{"id":52522,"date":"2019-07-30T20:01:46","date_gmt":"2019-07-30T20:01:46","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=52522"},"modified":"2026-03-25T10:51:26","modified_gmt":"2026-03-25T15:51:26","slug":"rational-expressions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/rational-expressions\/","title":{"rendered":"Rational Expressions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_fNPO4PTShak\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_fNPO4PTShak\" data-source-videoID=\"fNPO4PTShak\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Rational Expressions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Rational Expressions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_fNPO4PTShak:hover {cursor:pointer;} img#videoThumbnailImage_fNPO4PTShak {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/364-Rational-Expressions-2-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_fNPO4PTShak\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_fNPO4PTShak\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Rational Expressions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_fNPO4PTShak\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_fNPO4PTShak\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_fNPO4PTShak\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction OmL_Function() {\n  var x = document.getElementById(\"OmL\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"OmL_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"OmL\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Reviewing_Terminology\" class=\"smooth-scroll\">Reviewing Terminology<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Addition_and_Subtraction_of_Rational_Expressions\" class=\"smooth-scroll\">Addition and Subtraction of Rational Expressions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Multiplication_and_Division_of_Rational_Expressions\" class=\"smooth-scroll\">Multiplication and Division of Rational Expressions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rational_Expression_Practice_Questions\" class=\"smooth-scroll\">Rational Expression Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about rational expressions.<\/p>\n<h2><span id=\"Reviewing_Terminology\" class=\"m-toc-anchor\"><\/span>Reviewing Terminology<\/h2>\n<p>\nBefore we talk about what rational expressions are and the operations that can be performed with them, it may be a good idea to review some terminology.<\/p>\n<h3><span id=\"Polynomials\" class=\"m-toc-anchor\"><\/span>Polynomials<\/h3>\n<p>\nA <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/polynomials\/\">polynomial<\/a> is a group of algebraic or numeric terms that are joined by the operations of addition or subtraction. There are different types of polynomials based on the number of terms that are present:<\/p>\n<h4 style=\"text-align: center; margin-bottom: 0.5em;\"><span id=\"Types_of_Polynomials\" class=\"m-toc-anchor\"><\/span>Types of Polynomials<\/h4>\n<table class=\"ATable\" style=\"margin: auto; width: 65%;\">\n<thead>\n<tr style=\"height: 40px\">\n<th style=\"vertical-align: middle;\"><strong>Examples<\/strong><\/th>\n<th style=\"vertical-align: middle; line-height: 20px;\"><strong># of Terms<\/strong><\/th>\n<th style=\"vertical-align: middle;\"><strong>Type<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height:55px\">\n<td>\\(4\\), \\(5x\\), \\(3x^4\\), \\(8xy^2\\)<\/td>\n<td>1<\/td>\n<td>Monomial<\/td>\n<\/tr>\n<tr style=\"height:55px\">\n<td>\\((x+3)\\), \\((x^2-1)\\), \\((3xy+2y)\\)<\/td>\n<td>2<\/td>\n<td>Binomial<\/td>\n<\/tr>\n<tr style=\"line-height:40px\">\n<td>\\((x^2+5x+6)\\),<br \/>\n\\((x^2-2xy+y^2)\\)<\/td>\n<td>3<\/td>\n<td>Trinomial<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Rational Expressions<\/h3>\n<p>\nA <strong>rational expression<\/strong> is nothing more than a ratio of polynomials. As you know from previous practice with <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/ratios\/\">ratios<\/a>, you cannot divide by 0. It is important to keep this in mind when dealing with rational expressions because allowing a value of 0 in the denominator would create an expression that is \u201cundefined.\u201d <\/p>\n<p>Using function notation for polynomials, such as \\(p(x)\\) and \\(g(x)\\), a rational expression can be defined like this:<\/p>\n<div class=\"examplesentence\"><span style=\"font-size: 120%;\">\\(\\frac{p(x)}{g(x)}\\)<\/span> where \\(g(x)\\neq 0\\)<\/div>\n<p>\n&nbsp;<br \/>\nHere\u2019s an example:<\/p>\n<div class=\"examplesentence\"><span style=\"font-size: 120%;\">\\(\\frac{5x}{x-2}\\)<\/span><\/div>\n<p>\n&nbsp;<br \/>\nThis example shows a rational expression with a monomial, \\(5x\\), in the numerator and a binomial, \\((x-2)\\), in the denominator. The value \\(x=2\\) is the excluded value, as it would result in a denominator of 0. This expression cannot be simplified further.  <\/p>\n<div class=\"examplesentence\"><span style=\"font-size: 120%;\">\\(\\frac{5x}{x-2}\\)<\/span>, \\(x\\neq 2\\)<\/span><\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Addition_and_Subtraction_of_Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Addition and Subtraction of Rational Expressions<\/h2>\n<p>\nRational expressions cannot be added or subtracted unless they share a common denominator. Algebraic rules allow us to adjust fractions to create common denominators as long as we make the same adjustment to the numerator. Let\u2019s look at an example with fractions:<\/p>\n<div class=\"examplesentence\"><span style=\"font-size: 120%;\">\\(\\frac{1}{5}+\\frac{3}{7}\\)<\/span><\/div>\n<p>\n&nbsp;<br \/>\nIn order to add \\(\\frac{1}{5} + \\frac{3}{7}\\), we must create a common denominator. Specifically, we need to determine the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/greatest-common-factor\/\">least common denominator<\/a>, meaning the smallest multiple of 5 and 7. In this case, that number is 35. The adjustment to each fraction that needs to be made to create the common denominator is:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{1}{5} (\\frac{7}{7}) + \\frac{3}{7} (\\frac{5}{5})\\)<br \/>\n\\(\\frac{7}{35}+\\frac{15}{35}\\)<br \/>\n\\(=\\frac{22}{35}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe need to multiply the first denominator of the first expression by 7 to get to 35, but we also must multiply the numerator by the same value. Because we have simply created an equivalent fraction to allow us to add. Likewise, the second expression must be multiplied by \\(\\frac{5}{5}\\), in order to create 35 in the denominator. After these adjustments are made and the denominators are the same, simplify the numerators:<\/p>\n<div class=\"examplesentence\"style=\"font-size: 120%;\">\\(\\frac{7+15}{35} \\rightarrow \\frac{22}{35}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRational expressions are added and subtracted the same way. Typically, the expressions need to be factored before the least common denominator can be determined and domain restrictions (excluded values) should be noted. Consider this example:<\/p>\n<div class=\"examplesentence\"style=\"font-size: 120%;\">\\(\\frac{3x}{x-2} + \\frac{5}{x^2-x-2}\\)<br \/>\n\\(\\frac{3x}{x-2} + \\frac{5}{(x-2)(x+1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we want to determine the lowest common denominator. What is the smallest multiple of \\((x-2)\\) and \\((x-2)(x+1)\\)?<\/p>\n<p>Alright now that we have our equations written out, we want to make sure that we don\u2019t have any domains that need to be excluded. Which, we do. Remember we don\u2019t want 0 in the denominator position. So, in these scenarios, we know that \\(x\\neq 2\\), or over here, -1. If \\(x=-1\\) this would end up being 0, multiplied by another term, still remains 0. The 0 in the denominator, we can\u2019t have that. Over here, if \\(x=2\\), \\(2-2=0\\), again, we can\u2019t have a 0 in the denominator, so these are our two terms, our domains that need to be excluded.<\/p>\n<p>Alright, now we need to adjust the first expression by multiplying by the factor needed to match the least common denominator. So if we want our first term here, to match this term over here in the denominator position, we\u2019re going to multiply by \\(x+1\\) in the numerator and the denominator. <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{3x}{x-2} \\frac{(x+1)}{(x+1)} + \\frac{5}{(x-2)(x+1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we\u2019re going to rewrite the expression as a fraction, and simplify the numerator. And now we have our answer:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{3x^{2}+3x}{(x+1)(x-2)}+\\frac{5}{(x-2)(x+1)}\\)\\(=\\frac{3x^{2}+3x+5}{(x-2)(x+1)}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Multiplication_and_Division_of_Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Multiplication and Division of Rational Expressions<\/h2>\n<h3><span id=\"Multiplying_Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Multiplying Rational Expressions<\/h3>\n<p>\nHere are the three steps to multiplying rational expressions. Now, remember, when multiplying fractions, numerators and denominators are multiplied straight across.<\/p>\n<div class=\"transcriptcallout\" style=\"text-align: left;\">\n<ol style=\"list-style-type: none; margin-left: 1.5em; margin-bottom: 0.5em;\">\n<li style=\"margin-bottom: 10px;\"><strong>Step #1:<\/strong> Factor the numerator and denominator of each expression being multiplied.<\/li>\n<li style=\"margin-bottom: 10px;\"><strong>Step #2:<\/strong> Simplify by canceling out common factors from the numerator and the denominator.<\/li>\n<li><strong>Step #3:<\/strong> The final answer is what is left after canceling. You may be asked to include domain restrictions with your solution.