{"id":122995,"date":"2022-05-30T10:42:47","date_gmt":"2022-05-30T15:42:47","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=122995"},"modified":"2026-03-28T11:52:22","modified_gmt":"2026-03-28T16:52:22","slug":"dividing-trinomials-by-binomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/dividing-trinomials-by-binomials\/","title":{"rendered":"Dividing Trinomials by Binomials"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_ZpejYgEkHPY\" 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_ZpejYgEkHPY\" data-source-videoID=\"ZpejYgEkHPY\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Dividing Trinomials by Binomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Dividing Trinomials by Binomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_ZpejYgEkHPY:hover {cursor:pointer;} img#videoThumbnailImage_ZpejYgEkHPY {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/2342-thumbnail-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_ZpejYgEkHPY\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_ZpejYgEkHPY\" 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 Trinomials by Binomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ZpejYgEkHPY\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ZpejYgEkHPY\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_ZpejYgEkHPY\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Mmn_Function() {\n  var x = document.getElementById(\"Mmn\");\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=\"Mmn_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=\"Mmn\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_2\" class=\"smooth-scroll\">Example #2<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_3\" class=\"smooth-scroll\">Example #3<\/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>Hello! Welcome to this video on <strong>dividing trinomials by binomials<\/strong>.<\/p>\n<p>When <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/dividing-polynomials\/\">dividing trinomials by binomials<\/a>, you often will need to use long division. This process may seem challenging at first, but it will get easier with practice.<\/p>\n<h2><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h2>\n<p>\nLet\u2019s jump right into an example.<\/p>\n<div class=\"examplesentence\">\\((x^{2}+4x-5)\\div (x+1)\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, \\(x^{2}+4x-5\\) is a trinomial because it has 3 terms, and \\(x+1\\) is a binomial because it has 2 terms. Set up your long division by putting the trinomial under the \u201chouse,\u201d and the binomial to the left.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}\\phantom{)}\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThen, we want to figure out how many times the first term of our binomial goes into the first term of our trinomial. \\(x\\) goes into \\(x^{2}\\), \\(x\\) times, because \\(x\\cdot x=x^{2}\\). So write an \\(x\\) right here, above our \\(x\\)-term in the trinomial.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x\\phantom{)}\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThen, multiply the value you just wrote down \\((x)\\) by the binomial \\((x+1)\\), and write this under the trinomial. Since \\(x(x+1)=x^{2}+x\\), we write this under the trinomial.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x\\phantom{)}\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\((x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x)\\phantom{-}\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThen, subtract these two polynomials, and bring down the \\(\u20135\\).<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x\\phantom{)}\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{(0}x)\\phantom{-}\\downarrow\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}3x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nFrom here, see how many times the first term of our divisor \\((x+1)\\) goes into the first term of the new binomial \\((3x-5)\\). So how many times does \\(x\\) fit into \\(3x\\)? \\(x\\) goes into \\(3x\\) three times, because \\(x\\cdot 3=3x\\), so write this number after \\(x\\) in our answer.\\<\/p>\n<p>Since 5 is positive, write \\(+3\\).<\/p>\n<p>Then, multiply the number you just wrote down \\((3)\\) by the divisor \\((x+1)\\), and write this binomial below the one we found a few steps earlier: \\(3(x+1)=3x+3\\).<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x\\phantom{)}+5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{(0}x)\\phantom{-}\\downarrow\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}3x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nAnd we&#8217;re going to subtract the two binomials, so \\((3x-5)-(3x+3)=-8\\).<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0}x\\phantom{)}+5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+1\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}4x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(+\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{(0}x)\\phantom{-}\\downarrow\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}3x\\phantom{)}-5\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(&#8211;\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((3x\\phantom{)}+3)\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(0x)}-8\\phantom{)}\\)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nNow we have a remainder here, so when you have a remainder in a division problem like this, you&#8217;re going to take the remainder and divide it by the divisor. And add it to the expression that we have up here \\((x+3)\\).<\/p>\n<p>So since we have \u20138, I&#8217;m going to put minus, 8 in our numerator, and then \\(x+1\\) in our denominator. If it was positive down here, we would put a plus sign, but since it&#8217;s negative, that&#8217;s why I put a minus sign right here. And then this is our final answer:<\/p>\n<div class=\"examplesentence\">\\(x+3-\\frac{8}{x+1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo when dividing you will sometimes have remainders like we did here, this is exactly how you would handle them and it&#8217;s okay that things don&#8217;t always divide out perfectly. So that&#8217;s what you do when it doesn&#8217;t.<\/p>\n<h2><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h2>\n<p>\nLet\u2019s try another one!<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{x^{2}+8x-9}{x+2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis problem is written in fraction form. Remember, a fraction bar always represents division, so this is just asking us to divide \\(x^{2}+8x-9\\) by \\(x+2\\).<\/p>\n<p>Let\u2019s follow the same steps we did in the last problem. First, set up the long division problem.