{"id":785,"date":"2024-04-29T19:09:25","date_gmt":"2024-04-29T19:09:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=785"},"modified":"2025-08-21T23:04:11","modified_gmt":"2025-08-21T23:04:11","slug":"polynomial-and-rational-expressions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-and-rational-expressions-background-youll-need-2\/","title":{"raw":"Polynomial and Rational Expressions: Background You'll Need 2","rendered":"Polynomial and Rational Expressions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Find the largest factor that common terms in an algebraic expression share.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Greatest Common Factor (GCF)<\/h2>\r\nIn algebra, simplifying expressions is a crucial skill that helps us solve complex problems more easily. One important technique in simplification is finding the largest common factor (LCF) of terms in an algebraic expression. This process allows us to factor out the greatest shared component, making the expression more compact and often easier to work with.\r\n\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\nCommon factors are factors that are shared by two or more terms in an expression. The largest factor that is common to all terms in an expression is called the Greatest Common Factor (GCF).\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>greatest common factor (GCF)<\/h3>\r\nThe <strong>Greatest Common Factor<\/strong> (also known as the Greatest Common Divisor, or GCD) is the largest factor that two or more numbers have in common. It's the highest number that divides each of the numbers without leaving a remainder.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the greatest common factor of terms in an algebraic expression\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Factor each coefficient into primes. Write all variables with exponents in expanded form.\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use factor trees to break down each number into its prime factors.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For variables, write out each instance (e.g., [latex]x^2[\/latex] as [latex]x \\cdot x[\/latex]).<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">List all factors - matching common factors in a column.\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Write out the prime factorization for each term.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">In each column, circle the common factors that are shared by all terms.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply the factors to get the final GCF.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">\r\n<ul>\r\n \t<li>Find the GCF for the numbers [latex]30[\/latex] and [latex]45.[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"332498\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"332498\"]For the numbers [latex]30[\/latex] and [latex]45[\/latex], the GCF is [latex]15[\/latex].\r\n<ul>\r\n \t<li>The factors of [latex]30[\/latex]\u00a0are [latex]1, 2, 3, 5, 6, 10, 15, 30[\/latex].<\/li>\r\n \t<li>The factors of [latex]45[\/latex]\u00a0are [latex]1, 3, 5, 9, 15, 45[\/latex].<\/li>\r\n<\/ul>\r\nNotice that [latex]15[\/latex] is the greatest common factor.[\/hidden-answer]\r\n<ul>\r\n \t<li>Find the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"887042\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"887042\"]\r\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" summary=\"Three columns are shown. The top row of the first column says, \">\r\n<tbody>\r\n<tr>\r\n<td 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;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"460\"]<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=\".\" width=\"460\" height=\"193\" \/> Factor trees for 24 and 36[\/caption]<\/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;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"459\"]<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=\".\" width=\"459\" height=\"65\" \/> List of all common factors[\/caption]<\/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;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"459\"]<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=\".\" width=\"459\" height=\"141\" \/> Common factors circled[\/caption]<\/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\n[\/hidden-answer]\r\n<ul>\r\n \t<li>Find the GCF for the expressions [latex]16x[\/latex] and [latex]20x^2[\/latex].<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"701608\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"701608\"][latex]\\begin{align*} \\text{Expressions: } &amp; 16x \\text{ and } 20x^2 \\\\ \\text{Numerical coefficients: } &amp; 16 \\text{ and } 20 \\\\ \\text{Factors of 16: } &amp; 1, 2, 4, 8, 16 \\\\ \\text{Factors of 20: } &amp; 1, 2, 4, 5, 10, 20 \\\\ \\text{Greatest common factor (numerical): } &amp; 4 \\\\ \\text{Variable parts: } &amp; x \\text{ in } 16x \\text{ (or } x^1\\text{)} \\text{ and } x^2 \\text{ in } 20x^2 \\\\ \\text{Greatest common factor (variables): } &amp; x \\\\ \\text{GCF of expressions: } &amp; 4x \\end{align*}[\/latex][\/hidden-answer]\r\n<ul>\r\n \t<li>Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex].<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"784867\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"784867\"]\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>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"296\"]<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=\".