{"id":4150,"date":"2022-08-03T18:44:32","date_gmt":"2022-08-03T18:44:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=4150"},"modified":"2026-02-12T00:46:45","modified_gmt":"2026-02-12T00:46:45","slug":"3-4-1-the-division-of-polynomials-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-4-1-the-division-of-polynomials-2\/","title":{"raw":"3.4.1: The Division of Polynomials","rendered":"3.4.1: The Division of Polynomials"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Divide a polynomial by a monomial<\/li>\r\n \t<li>Use long division to divide polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Dividing a Monomial by a Monomial<\/h2>\r\nConsider the example, [latex]\\dfrac{x^3}{x^2}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x^3}{x^2}&amp;=\\dfrac{x\\cdot x\\cdot x}{x\\cdot x}\\\\&amp;=\\dfrac{\\cancel{x} \\cdot \\cancel{x} \\cdot x}{\\cancel{x} \\cdot \\cancel{x}}\\\\&amp;=\\dfrac{1 \\cdot 1 \\cdot x}{1 \\cdot 1}\\\\&amp;=\\dfrac{x}{1}\\\\&amp;=x\\end{aligned}[\/latex].<\/p>\r\nSince, [latex]\\dfrac{x}{x}=1,\\;x\\neq0[\/latex], we cancel common factors on the numerator and denominator to 1. Likewise, [latex]\\dfrac{x^2}{x^2}=1,\\;\\dfrac{x^3}{x^3}=1,\\;,...\\dfrac{x^n}{x^n}=1[\/latex], provided that [latex]x\\neq0[\/latex].\r\n\r\nSimilarly, for the division\u00a0[latex]\\dfrac{x}{x^2}[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x}{x^2}&amp;=\\dfrac{x}{x\\cdot x}\\\\&amp;=\\dfrac{\\cancel{x}}{\\cancel{x} \\cdot x}\\\\&amp;=\\dfrac{1}{1 \\cdot x}\\\\&amp;=\\dfrac{1}{x}\\end{aligned}[\/latex].<\/p>\r\nIn summary, to simplify a fraction where the numerator and denominator have the same base but different exponents, we can cancel out all common factors. This is equivalent to keeping the common base and subtracting the exponents i.e. to divide a monomial by a monomial, we employ the division property of exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>The division property of exponents<\/h3>\r\n<p style=\"text-align: center;\">For all [latex]m,\\;n,\\; x \\in\\mathbb{R}[\/latex] and [latex]x\\neq0[\/latex],<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{x^m}{x^n}=x^{m-n}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\dfrac{x^8}{x^3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{y^4}{y^4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{x^3}{x^7}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\dfrac{x^8}{x^3}=x^{8-3}=x^5[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{y^4}{y^4}=y^{4-4}=y^0=1[\/latex]\u00a0 \u00a0Recall that [latex]y^0=1[\/latex] for all [latex]y\\neq0[\/latex]; [latex]0^0[\/latex] is undefined.<\/li>\r\n \t<li><span style=\"font-size: 1rem; text-align: initial;\">[latex]\\dfrac{x^3}{x^7}=x^{3-7}=x^{-4}=\\dfrac{1}{x^4}[\/latex]\u00a0 \u00a0Recall that, [latex]x^{-n}=\\dfrac{1}{x^n}[\/latex]<\/span><\/li>\r\n<\/ol>\r\n<\/div>\r\nIf the exponential terms have coefficients other than 1, we divide the coefficients.\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\dfrac{9x^7}{3x^3}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{15y^8}{5y^8}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{7x^3}{14x^7}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\dfrac{9x^7}{3x^3}=\\dfrac{9}{3}x^{7-3}=3x^4[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{15y^8}{5y^8}=-\\dfrac{15}{5}y^{8-8}=-3y^0=-3\\cdot 1=-3[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{7x^3}{14x^7}-\\dfrac{7}{14}x^{3-7}=-\\dfrac{1}{2}x^{-4}=-\\dfrac{1}{2}\\cdot\\dfrac{1}{x^4}=-\\dfrac{1}{2x^4}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\n<ol>\r\n \t<li>[latex]\\dfrac{x^8}{x^6}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{20x^{12}}{5x^7}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{12y^5}{4y^5}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{16x^4}{8x^9}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm146\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm146\"]\r\n<ol>\r\n \t<li>[latex]\\dfrac{x^8}{x^6}=x^2[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{20x^{12}}{5x^7}=4x^5[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{12y^5}{4y^5}=-3[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{16x^4}{8x^9}=-\\dfrac{2}{x^5}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing a Polynomial by a Monomial<\/h2>\r\nThe distributive property of multiplication over addition also works for division, since division is just multiplication by the reciprocal.\r\n<div class=\"textbox shaded\">\r\n<h3>The distributive property<\/h3>\r\n<p style=\"text-align: center;\">[latex]a(b+c)=ab+ac[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{a+b}{c}=\\dfrac{1}{c}(a+b)=\\dfrac{a}{c}+\\dfrac{b}{c}[\/latex]<\/p>\r\nwhere [latex]a,\\;b,[\/latex] and [latex]c[\/latex] are algebraic terms.