{"id":38230,"date":"2018-03-07T19:32:00","date_gmt":"2018-03-07T19:32:00","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=38230"},"modified":"2026-03-26T11:59:05","modified_gmt":"2026-03-26T16:59:05","slug":"reducing-rational-expressions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/reducing-rational-expressions\/","title":{"rendered":"Reducing Rational Expressions &#8211; Polynomials"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_nrMdIEB422c\" 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_nrMdIEB422c\" data-source-videoID=\"nrMdIEB422c\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Reducing Rational Expressions &#8211; Polynomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Reducing Rational Expressions &#8211; Polynomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_nrMdIEB422c:hover {cursor:pointer;} img#videoThumbnailImage_nrMdIEB422c {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/183-reducing-rational-expressions-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_nrMdIEB422c\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_nrMdIEB422c\" 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\",\"Reducing Rational Expressions &#8211; Polynomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_nrMdIEB422c\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_nrMdIEB422c\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_nrMdIEB422c\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction qkW_Function() {\n  var x = document.getElementById(\"qkW\");\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=\"qkW_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=\"qkW\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_a_Rational_Expression\" class=\"smooth-scroll\">What is a Rational Expression?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_to_Reduce_Rational_Expressions\" class=\"smooth-scroll\">How to Reduce 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>Hey guys! Welcome to this video on simplifying rational expressions.<\/p>\n<h2><span id=\"What_is_a_Rational_Expression\" class=\"m-toc-anchor\"><\/span>What is a Rational Expression?<\/h2>\n<p>\nA <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/rational-expressions\/\">rational expression<\/a> just refers to a <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/fractions\/\">fraction<\/a> with a polynomial in the numerator, and a polynomial in the denominator.<\/p>\n<h3><span id=\"Rational_Expression_Examples\" class=\"m-toc-anchor\"><\/span>Rational Expression Examples<\/h3>\n<p>\nHere are a few examples:<\/p>\n<ol>\n<li style=\"margin-bottom: 1.25em;\"><span style=\"font-size: 125%;\">\\(\\frac{x^{2}-16}{x+4}\\)<\/span><\/li>\n<li style=\"margin-bottom: 1.25em;><span style=\"font-size: 125%;\">\\(\\frac{x^{2}-2x-8}{x^{2}-9x+20}\\)<\/span><\/li>\n<li><span style=\"font-size: 125%;\">\\(\\frac{4x+4}{x^{4}-x^{2}}\\)<\/span><\/li>\n<\/ol>\n<p>One thing that we need to keep in mind when working with rational expression is that divisibility by 0 is not allowed. Just like when dealing with regular numbers, you cannot divide by 0. So, when dealing with a rational expression, we always assume that whatever \\(x\\) is, it will not give us division by 0.<\/p>\n<h2><span id=\"How_to_Reduce_Rational_Expressions\" class=\"m-toc-anchor\"><\/span>How to Reduce Rational Expressions<\/h2>\n<p>\nAlright let\u2019s take a look at how to reduce a rational expression. We&#8217;re actually doing the same thing we would do when reducing a regular fraction.<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nSo, let\u2019s say we have \\(\\frac{18}{8}\\). When we reduce this, we can cancel our like terms. So we can rewrite this as:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{9(2)}{4(2)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can cancel our 2s here giving us:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{9}{4}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo now we have a fraction reduced down to its simplest form. There is not another number that both our numerator and denominator are divisible by.<\/p>\n<p>It works the same way with a rational expression.<\/p>\n<p>Let\u2019s try reducing our first example.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{x^{2}-16}{x+4}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can rewrite our numerator, once we factor this out, as:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{(x+4)(x-4)}{(x+4)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd once we do this, we can see that our \\((x+4)\\)s will cancel out. So we cancel that out, leaving us with \\(x-4\\)<\/p>\n<p>Now, we need to be careful when canceling terms. The only reason we were able to cancel out our \\((x+4)\\)s here was because they are both being multiplied in the numerator and the denominator. This would not work if our top was: \\(\\frac{(x+4)+(x-4)}{(x+4)}\\).<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s now move on to our second example, which is a bit trickier.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{x^{2}-2x-8}{x^{2}-9x+20}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can do the same thing that we did in our first example by rewriting our numerator and denominator. So that would give us:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(=\\frac{(x-4)(x+2)}{(x-5)(x-4)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo, we can go ahead here and cancel our \\((x-4)\\)s, which would leave us with:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(=\\frac{x+2}{x-5}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nFor our last example we have:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(\\frac{4x+4}{x^{4}-x^{2}}\\)<\/div>\n<p>\n&nbsp;<br \/>\nTo reduce it, we can rewrite our numerator by factoring out a 4, which would give us \\(4(x+1)\\). In the denominator we can factor out an \\(x^{2}\\), which would give us \\(x^{2}(x^{2}-1)\\).