{"id":121222,"date":"2022-05-06T13:26:15","date_gmt":"2022-05-06T18:26:15","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=121222"},"modified":"2026-03-28T11:51:48","modified_gmt":"2026-03-28T16:51:48","slug":"dividing-polynomials-by-monomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/dividing-polynomials-by-monomials\/","title":{"rendered":"Dividing Polynomials by Monomials"},"content":{"rendered":"<h1>Dividing Polynomials by Monomials<\/h1>\n\n\t\t\t<div id=\"mmDeferVideoEncompass_M8Ws_JL5dTA\" 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_M8Ws_JL5dTA\" data-source-videoID=\"M8Ws_JL5dTA\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Dividing Polynomials by Monomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Dividing Polynomials by Monomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_M8Ws_JL5dTA:hover {cursor:pointer;} img#videoThumbnailImage_M8Ws_JL5dTA {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/2341-thumbnail-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_M8Ws_JL5dTA\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_M8Ws_JL5dTA\" 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\",\"Dividing Polynomials by Monomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_M8Ws_JL5dTA\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_M8Ws_JL5dTA\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_M8Ws_JL5dTA\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello! Welcome to this video on <strong>dividing polynomials by monomials<\/strong>. Remember, <strong>monomials<\/strong> are mathematical expressions with one term, like \\(2x^{2}\\), and <strong>polynomials<\/strong> are mathematical expressions with two or more terms, like \\(4y^{2}+3y\\). In this video, we will walk through a few examples of how to divide polynomials by monomials.<\/p>\n<p>Let\u2019s start with a simple example.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{3x^{2}+4x}{x}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Remember, fraction bars always mean divide, so this just means \\((3x^{2}+4x)\\div x\\).<\/p>\n<p>When dividing polynomials by monomials, we want to divide each term of the polynomial by the monomial. Let\u2019s start with \\(3x^{2}\\). Remember, to divide monomials by one another, you first divide the coefficients. It doesn&#8217;t look like there&#8217;s a coefficient here \\((x)\\), but remember, if there&#8217;s no coefficient in front of a variable it&#8217;s assumed to be 1. <\/p>\n<p>Remember when we have variables, the same variable raised to exponents, you simply subtract the exponents when dividing them. Remember, any variable that&#8217;s not raised to a power is really raised to the first power, so this is the same as, \\(x^{2}\\div x^{1}\\), and \\(2-1=1\\), so we simply have \\(x^{1}\\). And \\(3x^{1}\\), of course, can simplify to \\(3x\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{3x^{2}}{x}=3x^{1}=3x\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Now let&#8217;s move on to \\(4\\)x.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{4x}{x}=4\\cdot 1=4\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Finally, we simply add the two terms.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{3x^{2}+4x}{x}=3x+4\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Let\u2019s try another example!<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\((28y^{5}-14y^{4}-21y^{3}+35y^{2})\\)\\(\\div 7y^{2}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Remember, we always want to divide each term of the polynomial by the monomial. So let&#8217;s start with \\(28y^{5}\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{28y^{5}}{7y^{2}}=4y^{3}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Now let&#8217;s move on to the next term. Signs are very important in math, so always grab the minus sign (or negative sign) if there is one.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{-14y^{4}}{7y^{2}}=-2y^{2}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Now let&#8217;s move on to the next term: \\(-21y^{3}\\) (remember to grab that negative sign).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{-21y^{3}}{7y^{2}}=-3y\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Finally, we\u2019ll move on to \\(35y^{2}\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{35y^{2}}{7y^{2}}=5\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>So that means that our answer is all of these terms added together. Remember adding a negative number is the same as subtracting a number.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(4y^{3}-2y^{2}-3y+5\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Before we go, let\u2019s try one that is slightly more challenging. <\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{24y^{6}+16y^{4} -2y^{5} +8y^{2}}{4y3}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>In this one, not all of the terms are going to divide evenly into whole numbers. Don\u2019t let that intimidate you! Simply follow the same steps we have been, even when things don\u2019t look as pretty. All we have to do is divide each term in the numerator by \\(4y^{3}\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{24y^{6}}{4y^{3}}=6y^{3}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Now we&#8217;ll move on to \\(16y^{4}\\).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{16y^{4}}{4y^{3}}=4y\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Then we&#8217;ll move on to \\(-2y^{3}\\) (don&#8217;t forget to grab the negative sign).<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{-2y^{3}}{4y^{3}}=-12\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Then finally, we have \\(8^{y}\\div 24y^{3}\\). Remember, anything to the negative power is in the denominator of a fraction.<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(\\frac{8y^{2}}{4y^{3}}=2y\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>Finally, add all the terms together. Our answer is:<\/p>\n<blockquote style=\"border: 0px; padding: 30px; background-color: #eee; box-shadow: 1.5px 1.5px 5px grey; width:80%; margin: auto;\">\n<div style=\"font-style:normal; font-size:90%; text-align:center;\">\\(6y^{3}+4y-\\frac{1}{2}+\\frac{2}{y}\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<\/p>\n<p>We have fractions in our answer, like I mentioned before. That is totally fine! This is the most simplified form of the answer.<\/p>\n<p>I hope that this video on dividing polynomials by monomials was helpful. Thanks for watching, and happy studying!<\/p>\n<\/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>Dividing Polynomials by Monomials Return to Algebra I Videos<\/p>\n","protected":false},"author":22,"featured_media":121225,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-121222","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-practice-question-videos","7":"page_type-video","8":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121222","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\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=121222"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121222\/revisions"}],"predecessor-version":[{"id":286438,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/121222\/revisions\/286438"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/121225"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=121222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}