\r\n\r\n<\/div>\r\nTo divide a polynomial by a monomial, we can write the division as a fraction, and then decompose the fraction into the sum of fractions, and then simplify each fraction. For example, to divide the polynomial [latex]3x^3-6x^2+18x-7[\/latex] by [latex]2x^2[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&amp;\\;\\;\\;\\;\\,\\dfrac{3x^3-6x^2+18x-7}{2x^2} \\\\ \\\\&amp;= \\dfrac{3x^3}{2x^2}-\\dfrac{6x^2}{2x^2}+\\dfrac{18x}{2x^2}-\\dfrac{7}{2x^2} \\\\ \\\\&amp;= \\dfrac{3}{2}x-3+\\dfrac{9}{x}-\\dfrac{7}{2x^2}\\end{aligned}[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nDivide [latex]12x^4-18x^3+36x^2-24x[\/latex] by [latex]-6x^2[\/latex].\r\n<h4>Solution<\/h4>\r\nDivide each term in the first polynomial by the monomial divisor:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&amp;\\;\\;\\;\\;\\,\\dfrac{12x^4-18x^3+36x^2-24x}{-6x^2}\\\\ \\\\&amp;=\\dfrac{12x^4}{-6x^2}-\\dfrac{18x^3}{-6x^2}+\\dfrac{36x^2}{-6x^2}-\\dfrac{24x}{-6x^2}\\\\ \\\\&amp;=-2x^2+3x-6+\\dfrac{4}{x}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDivide [latex]60x^5-30x^4+25x^3-35x+10[\/latex] by [latex]10x^3[\/latex].\r\n\r\n[reveal-answer q=\"hjm096\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm096\"]\r\n\r\n[latex]6x^2-3x+\\dfrac{5}{2}-\\dfrac{7}{2x^2}+\\dfrac{1}{x^3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing Polynomials using Long Division<\/h2>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-428\" class=\"standard post-428 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>long division revision<\/h3>\r\nWhen we divide two numbers, we often end up with a remainder. For example, [latex]17\\div 3 = 5\\;\\text{R}(2)[\/latex].\r\n\r\nThis can also be thought of as:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{17}{3}=5+\\dfrac{2}{3}[\/latex]<\/p>\r\nEquivalently, multiplying both sides by 3:\r\n<p style=\"text-align: center;\">[latex]17=3\\cdot 5 + 2[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe process used to accomplish division of polynomials is similar to long division of whole numbers, so it is essential that we are confident with long division of whole numbers. If you haven\u2019t performed long division for a while, take a moment to refresh before reading this section.\r\n\r\n[reveal-answer q=\"hjm954\"]Read more[\/reveal-answer]\r\n[hidden-answer a=\"hjm954\"]\r\n<h2>The Long Division Algorithm for Arithmetic<\/h2>\r\nWe begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204318\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division of 178 divided by 3. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/>\r\n\r\nAnother way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\text{Divisor }\\cdot \\text{Quotient}\\right)\\text{ + Remainder}&amp;=\\text{Dividend}\\\\ \\left(3\\cdot 59\\right)+1&amp;= 178\\end{aligned}[\/latex]<\/p>\r\nWe call this the <strong>Division Algorithm<\/strong>:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{Dividend}}{\\text{Divisor}}=\\text{Quotient}+\\dfrac{\\text{Remainder}}{\\text{Divisor}}[\/latex]<\/p>\r\nEquivalently:\r\n<p style=\"text-align: center;\">[latex]\\text{Dividend}=\\text{Quotient}\\cdot\\text{Divisor}+\\text{Remainder}[\/latex]<\/p>\r\nThe dividend, divisor, and remainder are whole numbers, and the remainder &lt; divisor.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nUse long division to divide: [latex]\\dfrac{242}{15}[\/latex]\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}16\\\\15) \\overline{242}\\\\15\\;\\;\\\\ \\overline{92}\\\\ 90\\\\ \\overline{2}\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nSo,\u00a0[latex]\\dfrac{242}{15}=16 +\\dfrac{2}{15}[\/latex]\r\n\r\n&nbsp;\r\n\r\nEquivalently, [latex]242=15\\cdot 16 + 2[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse long division to divide: [latex]\\dfrac{463}{12}[\/latex]\r\n\r\n[reveal-answer q=\"hjm761\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm761\"][latex]\\dfrac{463}{12}=38+\\dfrac{7}{12}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>The Division Algorithm<\/h3>\r\nThe division algorithm for arithmetic can be adapted to polynomials.\r\n<div class=\"textbox shaded\">\r\n<h3>division algorithm for polynomials<\/h3>\r\nLet [latex]p(x)[\/latex] and [latex]d(x)[\/latex] be polynomials where the degree of [latex]p(x)[\/latex] \u2265 the degree of [latex]d(x)[\/latex], then\r\n<p style=\"text-align: center;\">[latex]\\dfrac{p(x)}{d(x)}=q(x)+\\dfrac{r(x)}{d(x)}[\/latex]<\/p>\r\nwhere the degree of [latex]r(x)[\/latex] &lt; the degree of [latex]d(x)[\/latex].