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+8}x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+2\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}+8x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nNow, figure out how many times \\(x\\) goes into \\(x^{2}\\). Well, just like last time, \\(x\\cdot x=x^{2}\\), so it goes into \\(x^{2}\\), \\(x\\) times.<\/p>\n<p>Now, multiply \\(x\\) by \\(x+2\\) and write it below the trinomial. Since \\(x(x+2)=x^{2}+2x\\), we write this below the trinomial.<\/p>\n<p>Then, subtract the two polynomials, and bring down the \\(\u20139\\).<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+8}x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+2\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}+8x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}+2x)\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{-1}\\downarrow \\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+}6x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nNow, figure out how many times \\(x\\) goes into \\(6x\\). Since \\(x\\cdot 6=6x\\), \\(x\\) goes into \\(6x\\) six times. Since 6 is positive, write \\(+6\\) after \\(x\\) in the solution.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+8}x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(+\\phantom{1}6\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+2\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}+8x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}+2x)\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{-1}\\downarrow \\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+}6x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{(x^{2}}-(6x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(+12)\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-21\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nFrom here, multiply 6 by \\(x+2\\), and write this below the new binomial we just created: \\(6(x+2)=6x+12\\).<\/p>\n<p>Subtract these two binomials.<\/p>\n<p>So here we have a remainder again. And remember, when we have a remainder, we just take this number \\((\u201321)\\) and that will be the numerator of our fraction. So we&#8217;ll have minus 21 over, the denominator will be the divisor over here, so over \\(x+2\\). So our answer for this problem is:<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+8}x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(+\\phantom{1}6\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-\\frac{21}{x+2}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x+2\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}x^{2}+8x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x+}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((x^{2}+2x)\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{-1}\\downarrow \\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(x^{2}+}6x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-\\phantom{1}9\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{(x^{2}}-(6x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(+12)\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-21\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<\/p>\n<div class=\"examplesentence\">\\(x+6-\\frac{21}{x+2}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h2>\n<p>\nBefore we go, I want to try one last example.<\/p>\n<div class=\"examplesentence\">\\((2x^{2}+4x-12)\\div (x-6)\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo again, we&#8217;re going to set up our long division problem, just like we have been.<\/p>\n<table class=\"ATable\" style=\"margin: auto; border:none; font-size:100%; width:13%; padding: 0px;\">\n<body><\/p>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\" ><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(2x^{2}+}2x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(+16\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(+\\frac{84}{x-6}\\)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right;border-left:0px; border-right: 0px; border-top:0px; border-bottom:0px; padding: 10px;\">\\(x-6\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(}2x^{2}+\\phantom{1}4x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\">\\(-12\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border:0px; padding: 0px;\">\\(\\phantom{x-}-\\)<\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\((2x^{2}-12x)\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{-}\\downarrow\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{(2x^{2}+}16x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(-12\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(\\phantom{2x^{2}}-(16x\\phantom{)}\\)<\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; padding: 0px;\">\\(-96)\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:right; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/td>\n<td style=\"text-align:center; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\">\\(\\phantom{-}84\\phantom{)}\\)<\/td>\n<td style=\"text-align:left; border-right: 0px; border-left:0px; border-top:0px; border-bottom:0px; padding: 0px;\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThen, see how many times \\(x\\) goes into \\(2x^{2}\\). \\(x\\) goes into \\(2x^{2} 2x\\) times, because \\(x\\cdot 2x=2x^{2}\\).<\/p>\n<p>Now, multiply \\(2x\\) by \\(x-6\\).<\/p>\n<p>From here, subtract the two polynomials, and bring down the \\(\u201312\\).<\/p>\n<p>Now, figure out how many times \\(x\\) goes into \\(16x\\). It goes in 16 times because \\(x\\cdot 16=16x\\), so we write plus 16 right here.<\/p>\n<p>And then we multiply 16 by our expression \\(x-6\\). So that gives us \\(16x-96\\). And then from here, we&#8217;re just going to subtract our expressions.<\/p>\n<p>And again, we&#8217;re going to do what we&#8217;ve been doing the last few times and add our remainder, so we&#8217;ll add 84x-6. So our answer for this question is:<\/p>\n<div class=\"examplesentence\">\\(2x+16+\\frac{84}{x-6}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow just as a disclaimer: for all of these examples we&#8217;ve had a remainder, but sometimes things will divide out nice and evenly and not have any remainder. So if you get a zero at the last step, you&#8217;re not going to have a remainder and you don&#8217;t have to worry about this fractional part. But if you do have a remainder, it&#8217;s nothing to be concerned about. It&#8217;s just a simple easy extra step that you now know how to do, and are hopefully comfortable with.<\/p>\n<p>I hope that this video on dividing trinomials by binomials 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>Return to Algebra I Videos<\/p>\n","protected":false},"author":22,"featured_media":122998,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-122995","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":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"When dividing a trinomial by a binomial, place the binomial under the &quot;house&quot; and the binomial to the left. 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