<\/li>\n<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s use these steps to solve an example problem:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\n\\(\\frac{3x}{4x-8}\\cdot \\frac{2x^{2}-4x}{9x}\\)<br \/>\n\\(\\frac{3x}{4(x-2)}\\cdot \\frac{2x(x-2)}{9x}\\)<br \/>\n\\(\\frac{6x^{2}(x-2)}{36x(x-2)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, because we have like terms in the numerator and the denominator position, we\u2019re able to cancel them out. That leaves us with: <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\n\\(\\frac{6x^2}{36x}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>But we can simplify this even further, remember, 6 is a <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/factors\/\">factor<\/a> of 36, so let\u2019s simplify: <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{x^2}{6x}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd yet, we can simplify this again, remember, you have an \\(x\\) in the numerator and an \\(x\\) in the denominator, so let\u2019s simplify: <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{x}{6}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd now we have our answer, \\(\\frac{x}{6}\\). But that\u2019s not the complete answer. Remember, we have some domain that we have to exclude. Up here, \\(x\\neq 2\\) because \\(2\\times 4-8=0\\). And we can\u2019t have a 0 in the denominator. So 2 is out, \\(x\\neq 2\\). Also, \\(x\\neq 0\\), because \\(0\\times 9=0\\), and again, give us a 0 in the denominator. So the domains we have to exclude from this answer are 2 and 0. So our answer is \\(\\frac{x}{6},x\\neq 2,0\\).<\/p>\n<h3><span id=\"Dividing_Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Dividing Rational Expressions<\/h3>\n<p>\nDividing rational expressions includes one extra step at the beginning of the process. When dividing by a fraction, it is the same as multiplying by the reciprocal of the second fraction. You can remember this rule as, \u201cKeep, Change, and Flip\u201d which translates to keep the first fraction, change the operation to multiplication, and take the reciprocal (or flip) of the second fraction.<\/p>\n<p>Keep in mind that domain restrictions must be considered from both the numerator and denominator of the second fraction because of the \u201cflip\u201d in the division process.<\/p>\n<p>Here\u2019s an example:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\n\\(\\frac{9x^2}{x^2+12x+36} \\div \\frac{12x}{x^2+6x} \\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, remember our three steps: <em>keep<\/em> the first fraction, <em>change<\/em> the operation, and then <em>flip<\/em>. Here we go: <\/p>\n<div class=\"examplesentence\">\n<span  style=\"font-size: 120%;\">\\(\\frac{9x^2}{x^2+12x+36} \\times \\frac{x^2+6x}{12x}\\)<\/span>, \\(x\\neq 0\\), \\(-6\\)<\/div>\n<p>\n&nbsp;<br \/>\nHere is now where we multiply, cause we kept the first fraction, we changed to multiplication, and then we flipped the fraction over here. So, time to multiply. <\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\n\\(\\frac{9x^{2}}{(x+6)(x+6)}\\cdot \\frac{(x+6)}{12}=\\frac{3x^{2}}{4(x+6)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo now we have our answer: \\(\\frac{3x^{2}}{4(x+6)}\\).<\/p>\n<p>But remember, that\u2019s not our complete answer if we don\u2019t include our restricted domain, we have \\(x\\neq 0\\), and \\(x\\neq -6\\). Remember we have to make sure that we don\u2019t have a 0 in the denominator or the numerator of our second term.<\/p>\n<p>I hope this review was helpful! See you next time!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Rational_Expression_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Rational Expression Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich polynomial is considered a binomial?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(2x\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(x^2+2x-4\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(100\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-4\">\\(3x-7\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A polynomial consists of one or more monomials combined by addition or subtraction. <\/p>\n<ul style=\"margin-left: 1.5em\">\n<li>A monomial has one term, such as \\(2x\\)<\/li>\n<li>A binomial has two terms, such as \\(3x\u22127\\)<\/li>\n<li>A trinomial has three terms, such as \\(x^2+2x-4\\)<\/li>\n<\/ul>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nAdd the following polynomials:<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{x+2}{3x}+\\dfrac{x-3}{6x}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(3x+1\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\"><span style=\"font-size: 120%\">\\(\\frac{3x+1}{6x}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-2-3\"><span style=\"font-size: 120%\">\\(\\frac{6x+1}{3x}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(6x\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, rewrite the expression with a common denominator. In this case, we need to multiply the numerator and denominator by 2 in the first fraction so that our denominator becomes \\(6x\\).<\/p>\n<p style=\"text-align: center; line-height: 70px\">\\(\\dfrac{x+2}{3x}+\\dfrac{x-3}{6x}\\)<br \/>\n\\(\\dfrac{(x+2)(2)}{(3x)(2)}+\\dfrac{(x+3)}{(6x)}\\)<br \/>\n\\(\\dfrac{2x+4}{6x}+\\dfrac{x-3}{6x}\\)<\/p>\n<p>Add the numerators, and combine like terms.