\r\n\r\nEquivalently, multiplying both sides by [latex]d(x)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]p(x)=d(x)q(x) + r(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe division algorithm allows us to divide two polynomials. To set up this division there are two basic rules that must be adhered to:\r\n<ol>\r\n \t<li><span style=\"color: #000000;\">The polynomials of the divisor and dividend need to be in <em><strong>descending order<\/strong><\/em>, which means the terms of a polynomial are written in the order of descending powers of variables.<\/span><\/li>\r\n \t<li><span style=\"color: #000000;\">There should be no gaps in the descending order of powers. Any gaps need to be filled with zeros.<\/span><\/li>\r\n<\/ol>\r\nFor example, if the dividend is given as [latex]4+5x-3x^3[\/latex], the first step tells us to write the polynomial in descending order:\r\n<p style=\"text-align: center;\">[latex]-3x^3+5x+4[\/latex]<\/p>\r\nSince there is no [latex]x^2[\/latex] term in this polynomial (i.e. there is a gap), we must include [latex]0x^2[\/latex] as a term to fill the gap:\r\n<p style=\"text-align: center;\">[latex]-3x^3\\color{blue}{+0x^2}+5x+4[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nWrite the polynomial in descending order, filling any gaps with zeros.\r\n<ol>\r\n \t<li>[latex]7x^3-8+2x^2[\/latex]<\/li>\r\n \t<li>[latex]6x-4x^2-3x^3+x^5[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1. First write\u00a0[latex]7x^3-8+2x^2[\/latex] in descending order:\u00a0[latex]7x^3+2x^2-8[\/latex]\r\n\r\nNow look for missing terms: there is no [latex]x[\/latex] term. Write [latex]0x[\/latex] where the [latex]x[\/latex] term should be:\r\n<p style=\"text-align: center;\">[latex]7x^3+2x^2\\color{blue}{+0x}-8[\/latex]<\/p>\r\n2. First write\u00a0[latex]6x-4x^2-3x^3+x^5[\/latex] in descending order:\u00a0[latex]x^5-3x^3-4x^2+6x[\/latex]\r\n\r\nNow look for missing terms: there is no [latex]x^4[\/latex] term and no constant term. Write [latex]0x^4[\/latex] where the [latex]x^4[\/latex] term should be, and write 0 for the constant term at the end of the polynomial:\r\n<p style=\"text-align: center;\">[latex]x^5\\color{blue}{+0x^4}-3x^3-4x^2+6x\\color{blue}{+0}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nWrite the polynomial in descending order, filling any gaps with zeros.\r\n<ol>\r\n \t<li>[latex]7x^4-8x+2x^2[\/latex]<\/li>\r\n \t<li>[latex]-2x-4x^4-3x^3+x^5[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm871\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm871\"]\r\n<ol>\r\n \t<li>[latex]7x^4\\color{blue}{+0x^3}+2x^2-8x\\color{blue}{+0}[\/latex]<\/li>\r\n \t<li>[latex]x^5-4x^4-3x^3\\color{blue}{+0x^2}-2x\\color{blue}{+0}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe process for dividing polynomials is basically the same as it was for whole numbers.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 66.3675%;\" colspan=\"2\"><span style=\"font-size: 16px; orphans: 1;\">Process for dividing polynomials<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 66.3675%;\"><span style=\"font-size: 16px; orphans: 1;\">As an example, let's divide [latex]x^2+x+1[\/latex] by [latex]x-1[\/latex]. <\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">We truncate the divisor\u00a0<\/span><span style=\"font-size: 16px; orphans: 1;\">[latex]x-1[\/latex] to [latex]x[\/latex] and divide the first term in the dividend by [latex]x[\/latex]:\u00a0 [latex]\\dfrac{x^2}{x}=x[\/latex]. This [latex]\\color{blue}{x}[\/latex] goes on the quotient line. <\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">Now multiply [latex]\\color{blue}{x}[\/latex] by the divisor [latex]x-1[\/latex]: [latex]\\color{blue}{x}\\cdot(x-1)=\\color{blue}{x^2-x}[\/latex]. This goes under the dividend.<\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">Now subtract:\u00a0 [latex](x^2+x+1)-(\\color{blue}{x^2-x})=(x^2-x^2)+(x-(-x))+1=\\color{red}{2x+1}[\/latex]. <\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">Divide [latex]2x[\/latex] by [latex]x[\/latex]:\u00a0 [latex]\\dfrac{2x}{x}=\\color{green}{2}[\/latex]. Add this [latex]\\color{green}{2}[\/latex] onto the quotient line. <\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">Multiply [latex]\\color{green}{2}[\/latex] by the\u00a0divisor [latex]x-1[\/latex]: [latex]\\color{green}{2}(x-1)=\\color{green}{2x-2}[\/latex]. <\/span>\r\n\r\n<span style=\"font-size: 16px; orphans: 1;\">FInally, subtract:\u00a0 [latex]\\color{red}{(2x+1)}-\\color{green}{(2x-2)}=(\\color{red}{2x}-\\color{green}{2x})+(\\color{red}{1}-(\\color{green}{-2}))=\\color{orange}{3}[\/latex].<\/span>\r\n\r\n<span style=\"font-size: 16px;\">We have found that\u00a0[latex]\\dfrac{x^2+x+1}{x-1}=x+2+\\dfrac{3}{x-1}[\/latex].