{"id":1227,"date":"2013-06-06T09:00:52","date_gmt":"2013-06-06T09:00:52","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=1227"},"modified":"2026-03-28T11:10:01","modified_gmt":"2026-03-28T16:10:01","slug":"adding-and-subtracting-polynomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/adding-and-subtracting-polynomials\/","title":{"rendered":"Adding and Subtracting Polynomials"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_PLc2_59emsM\" 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_PLc2_59emsM\" data-source-videoID=\"PLc2_59emsM\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Adding and Subtracting Polynomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Adding and Subtracting Polynomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_PLc2_59emsM:hover {cursor:pointer;} img#videoThumbnailImage_PLc2_59emsM {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1264-adding-and-subtracting-polynomials-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_PLc2_59emsM\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_PLc2_59emsM\" 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\",\"Adding and Subtracting Polynomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PLc2_59emsM\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PLc2_59emsM\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_PLc2_59emsM\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction uGe_Function() {\n  var x = document.getElementById(\"uGe\");\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=\"uGe_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=\"uGe\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Polynomials_in_the_Real_World\" class=\"smooth-scroll\">Polynomials in the Real World<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Simplifying_Polynomials\" class=\"smooth-scroll\">Simplifying Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Adding_Polynomials\" class=\"smooth-scroll\">Adding Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Subtracting_Polynomials\" class=\"smooth-scroll\">Subtracting Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Writing_Polynomials_in_Proper_Form\" class=\"smooth-scroll\">Writing Polynomials in Proper Form<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Practice_Problems\" class=\"smooth-scroll\">Practice Problems<\/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>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this review of polynomials! Today we are going to dive into the topic of adding and subtracting them. <\/p>\n<p>A polynomial can be monomial such as \\(3x^2\\), binomial such as \\(3x^2 &#8211; 4x\\), or trinomial such as \\(3x^2 &#8211; 4x + 5\\).<\/p>\n<h2><span id=\"Polynomials_in_the_Real_World\" class=\"m-toc-anchor\"><\/span>Polynomials in the Real World<\/h2>\n<p>\nWorking with polynomials is not just a skill we need to master for algebra class; we see polynomials used in real-world situations all the time. Economists use polynomials to model economic growth patterns, medical researchers use polynomials to describe growth patterns of bacterial colonies, and rollercoaster designers use polynomials to graph out the design of a ride. <\/p>\n<p><H2>Simplifying Polynomials<\/h2>\n<p>\nBut before we jump into the process of adding and subtracting polynomials, it is important to remember how to simplify polynomials. The process for this revolves around the skill of combining like terms. When we combine like terms, we are essentially combining or consolidating all of the terms that have the same variable, raised to the same power. <\/p>\n<p>For example, let\u2019s quickly look at this expression that needs to be simplified. <\/p>\n<div class=\"examplesentence\">\\(-5x^2 + 5x &#8211; 2x^2 &#8211; 8x +11 &#8211; x &#8211; 15\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn order to simplify this expression, we need to combine terms that have the same variable raised to the same power. This means we are going to combine all of the terms that contain x2, all of the terms that contain x, and all of the terms that are constants, which means they have no variable.<\/p>\n<p>With each section simplified, we can rewrite our expression as \\(-7x^2 &#8211; 4x &#8211; 4\\). <\/p>\n<p>Now that we have a solid foundation with the process of simplifying expressions, we can move into the process of adding and subtracting polynomials!