;\\frac{14x^6+21x^5-63x^4-77x^3}{7x^3}\\\\&=\\frac{14x^6}{7x^3}+\\frac{21x^5}{7x^3}-\\frac{63x^4}{7x^3}-\\frac{77x^3}{7x^3}\\\\&=2x^3+3x^2-9x-11\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Write the polynomial as the product of factors:<\/p>\n<p>[latex]14x^6+21x^5-63x^4-77x^3=7x^3\\left (2x^3+3x^2-9x-11\\right )[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Factor the polynomial [latex]12x^4-21x^3+15x^2[\/latex] using the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm749\">Show Answer<\/span><\/p>\n<div id=\"qhjm749\" class=\"hidden-answer\" style=\"display: none\">[latex]12x^4-21x^3+15x^2=3x^2\\left (4x^2-7x+5\\right )[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Factor the polynomial [latex]18x^8+54x^5-27x^4+45x^3[\/latex] using the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm992\">Show Answer<\/span><\/p>\n<div id=\"qhjm992\" class=\"hidden-answer\" style=\"display: none\">[latex]18x^8+54x^5-27x^4+45x^3=9x^3\\left (2x^5+6x^2-3x+5\\right )[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>Notice that every time, the exponent on the variable in the GCF is always the lowest exponent in the polynomial. This will always be the case.<\/p>\n<p>When the leading coefficient of a polynomial is negative, we include the negative sign as part of the GCF.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Factor the polynomial [latex]-5x^6+15x^5-35x^4+45x^3[\/latex] using the GCF.<\/p>\n<h4>Solution<\/h4>\n<p>Find the GCF of the monomials that make up the polynomial:<\/p>\n<p>[latex]5x^6=5\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]15x^5=3\\cdot 5\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]35x^4=5\\cdot 7\\cdot x\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>[latex]45x^3=3\\cdot 3\\cdot 5\\cdot x\\cdot x\\cdot x[\/latex]<\/p>\n<p>GCF = [latex]5x^3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Divide the polynomial by the GCF and <strong>include the negative sign of the leading coefficient<\/strong>:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\;\\frac{-5x^6+15x^5-35x^4+45x^3}{-5x^3}\\\\&=\\frac{-5x^6}{-5x^3}+\\frac{15x^5}{-5x^3}-\\frac{35x^4}{-5x^3}+\\frac{45x^3}{-5x^3}\\\\&=x^3-3x^2+7x-9\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Write the polynomial as the product of factors:<\/p>\n<p>[latex]-5x^6+15x^5-35x^4+45x^3=-5x^3\\left (x^3-3x^2+7x-9\\right )[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Factor the polynomial [latex]-6x^6+18x^5-24x^4[\/latex] using the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm781\">Show Answer<\/span><\/p>\n<div id=\"qhjm781\" class=\"hidden-answer\" style=\"display: none\">[latex]-6x^6+18x^5-24x^4=-6x^4\\left (x^2-3x+4\\right )[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>As you gain confidence with factoring you will find that you can skip the division step and simply use the distributive property to factor.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Factor the polynomial [latex]18x^3-12x^2[\/latex] using the GCF.<\/p>\n<h4>Solution<\/h4>\n<p>The GCF of the monomials is 2.<\/p>\n<p>So,\u00a0[latex]18x^3-12x^4=2(\\text{a binomial})[\/latex] and we need to find that trinomial.<\/p>\n<p>We ask ourselves, [latex]2[\/latex] times what equals [latex]18x^3[\/latex]?\u00a0 [latex]2(9x^3)=18x^3[\/latex] so the first term of the binomial is [latex]9x^3[\/latex].<\/p>\n<p>[latex]18x^3-12x^2=2(9x^3+...)[\/latex]<\/p>\n<p>Now we ask, what times[latex]2[\/latex] equals [latex]-12x^2[\/latex]?\u00a0 Well,\u00a0[latex]2(-6x^2)=-12x^2[\/latex], so the second term of the binomial is [latex]-6x^2[\/latex].<\/p>\n<p>[latex]18x^3-12x^2=2(9x^3-6x^2)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]18x^3-12x^2=2(9x^3-6x^2)[\/latex]TRY\u00a0IT<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/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<p>&nbsp;<\/p>\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>The following videos provide more examples of factoring a polynomial using the distributive property.<\/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<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-3.docx\">Transcript-9.6.1-3<\/a><\/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<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-9.6.1-4.docx\">Transcript-9.6.1-4<\/a><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Factor the polynomial [latex]14x^4-21x^3-7x^2[\/latex] using the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm268\">Show Answer<\/span><\/p>\n<div id=\"qhjm268\" class=\"hidden-answer\" style=\"display: none\">[latex]14x^4-21x^3-7x^2=7x^2\\left (2x^2-3x-1\\right )[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Factor the polynomial [latex]-6x^4+21x^3-18x^2[\/latex] using the GCF.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm494\">Show Answer<\/span><\/p>\n<div id=\"qhjm494\" class=\"hidden-answer\" style=\"display: none\">[latex]-6x^4+21x^3-18x^2=-3x^2\\left (2x^2-7x+9\\right )[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Sometimes the GCF of the monomials in a polynomial is [latex]1[\/latex]. For example, the polynomial [latex]6x^2-7x+2[\/latex] has a greatest common factor of [latex]1[\/latex] across the monomials making up the polynomial. Factoring the polynomial as [latex]6x^2-7x+2=1\\left (6x^2-7x+2\\right )[\/latex] isn&#8217;t of much use. However, we will see in the next sections that there are other ways to factor a polynomial. Indeed,\u00a0[latex]6x^2-7x+2[\/latex] factors into binomial factors [latex](3x-2)(2x-1)[\/latex].<\/p>\n","protected":false},"author":370291,"menu_order":9,"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-2936","chapter","type-chapter","status-publish","hentry"],"part":2917,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2936","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2936\/revisions"}],"predecessor-version":[{"id":3180,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2936\/revisions\/3180"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2917"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2936\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2936"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2936"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2936"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2936"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}