<\/span><\/td>\r\n<td style=\"width: 33.6325%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]\\begin{aligned}\\color{blue}{x}+\\color{green}{2}\\\\x-1 )\\overline{x^2+x+1}\\\\ \\color{blue}{x^2-x}\\;\\;\\;\\;\\;\\;\\\\ \\overline{\\color{red}{2x+1}}\\\\\\underline{\\color{green}{2x-2}}\\\\ \\color{orange}{3}\\end{aligned}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can identify the dividend, divisor, quotient, and remainder by writing this answer in its equivalent form:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllll}&amp;x^2+x+1&amp;=&amp;(x-1)&amp;(x+2)&amp;+&amp;3\\\\&amp;\\text{Dividend}&amp;&amp;\\text{Divisor}&amp;\\text{Quotient}&amp;&amp;\\text{Remainder}\\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nDivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.\r\n<h4>Solution<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204321\/CNX_Precalc_revised_eq_12.png\" alt=\"Long Division of 2x cubed minus 3x squared plus 4x plus 5 divided by the binomial x plus 2. The answer is 2x squared minus 7x plus 18 with a remainder of -31.\" width=\"522\" height=\"462\" \/>\r\n<h4>Answer<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18+\\dfrac{-31}{x+2}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">or equivalently,<\/p>\r\n<p style=\"text-align: center;\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)+(-31)[\/latex]<\/p>\r\nWe can identify the\u00a0<strong>dividend<\/strong>,\u00a0<strong>divisor<\/strong>,\u00a0<strong>quotient<\/strong>, and\u00a0<strong>remainder<\/strong>.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204324\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/><\/div>\r\nNotice that the remainder in this example is negative. Even when the remainder is negative, we add it as a negative remainder.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>using long division to divide polynomials<\/h3>\r\n<ol>\r\n \t<li>Set up the division problem based on the rules of descending powers and no gaps.<\/li>\r\n \t<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\r\n \t<li>Put the answer as the first term in the quotient.<\/li>\r\n \t<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t<li>Subtract the product in step 4 from all the terms above it.<\/li>\r\n \t<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\r\n \t<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nDivide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].\r\n<h4>Solution<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Long Division of the trinomial 5x squared plus 3x minus 2 the binomial x plus 1. The answer is 5x minus 2 with remainder of zero.\" width=\"426\" height=\"288\" \/>\r\nThe quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\r\n&nbsp;\r\n<h4>Analysis of the Solution<\/h4>\r\nThis division problem had a remainder of 0. This tells us that the dividend is divided exactly by the divisor and that, therefore,\u00a0 the divisor and the quotient are factors of the dividend.\u00a0\u00a0<span style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nDivide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].\r\n<h4>Solution<\/h4>\r\n<div class=\"wp-nocaption aligncenter wp-image-11885\">\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"Long division of the polynomial 6x to the third power plus 11x squared minus 31x plus 1 by the binomial 3x minus 2. \" width=\"621\" height=\"153\" \/><\/a>\r\n\r\n<\/div>\r\nThere is a remainder of 1. We can express the result as:\r\n\r\n[latex]\\dfrac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\dfrac{1}{3x - 2}[\/latex]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can check our work by using the Division Algorithm to rewrite the solution then multiplying.\r\n\r\n[latex]\\;\\;\\;\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1\\\\=3x\\left(2{x}^{2}+5x - 7\\right)-2\\left(2{x}^{2}+5x - 7\\right)+1\\\\=6x^3+15x^2-21x-4x^2-10x+14+1\\\\=6{x}^{3}+11{x}^{2}-31x+15[\/latex]\r\n\r\nNotice, as we write our result,\r\n<ul id=\"fs-id1165135152079\">\r\n \t<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n \t<li>the divisor is [latex]3x - 2[\/latex]<\/li>\r\n \t<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\r\n \t<li>the remainder is\u00a01<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nDivide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].\r\n\r\n[reveal-answer q=\"hjm202\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm202\"]\r\n\r\n[latex]\\dfrac{16x^3-12x^2+20x-3}{4x+5}=4x^2-8x+15+\\left (\\dfrac{-78}{4x+5}\\right )[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Divide a polynomial by a monomial<\/li>\n<li>Use long division to divide polynomials<\/li>\n<\/ul>\n<\/div>\n<h2>Dividing a Monomial by a Monomial<\/h2>\n<p>Consider the example, [latex]\\dfrac{x^3}{x^2}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x^3}{x^2}&=\\dfrac{x\\cdot x\\cdot x}{x\\cdot x}\\\\&=\\dfrac{\\cancel{x} \\cdot \\cancel{x} \\cdot x}{\\cancel{x} \\cdot \\cancel{x}}\\\\&=\\dfrac{1 \\cdot 1 \\cdot x}{1 \\cdot 1}\\\\&=\\dfrac{x}{1}\\\\&=x\\end{aligned}[\/latex].