<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(=\\frac{4(x+1)}{x^{2}(x^{2}-1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nBut, notice, we can factor this out even further so we can get something to cancel out with our numerator here.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(=\\frac{4(x+1)}{x^{2}(x+1)(x-1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAt this point, we can cancel out our \\((x+1)\\)s here, leaving us with:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 125%;\">\\(=\\frac{4}{x^{2}(x-1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThere is no further reduction we can do, so we now have it in our simplest form.<\/p>\n<hr>\n<p>\nI hope that this video has been helpful for you!<\/p>\n<p>See you guys 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 \/>\nCan the fraction below be reduced?<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{x+8}{x-8}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">Yes<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">No<\/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>You cannot cancel out anything between the numerator and the denominator because we\u2019re adding 8 in the numerator and subtracting by 8 in the denominator. This means that \\(\\large{\\frac{x+8}{x-8}}\\) is as simplified as it\u2019s going to get.<\/p>\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 \/>\nDoes \\(\\large{\\frac{(x+5)-(2x+1)}{10(2x+1)}}\\) reduce to \\(\\large{\\frac{x+5}{10}}\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">Yes<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">No<\/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>Because we are subtracting \\(2x+1\\) from \\(x+5\\), we have to distribute the negative sign and simplify the numerator first:<\/p>\n<p style=\"text-align: center; line-height: 60px\">\\(\\dfrac{(x+5)-(2x+1)}{10(2x+1)}=\\dfrac{x+5-2x-1}{20x+10}\\)\\(\\:=\\dfrac{-x+4}{20x+10}=\\dfrac{-x+4}{10(2x+1)}\\)<\/p>\n<p>This expression cannot be simplified any further or reduced.<\/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 \/>\nBrandon is solving a problem on his homework. He\u2019s been asked to reduce the expression \\(\\large{\\frac{7x^2+16x}{4x}}\\). Here are the steps that he took and his final answer:<\/p>\n<h4 style=\"font-weight: 600 !important;\">Step 1:<\/h4>\n<p style=\"margin-left: 0.75em\">\\(\\dfrac{7x^2+16x}{4x}=\\dfrac{x(7x+16)}{4x}\\)<\/p>\n<p><h4 style=\"font-weight: 600 !important;\">Step 2:<\/h4>\n<p style=\"margin-left: 0.75em\">\\(\\dfrac{7x+16}{4}\\)<\/p>\n<h4 style=\"font-weight: 600 !important;\">Step 3:<\/h4>\n<p style=\"margin-left: 0.75em\">\\(\\dfrac{7x+4\\times4}{4}=7x+4\\)<\/p>\n<p>Where did Brandon first make a mistake while solving?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">Between step 1 and step 2<\/div><div class=\"PQ\"  id=\"PQ-3-2\">Between step 2 and step 3<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\">Between step 3 and his final answer<\/div><div class=\"PQ\"  id=\"PQ-3-4\">He did not make any mistakes<\/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>In hopes to reduce the original expression to the point where the expression no longer looks like a fraction, Brandon made an &#8220;illegal move.&#8221; Even though it is technically correct that 16 can be written as \\(4\\times4\\), he cannot cancel out one of these 4s without canceling a 4 from the additive term (\\(7x\\)).<\/p>\n<p>In other words, if the expression were \\(\\large{\\frac{8x+16}{4}}\\) instead, he could have done the following:<\/p>\n<p style=\"text-align: center;\">\\(\\dfrac{8x+16}{4}=\\dfrac{4\\times2x+4\\times4}{4}\\)\\(\\:=2x+4\\)<\/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 \/>\nSimplify the following expression:<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{x^3+6x^2-16x}{x-6x-40}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">\\(\\large{\\frac{x(x-8)(x+2)}{-7x+40}}\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(\\large{\\frac{x(x+8)(x-2)}{-5(x-8)}}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">\\(\\large{\\frac{x(x-2)}{-5}}\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(\\large{\\frac{x(x+2)}{x-1}}\\)<\/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>First, simplify the numerator:<\/p>\n<p style=\"text-align: center; line-height: 65px\">\\(\\dfrac{x^3+6x^2-16x}{x-6x-40}=\\dfrac{x(x^2+6x-16)}{x-6x-40}\\)\\(\\:=\\dfrac{x(x+8)(x-2)}{x-6x-40}\\)<\/p>\n<p>Now, work on the denominator:<\/p>\n<p style=\"text-align: center; line-height: 65px\">\\(\\dfrac{x(x+8)(x-2)}{x-6x-40}=\\dfrac{x(x+8)(x-2)}{-5x-40}\\)\\(\\:=\\dfrac{x(x+8)(x-2)}{-5(x+8)}\\)<\/p>\n<p>Finally, notice that we can cancel out the \\((x+8)\\) term:<\/p>\n<p style=\"text-align: center; line-height: 65px\">\\(\\dfrac{x(x+8)(x-2)}{-5(x+8)}=\\dfrac{x(x-2)}{-5}\\)<\/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 \/>\nSimplify the following expression:<\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{(x+7)+(5x+5)}{x^2+2x}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\(\\large{\\frac{6(6x+1)}{x}}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(\\large{\\frac{x+6}{2}}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(\\large{\\frac{5(x+7)(x+1)}{x(x+2)}}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">\\(\\large{\\frac{6}{x}}\\)<\/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>First, simplify the numerator:<\/p>\n<p style=\"text-align: center; line-height: 65px\">\\(\\dfrac{(x+7)+(5x+5)}{x^2+2x}=\\dfrac{x+7+5x+5}{x^2+2x}\\)\\(\\:=\\dfrac{6x+12}{x^2+2x}=\\dfrac{6(x+2)}{x^2+2x}\\)<\/p>\n<p>Now, we\u2019ll simplify the denominator and see if we can reduce the expression:<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{6(x+2)}{x^2+2x}=\\dfrac{6(x+2)}{x(x+2)}\\)\\(\\:=\\dfrac{6}{x}\\)<\/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":91225,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-38230","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-manipulating-expressions-1","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"A rational expression has polynomials in both the numerator and denominator. 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