<\/p>\n<p style=\"text-align: center; line-height: 70px\">\\(\\dfrac{2x+4}{6x}+\\dfrac{x-3}{6x}\\)<br \/>\n\\(\\dfrac{3x+1}{6x}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat are the domain restrictions for the following expression? <\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{7x+2}{x^2-4}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">\\(x\\neq2\\) and \\(x\\neq-2\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(x\\neq2\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(x\\neq-2\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(x\\neq7\\) and \\(x\\neq-4\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A rational expression cannot have a zero in the denominator. For our expression, there are two values for \\(x\\) that will produce a zero in the denominator. It can be helpful to break apart the denominator \\((x^2-4)\\) into \\((x+2)(x-2)\\) in order to identify these domain restrictions.<\/p>\n<p>Now that the denominator is factored, we can see that if \\(x\\) is 2, the term on the left will be zero. Similarly, if \\(x\\) is \u22122, the term on the right will be zero. Therefore, \\(x\\) cannot be 2 or \u22122. These are referred to as domain restrictions:<\/p>\n<p style=\"text-align: center\">\\(x\\neq2\\) and \\(x\\neq-2\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMultiply the following polynomials:<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{12x^2}{5y^3}\\times\\dfrac{20y^4}{6x^3}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\"><span style=\"font-size: 120%\">\\(\\frac{8y}{x^2}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-4-2\"><span style=\"font-size: 120%\">\\(\\frac{6y}{x}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-4-3\"><span style=\"font-size: 120%\">\\(\\frac{8y}{y^2}\\)<\/span><\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\"><span style=\"font-size: 120%\">\\(\\frac{8y}{x}\\)<\/span><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When multiplying fractions, simply multiply straight across:<\/p>\n<p style=\"text-align: center; line-height: 70px\">\\(\\dfrac{12x^2}{5y^3}\\times\\dfrac{20y^4}{6x^3}\\)\\(\\:=\\dfrac{240x^2y^4}{30x^3y^3}\\)<\/p>\n<p>Simplify by canceling out common factors that appear in the numerator and denominator.<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{240x^2y^4}{30x^3y^3}\\)\\(\\:=\\dfrac{8y}{x}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDivide the following polynomials:<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{x^2-x-12}{3x-15}\\div\\dfrac{x^2-9}{24x-72}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\"><span style=\"font-size: 120%\">\\(\\frac{8(x-4)}{x-5}\\)<\/span>\\(x\\neq 5,3,-3\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\"><span style=\"font-size: 120%\">\\(\\frac{x-3}{x-5}\\)<\/span>\\( x\\neq 5\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\"><span style=\"font-size: 120%\">\\(\\frac{8(x+4)}{x+5}\\)<\/span> \\(x\\neq -3\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\"><span style=\"font-size: 120%\">\\(\\frac{x+4}{x-5}\\)<\/span> \\(x\\neq 5,4,6\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p><strong>Step 1:<\/strong> Keep, change, flip<br \/>\nKeep the fraction on the left in its original form. Then, change the division sign to a multiplication sign. Finally, flip the second fraction.<\/p>\n<p style=\"text-align: center; line-height: 70px\">\\(\\dfrac{x^2-x-12}{3x-15}\\times\\dfrac{24x-72}{x^2-9}\\)<\/p>\n<p><strong>Step 2:<\/strong> Factor<br \/>\nFactoring makes the process of canceling more straight forward later on.<\/p>\n<p style=\"text-align: center; line-height: 70px\">\\(\\dfrac{(x-4)(x+3)}{3(x-5)}\\times\\dfrac{24(x-3)}{(x+3)(x-3)}\\)<\/p>\n<p><strong>Step 3:<\/strong> Identify the restrictions<br \/>\nRestrictions are the values for \\(x\\) that will produce a zero in the denominator.<\/p>\n<p>We know that \\(x\\) cannot be 5 because that would produce a zero in the denominator of the first fraction. Similarly, \\(x\\) cannot be 3 or \u22123 because that would produce a zero in the denominator of the second fraction. The restrictions are \\(x\u22605, 3, -3\\).<\/p>\n<p><strong>Step 4:<\/strong> Simplify<br \/>\nCancel out common factors in the numerator and denominator. <\/p>\n<p style=\"text-align: center\">\\(\\dfrac{8(x-4)}{x-5}\\)<\/p>\n<p>This gives us the final answer:<\/p>\n<p style=\"text-align: center\"><span style=\"font-size: 120%\">\\(\\frac{8(x-4)}{x-5}\\)<\/span> \\(x \u2260 5,3,-3\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":91735,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-52522","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-algebra-basics-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52522","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=52522"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52522\/revisions"}],"predecessor-version":[{"id":278971,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52522\/revisions\/278971"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/91735"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=52522"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}