<\/p>\n<p>Since, [latex]\\dfrac{x}{x}=1,\\;x\\neq0[\/latex], we cancel common factors on the numerator and denominator to 1. Likewise, [latex]\\dfrac{x^2}{x^2}=1,\\;\\dfrac{x^3}{x^3}=1,\\;,...\\dfrac{x^n}{x^n}=1[\/latex], provided that [latex]x\\neq0[\/latex].<\/p>\n<p>Similarly, for the division\u00a0[latex]\\dfrac{x}{x^2}[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\dfrac{x}{x^2}&=\\dfrac{x}{x\\cdot x}\\\\&=\\dfrac{\\cancel{x}}{\\cancel{x} \\cdot x}\\\\&=\\dfrac{1}{1 \\cdot x}\\\\&=\\dfrac{1}{x}\\end{aligned}[\/latex].<\/p>\n<p>In summary, to simplify a fraction where the numerator and denominator have the same base but different exponents, we can cancel out all common factors. This is equivalent to keeping the common base and subtracting the exponents i.e. to divide a monomial by a monomial, we employ the division property of exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>The division property of exponents<\/h3>\n<p style=\"text-align: center;\">For all [latex]m,\\;n,\\; x \\in\\mathbb{R}[\/latex] and [latex]x\\neq0[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{x^m}{x^n}=x^{m-n}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\dfrac{x^8}{x^3}[\/latex]<\/li>\n<li>[latex]\\dfrac{y^4}{y^4}[\/latex]<\/li>\n<li>[latex]\\dfrac{x^3}{x^7}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\dfrac{x^8}{x^3}=x^{8-3}=x^5[\/latex]<\/li>\n<li>[latex]\\dfrac{y^4}{y^4}=y^{4-4}=y^0=1[\/latex]\u00a0 \u00a0Recall that [latex]y^0=1[\/latex] for all [latex]y\\neq0[\/latex]; [latex]0^0[\/latex] is undefined.<\/li>\n<li><span style=\"font-size: 1rem; text-align: initial;\">[latex]\\dfrac{x^3}{x^7}=x^{3-7}=x^{-4}=\\dfrac{1}{x^4}[\/latex]\u00a0 \u00a0Recall that, [latex]x^{-n}=\\dfrac{1}{x^n}[\/latex]<\/span><\/li>\n<\/ol>\n<\/div>\n<p>If the exponential terms have coefficients other than 1, we divide the coefficients.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\dfrac{9x^7}{3x^3}[\/latex]<\/li>\n<li>[latex]-\\dfrac{15y^8}{5y^8}[\/latex]<\/li>\n<li>[latex]-\\dfrac{7x^3}{14x^7}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\dfrac{9x^7}{3x^3}=\\dfrac{9}{3}x^{7-3}=3x^4[\/latex]<\/li>\n<li>[latex]-\\dfrac{15y^8}{5y^8}=-\\dfrac{15}{5}y^{8-8}=-3y^0=-3\\cdot 1=-3[\/latex]<\/li>\n<li>[latex]-\\dfrac{7x^3}{14x^7}-\\dfrac{7}{14}x^{3-7}=-\\dfrac{1}{2}x^{-4}=-\\dfrac{1}{2}\\cdot\\dfrac{1}{x^4}=-\\dfrac{1}{2x^4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<ol>\n<li>[latex]\\dfrac{x^8}{x^6}[\/latex]<\/li>\n<li>[latex]\\dfrac{20x^{12}}{5x^7}[\/latex]<\/li>\n<li>[latex]-\\dfrac{12y^5}{4y^5}[\/latex]<\/li>\n<li>[latex]-\\dfrac{16x^4}{8x^9}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm146\">Show Answer<\/span><\/p>\n<div id=\"qhjm146\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{x^8}{x^6}=x^2[\/latex]<\/li>\n<li>[latex]\\dfrac{20x^{12}}{5x^7}=4x^5[\/latex]<\/li>\n<li>[latex]-\\dfrac{12y^5}{4y^5}=-3[\/latex]<\/li>\n<li>[latex]-\\dfrac{16x^4}{8x^9}=-\\dfrac{2}{x^5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Dividing a Polynomial by a Monomial<\/h2>\n<p>The distributive property of multiplication over addition also works for division, since division is just multiplication by the reciprocal.<\/p>\n<div class=\"textbox shaded\">\n<h3>The distributive property<\/h3>\n<p style=\"text-align: center;\">[latex]a(b+c)=ab+ac[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{a+b}{c}=\\dfrac{1}{c}(a+b)=\\dfrac{a}{c}+\\dfrac{b}{c}[\/latex]<\/p>\n<p>where [latex]a,\\;b,[\/latex] and [latex]c[\/latex] are algebraic terms.<\/p>\n<\/div>\n<p>To divide a polynomial by a monomial, we can write the division as a fraction, and then decompose the fraction into the sum of fractions, and then simplify each fraction. For example, to divide the polynomial [latex]3x^3-6x^2+18x-7[\/latex] by [latex]2x^2[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&\\;\\;\\;\\;\\,\\dfrac{3x^3-6x^2+18x-7}{2x^2} \\\\ \\\\&= \\dfrac{3x^3}{2x^2}-\\dfrac{6x^2}{2x^2}+\\dfrac{18x}{2x^2}-\\dfrac{7}{2x^2} \\\\ \\\\&= \\dfrac{3}{2}x-3+\\dfrac{9}{x}-\\dfrac{7}{2x^2}\\end{aligned}[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Divide [latex]12x^4-18x^3+36x^2-24x[\/latex] by [latex]-6x^2[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Divide each term in the first polynomial by the monomial divisor:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&\\;\\;\\;\\;\\,\\dfrac{12x^4-18x^3+36x^2-24x}{-6x^2}\\\\ \\\\&=\\dfrac{12x^4}{-6x^2}-\\dfrac{18x^3}{-6x^2}+\\dfrac{36x^2}{-6x^2}-\\dfrac{24x}{-6x^2}\\\\ \\\\&=-2x^2+3x-6+\\dfrac{4}{x}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Divide [latex]60x^5-30x^4+25x^3-35x+10[\/latex] by [latex]10x^3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm096\">Show Answer<\/span><\/p>\n<div id=\"qhjm096\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6x^2-3x+\\dfrac{5}{2}-\\dfrac{7}{2x^2}+\\dfrac{1}{x^3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Dividing Polynomials using Long Division<\/h2>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-428\" class=\"standard post-428 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>long division revision<\/h3>\n<p>When we divide two numbers, we often end up with a remainder. For example, [latex]17\\div 3 = 5\\;\\text{R}(2)[\/latex].