\" width=\"296\" height=\"118\" \/> Diagram showing how to determine GCF[\/caption]<\/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<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18909[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18910[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18911[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the largest factor that common terms in an algebraic expression share.<\/li>\n<\/ul>\n<\/section>\n<h2>Greatest Common Factor (GCF)<\/h2>\n<p>In algebra, simplifying expressions is a crucial skill that helps us solve complex problems more easily. One important technique in simplification is finding the largest common factor (LCF) of terms in an algebraic expression. This process allows us to factor out the greatest shared component, making the expression more compact and often easier to work with.<\/p>\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>Common factors are factors that are shared by two or more terms in an expression. The largest factor that is common to all terms in an expression is called the Greatest Common Factor (GCF).<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>greatest common factor (GCF)<\/h3>\n<p>The <strong>Greatest Common Factor<\/strong> (also known as the Greatest Common Divisor, or GCD) is the largest factor that two or more numbers have in common. It&#8217;s the highest number that divides each of the numbers without leaving a remainder.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the greatest common factor of terms in an algebraic expression<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Factor each coefficient into primes. Write all variables with exponents in expanded form.\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use factor trees to break down each number into its prime factors.<\/li>\n<li class=\"whitespace-normal break-words\">For variables, write out each instance (e.g., [latex]x^2[\/latex] as [latex]x \\cdot x[\/latex]).<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">List all factors &#8211; matching common factors in a column.\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Write out the prime factorization for each term.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">In each column, circle the common factors that are shared by all terms.<\/li>\n<li class=\"whitespace-normal break-words\">Multiply the factors to get the final GCF.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<ul>\n<li>Find the GCF for the numbers [latex]30[\/latex] and [latex]45.[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q332498\">Show Answer<\/button><\/p>\n<div id=\"q332498\" class=\"hidden-answer\" style=\"display: none\">For the numbers [latex]30[\/latex] and [latex]45[\/latex], the GCF is [latex]15[\/latex].<\/p>\n<ul>\n<li>The factors of [latex]30[\/latex]\u00a0are [latex]1, 2, 3, 5, 6, 10, 15, 30[\/latex].<\/li>\n<li>The factors of [latex]45[\/latex]\u00a0are [latex]1, 3, 5, 9, 15, 45[\/latex].<\/li>\n<\/ul>\n<p>Notice that [latex]15[\/latex] is the greatest common factor.<\/p><\/div>\n<\/div>\n<ul>\n<li>Find the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q887042\">Show Answer<\/button><\/p>\n<div id=\"q887042\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" 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;\">\n<figure style=\"width: 460px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\".\" width=\"460\" height=\"193\" \/><figcaption class=\"wp-caption-text\">Factor trees for 24 and 36<\/figcaption><\/figure>\n<\/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;\">\n<figure style=\"width: 459px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\".\" width=\"459\" height=\"65\" \/><figcaption class=\"wp-caption-text\">List of all common factors<\/figcaption><\/figure>\n<\/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;\">\n<figure style=\"width: 459px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\".\" width=\"459\" height=\"141\" \/><figcaption class=\"wp-caption-text\">Common factors circled<\/figcaption><\/figure>\n<\/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<\/div>\n<\/div>\n<ul>\n<li>Find the GCF for the expressions [latex]16x[\/latex] and [latex]20x^2[\/latex].<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q701608\">Show Answer<\/button><\/p>\n<div id=\"q701608\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align*} \\text{Expressions: } & 16x \\text{ and } 20x^2 \\\\ \\text{Numerical coefficients: } & 16 \\text{ and } 20 \\\\ \\text{Factors of 16: } & 1, 2, 4, 8, 16 \\\\ \\text{Factors of 20: } & 1, 2, 4, 5, 10, 20 \\\\ \\text{Greatest common factor (numerical): } & 4 \\\\ \\text{Variable parts: } & x \\text{ in } 16x \\text{ (or } x^1\\text{)} \\text{ and } x^2 \\text{ in } 20x^2 \\\\ \\text{Greatest common factor (variables): } & x \\\\ \\text{GCF of expressions: } & 4x \\end{align*}[\/latex]<\/div>\n<\/div>\n<ul>\n<li>Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[\/latex].<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q784867\">Show Answer<\/button><\/p>\n<div id=\"q784867\" class=\"hidden-answer\" style=\"display: none\">\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>\n<figure style=\"width: 296px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" 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=\".\" width=\"296\" height=\"118\" \/><figcaption class=\"wp-caption-text\">Diagram showing how to determine GCF<\/figcaption><\/figure>\n<\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18909\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18909&theme=lumen&iframe_resize_id=ohm18909&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" 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