<\/p>\n<h2><span id=\"Adding_Polynomials\" class=\"m-toc-anchor\"><\/span>Adding Polynomials<\/h2>\n<p>\nLet\u2019s add these two polynomials together.<\/p>\n<div class=\"examplesentence\">\\((x^3 + 2x^2 + 8x) + (2x &#8211; 3x^2 &#8211; x^3)\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe first thing we need to do here is simplify like terms from both sides.<\/p>\n<p>Here, we have \\(x^3\\) and \\(-x^3\\) as like terms, \\(2x^2\\) and \\(-3x^2\\) are like terms, and \\(8x\\) and \\(2x\\) are like terms.<\/p>\n<p>Now, we can combine these terms, and the result will be our answer.<\/p>\n<p>\\(x^3 &#8211; x^3\\) equals zero, so it\u2019s canceled out. \\(2x^2 &#8211; 3x^2 = -x^2\\). And \\(8x + 2x = 10x\\).<\/p>\n<p>Now we can simply rewrite our answer as \\(-x^2 +10x\\).<\/p>\n<h2><span id=\"Subtracting_Polynomials\" class=\"m-toc-anchor\"><\/span>Subtracting Polynomials<\/h2>\n<p>\nLet\u2019s try an example where we have to distribute a negative sign into our second set of grouping symbols.<\/p>\n<p>In this example, we will need to remember that when you subtract a negative from a negative, the minus sign flips and becomes a positive. So, for example, \\(-6 &#8211; -9 = 3\\). All right, let\u2019s subtract these two polynomials:<\/p>\n<div class=\"examplesentence\">\\((-6x^3 + 5x^2 &#8211; 3) &#8211; (2x^3 -4x^2 &#8211; 3x +1)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFirst, just like before, let\u2019s combine our like terms so we can simplify them. \\(-6x^3\\) and \\(2x^3\\) are like terms, \\(5x^2\\) and \\(-4x^2\\) are like terms, and -3 and 1 are like terms. \\(-3x\\) will become \\(3x\\) because it is a negative term being subtracted from another expression, and nothing minus a negative becomes a positive.<\/p>\n<p>\\(-6x^3 &#8211; 2x^3 = -8x^3\\). \\(5x^2 &#8211; -4x^2\\) becomes \\(5x^2 + 4x^2\\) since we\u2019re subtracting a negative, and that gives us \\(9x^2\\). -3 minus 1 is -4. Now, we put all of these pieces together for our final answer: \\(-8x^3 + 9x^2 + 3x -4\\).<\/p>\n<h2><span id=\"Writing_Polynomials_in_Proper_Form\" class=\"m-toc-anchor\"><\/span>Writing Polynomials in Proper Form<\/h2>\n<p>\nIt is helpful to know that when you write your answer, it is considered to be in \u201cproper form\u201d when the exponents are arranged by descending powers. For example: \\(x^3\\), and then \\(x^2\\), and then \\(x\\), and then your constant. <\/p>\n<p>So if our answer produced \\(5x + 3 + 8x^2 -x^3\\), we would want to rearrange this so that it\u2019s written as \\(-x^3 +8x^2 +5x +3\\). <\/p>\n<hr>\n<h2><span id=\"Practice_Problems\" class=\"m-toc-anchor\"><\/span>Practice Problems<\/h2>\n<p>\nOkay, before we go, try out a couple of example problems on your own! Here\u2019s the first one:<\/p>\n<div class=\"examplesentence\">\\((7 + 3x^2 + 9x^3 + x) + (-4 &#8211; 2x^2 &#8211; 2x^3 + 6x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nPause the video and see what answer you get!<\/p>\n<p>Okay, let\u2019s look at it together. First, you should have combined like terms, so let\u2019s do that.<\/p>\n<p>When we put these pieces together, we get \\(3 + x^2 + 7x^3 + 7x\\). In proper form, this would be written as \\(7x^3 + x^2 + 7x + 3\\).<\/p>\n<p>Let\u2019s try one more! Pause the video and see if you can subtract these two polynomials:<\/p>\n<div class=\"examplesentence\">\\((14 + 9x +2x^3 + x^2) &#8211; (-8 + 8x &#8211; 6x^3 &#8211; 5x^2)\\)<\/div>\n<p>\n&nbsp;<br \/>\nThink you got it? Let\u2019s take a look.<\/p>\n<p>First, let\u2019s combine the like terms.<\/p>\n<p>Now, we can put this together to get \\(22 + x + 8x^3 + 6x^2\\). In proper form, this would be \\(8x^3 +  6x^2 + x + 22\\).<\/p>\n<p>All right, that\u2019s all for this review! 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>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":100363,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-1227","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-polynomial-videos","8":"page_category-quadratics-videos","9":"page_type-video","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1227","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=1227"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1227\/revisions"}],"predecessor-version":[{"id":280832,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1227\/revisions\/280832"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100363"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=1227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}