<\/p>\n<p>This can also be thought of as:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{17}{3}=5+\\dfrac{2}{3}[\/latex]<\/p>\n<p>Equivalently, multiplying both sides by 3:<\/p>\n<p style=\"text-align: center;\">[latex]17=3\\cdot 5 + 2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The process used to accomplish division of polynomials is similar to long division of whole numbers, so it is essential that we are confident with long division of whole numbers. If you haven\u2019t performed long division for a while, take a moment to refresh before reading this section.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm954\">Read more<\/span><\/p>\n<div id=\"qhjm954\" class=\"hidden-answer\" style=\"display: none\">\n<h2>The Long Division Algorithm for Arithmetic<\/h2>\n<p>We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204318\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division of 178 divided by 3. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/><\/p>\n<p>Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\left(\\text{Divisor }\\cdot \\text{Quotient}\\right)\\text{ + Remainder}&=\\text{Dividend}\\\\ \\left(3\\cdot 59\\right)+1&= 178\\end{aligned}[\/latex]<\/p>\n<p>We call this the <strong>Division Algorithm<\/strong>:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{Dividend}}{\\text{Divisor}}=\\text{Quotient}+\\dfrac{\\text{Remainder}}{\\text{Divisor}}[\/latex]<\/p>\n<p>Equivalently:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Dividend}=\\text{Quotient}\\cdot\\text{Divisor}+\\text{Remainder}[\/latex]<\/p>\n<p>The dividend, divisor, and remainder are whole numbers, and the remainder &lt; divisor.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Use long division to divide: [latex]\\dfrac{242}{15}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}16\\\\15) \\overline{242}\\\\15\\;\\;\\\\ \\overline{92}\\\\ 90\\\\ \\overline{2}\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>So,\u00a0[latex]\\dfrac{242}{15}=16 +\\dfrac{2}{15}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Equivalently, [latex]242=15\\cdot 16 + 2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use long division to divide: [latex]\\dfrac{463}{12}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm761\">Show Answer<\/span><\/p>\n<div id=\"qhjm761\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<\/div>\n<p>[latex]\\dfrac{463}{12}=38+\\dfrac{7}{12}[\/latex]<\/p><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>The Division Algorithm<\/h3>\n<p>The division algorithm for arithmetic can be adapted to polynomials.<\/p>\n<div class=\"textbox shaded\">\n<h3>division algorithm for polynomials<\/h3>\n<p>Let [latex]p(x)[\/latex] and [latex]d(x)[\/latex] be polynomials where the degree of [latex]p(x)[\/latex] \u2265 the degree of [latex]d(x)[\/latex], then<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{p(x)}{d(x)}=q(x)+\\dfrac{r(x)}{d(x)}[\/latex]<\/p>\n<p>where the degree of [latex]r(x)[\/latex] &lt; the degree of [latex]d(x)[\/latex].<\/p>\n<p>Equivalently, multiplying both sides by [latex]d(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]p(x)=d(x)q(x) + r(x)[\/latex]<\/p>\n<\/div>\n<p>The division algorithm allows us to divide two polynomials. To set up this division there are two basic rules that must be adhered to:<\/p>\n<ol>\n<li><span style=\"color: #000000;\">The polynomials of the divisor and dividend need to be in <em><strong>descending order<\/strong><\/em>, which means the terms of a polynomial are written in the order of descending powers of variables.<\/span><\/li>\n<li><span style=\"color: #000000;\">There should be no gaps in the descending order of powers. Any gaps need to be filled with zeros.<\/span><\/li>\n<\/ol>\n<p>For example, if the dividend is given as [latex]4+5x-3x^3[\/latex], the first step tells us to write the polynomial in descending order:<\/p>\n<p style=\"text-align: center;\">[latex]-3x^3+5x+4[\/latex]<\/p>\n<p>Since there is no [latex]x^2[\/latex] term in this polynomial (i.e. there is a gap), we must include [latex]0x^2[\/latex] as a term to fill the gap:<\/p>\n<p style=\"text-align: center;\">[latex]-3x^3\\color{blue}{+0x^2}+5x+4[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Write the polynomial in descending order, filling any gaps with zeros.<\/p>\n<ol>\n<li>[latex]7x^3-8+2x^2[\/latex]<\/li>\n<li>[latex]6x-4x^2-3x^3+x^5[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1. First write\u00a0[latex]7x^3-8+2x^2[\/latex] in descending order:\u00a0[latex]7x^3+2x^2-8[\/latex]<\/p>\n<p>Now look for missing terms: there is no [latex]x[\/latex] term. Write [latex]0x[\/latex] where the [latex]x[\/latex] term should be:<\/p>\n<p style=\"text-align: center;\">[latex]7x^3+2x^2\\color{blue}{+0x}-8[\/latex]<\/p>\n<p>2. First write\u00a0[latex]6x-4x^2-3x^3+x^5[\/latex] in descending order:\u00a0[latex]x^5-3x^3-4x^2+6x[\/latex]<\/p>\n<p>Now look for missing terms: there is no [latex]x^4[\/latex] term and no constant term. Write [latex]0x^4[\/latex] where the [latex]x^4[\/latex] term should be, and write 0 for the constant term at the end of the polynomial:<\/p>\n<p style=\"text-align: center;\">[latex]x^5\\color{blue}{+0x^4}-3x^3-4x^2+6x\\color{blue}{+0}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Write the polynomial in descending order, filling any gaps with zeros.<\/p>\n<ol>\n<li>[latex]7x^4-8x+2x^2[\/latex]<\/li>\n<li>[latex]-2x-4x^4-3x^3+x^5[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm871\">Show Answer<\/span><\/p>\n<div id=\"qhjm871\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]7x^4\\color{blue}{+0x^3}+2x^2-8x\\color{blue}{+0}[\/latex]<\/li>\n<li>[latex]x^5-4x^4-3x^3\\color{blue}{+0x^2}-2x\\color{blue}{+0}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The process for dividing polynomials is basically the same as it was for whole numbers.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 66.3675%;\" colspan=\"2\"><span style=\"font-size: 16px; orphans: 1;\">Process for dividing polynomials<\/span><\/th>\n<\/tr>\n<tr>\n<td style=\"width: 66.3675%;\"><span style=\"font-size: 16px; orphans: 1;\">As an example, let&#8217;s divide [latex]x^2+x+1[\/latex] by [latex]x-1[\/latex]. <\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">We truncate the divisor\u00a0<\/span><span style=\"font-size: 16px; orphans: 1;\">[latex]x-1[\/latex] to [latex]x[\/latex] and divide the first term in the dividend by [latex]x[\/latex]:\u00a0 [latex]\\dfrac{x^2}{x}=x[\/latex]. This [latex]\\color{blue}{x}[\/latex] goes on the quotient line. <\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">Now multiply [latex]\\color{blue}{x}[\/latex] by the divisor [latex]x-1[\/latex]: [latex]\\color{blue}{x}\\cdot(x-1)=\\color{blue}{x^2-x}[\/latex]. This goes under the dividend.<\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">Now subtract:\u00a0 [latex](x^2+x+1)-(\\color{blue}{x^2-x})=(x^2-x^2)+(x-(-x))+1=\\color{red}{2x+1}[\/latex]. <\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">Divide [latex]2x[\/latex] by [latex]x[\/latex]:\u00a0 [latex]\\dfrac{2x}{x}=\\color{green}{2}[\/latex]. Add this [latex]\\color{green}{2}[\/latex] onto the quotient line. <\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">Multiply [latex]\\color{green}{2}[\/latex] by the\u00a0divisor [latex]x-1[\/latex]: [latex]\\color{green}{2}(x-1)=\\color{green}{2x-2}[\/latex]. <\/span><\/p>\n<p><span style=\"font-size: 16px; orphans: 1;\">FInally, subtract:\u00a0 [latex]\\color{red}{(2x+1)}-\\color{green}{(2x-2)}=(\\color{red}{2x}-\\color{green}{2x})+(\\color{red}{1}-(\\color{green}{-2}))=\\color{orange}{3}[\/latex].<\/span><\/p>\n<p><span style=\"font-size: 16px;\">We have found that\u00a0[latex]\\dfrac{x^2+x+1}{x-1}=x+2+\\dfrac{3}{x-1}[\/latex].<\/span><\/td>\n<td style=\"width: 33.6325%; text-align: center;\"><span style=\"font-size: 16px; orphans: 1; text-align: center;\">[latex]\\begin{aligned}\\color{blue}{x}+\\color{green}{2}\\\\x-1 )\\overline{x^2+x+1}\\\\ \\color{blue}{x^2-x}\\;\\;\\;\\;\\;\\;\\\\ \\overline{\\color{red}{2x+1}}\\\\\\underline{\\color{green}{2x-2}}\\\\ \\color{orange}{3}\\end{aligned}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can identify the dividend, divisor, quotient, and remainder by writing this answer in its equivalent form:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllll}&x^2+x+1&=&(x-1)&(x+2)&+&3\\\\&\\text{Dividend}&&\\text{Divisor}&\\text{Quotient}&&\\text{Remainder}\\end{array}[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\n<h4>Solution<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204321\/CNX_Precalc_revised_eq_12.png\" alt=\"Long Division of 2x cubed minus 3x squared plus 4x plus 5 divided by the binomial x plus 2. The answer is 2x squared minus 7x plus 18 with a remainder of -31.\" width=\"522\" height=\"462\" \/><\/p>\n<h4>Answer<\/h4>\n<p style=\"text-align: center;\">[latex]\\dfrac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18+\\dfrac{-31}{x+2}[\/latex]<\/p>\n<p style=\"text-align: left;\">or equivalently,<\/p>\n<p style=\"text-align: center;\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)+(-31)[\/latex]<\/p>\n<p>We can identify the\u00a0<strong>dividend<\/strong>,\u00a0<strong>divisor<\/strong>,\u00a0<strong>quotient<\/strong>, and\u00a0<strong>remainder<\/strong>.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204324\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/><\/div>\n<p>Notice that the remainder in this example is negative. Even when the remainder is negative, we add it as a negative remainder.<\/p>\n<\/div>\n<\/div>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>using long division to divide polynomials<\/h3>\n<ol>\n<li>Set up the division problem based on the rules of descending powers and no gaps.<\/li>\n<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n<li>Put the answer as the first term in the quotient.<\/li>\n<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n<li>Subtract the product in step 4 from all the terms above it.<\/li>\n<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Long Division of the trinomial 5x squared plus 3x minus 2 the binomial x plus 1. The answer is 5x minus 2 with remainder of zero.\" width=\"426\" height=\"288\" \/><br \/>\nThe quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This division problem had a remainder of 0. This tells us that the dividend is divided exactly by the divisor and that, therefore,\u00a0 the divisor and the quotient are factors of the dividend.\u00a0\u00a0<span style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\n<h4>Solution<\/h4>\n<div class=\"wp-nocaption aligncenter wp-image-11885\">\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"Long division of the polynomial 6x to the third power plus 11x squared minus 31x plus 1 by the binomial 3x minus 2.\" width=\"621\" height=\"153\" \/><\/a><\/p>\n<\/div>\n<p>There is a remainder of 1. We can express the result as:<\/p>\n<p>[latex]\\dfrac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\dfrac{1}{3x - 2}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our work by using the Division Algorithm to rewrite the solution then multiplying.<\/p>\n<p>[latex]\\;\\;\\;\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1\\\\=3x\\left(2{x}^{2}+5x - 7\\right)-2\\left(2{x}^{2}+5x - 7\\right)+1\\\\=6x^3+15x^2-21x-4x^2-10x+14+1\\\\=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/p>\n<p>Notice, as we write our result,<\/p>\n<ul id=\"fs-id1165135152079\">\n<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\n<li>the divisor is [latex]3x - 2[\/latex]<\/li>\n<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\n<li>the remainder is\u00a01<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm202\">Show Answer<\/span><\/p>\n<div id=\"qhjm202\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{16x^3-12x^2+20x-3}{4x+5}=4x^2-8x+15+\\left (\\dfrac{-78}{4x+5}\\right )[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4150\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Two Rules for Setting Up a Long Division. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Divide a Polynomial by a Monomial. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Division algorithm for polynomials; Examples and Try Its: hjm202, hjm871; hjm761; hjm096. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 29482. <strong>Authored by<\/strong>: McClure,Caren. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program.. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext.%20\">http:\/\/nrocnetwork.org\/dm-opentext.%20<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Two Rules for Setting Up a Long Division\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 29482\",\"author\":\"McClure,Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Hazel McKenna\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program.\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext. \",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Divide a Polynomial by a Monomial\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Division algorithm for polynomials; Examples and Try Its: hjm202, hjm871; hjm761; hjm096\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4150","chapter","type-chapter","status-publish","hentry"],"part":4124,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4150","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4150\/revisions"}],"predecessor-version":[{"id":4880,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4150\/revisions\/4880"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/4124"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4150\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=4150"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4150"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=4150"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=4150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}