{"id":18365,"date":"2022-04-15T21:16:38","date_gmt":"2022-04-15T21:16:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18365"},"modified":"2022-04-28T23:16:53","modified_gmt":"2022-04-28T23:16:53","slug":"cr-16-dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-16-dividing-polynomials\/","title":{"raw":"CR.16: Dividing Polynomials","rendered":"CR.16: Dividing Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify a polynomial expression using the quotient property of exponents<\/li>\r\n \t<li>Divide a polynomial by a monomial<\/li>\r\n \t<li>Use long division to divide polynomials.<\/li>\r\n \t<li>Use synthetic division to divide polynomials.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\r\nNow we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]\\frac{a}{b}=\\frac{a\\cdot c}{b\\cdot c}\\text{ and }\\frac{a\\cdot c}{b\\cdot c}=\\frac{a}{b}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nAs before, we'll try to discover a property by looking at some examples.\r\n[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill &amp; &amp; &amp; \\hfill \\frac{{x}^{5}}{{x}^{2}}\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ \\text{What do they mean?}\\hfill &amp; &amp; &amp; \\hfill \\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{x\\cdot x}{x\\cdot x\\cdot x}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {x}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]\r\nNotice that in each case the bases were the same and we subtracted the exponents.\r\n<ul id=\"fs-id916657\">\r\n \t<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\r\n \t<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\r\n<\/ul>\r\nWe write:\r\n\r\n[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then\r\n\r\n[latex]\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m&gt;n\\text{ and }\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}},n&gt;m[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nA couple of examples with numbers may help to verify this property.\r\n\r\n[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill &amp; &amp; &amp; \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill &amp; &amp; &amp; \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill &amp; &amp; &amp; \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]\r\n\r\nWhen we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{{x}^{10}}{{x}^{8}}[\/latex]\r\n2. [latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\r\n<td>[latex]\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex] .<\/td>\r\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\r\n<td>[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex].<\/td>\r\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that when the larger exponent is in the numerator, we are left with factors in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146219[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{{b}^{10}}{{b}^{15}}[\/latex]\r\n2. [latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]\r\n[reveal-answer q=\"738923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"738923\"]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">1.<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.34375px;\">\r\n<td style=\"height: 15.34375px;\">Since [latex]15&gt;10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\r\n<td style=\"height: 15.34375px;\">[latex]\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Use the quotient property with [latex]n&gt;m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{1}{{b}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]5&gt;3[\/latex], there are more factors of [\/latex]3[\/latex] in the denominator.<\/td>\r\n<td>[latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]n&gt;m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td>[latex]\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{1}{{3}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Apply the exponent.<\/td>\r\n<td>[latex]\\frac{1}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146220[\/ohm_question]\r\n\r\n<\/div>\r\nNow let's see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{{a}^{5}}{{a}^{9}}[\/latex]\r\n2. [latex]\\frac{{x}^{11}}{{x}^{7}}[\/latex]\r\n[reveal-answer q=\"903400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903400\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15.2344px;\">\r\n<td style=\"height: 15.2344px;\">1.<\/td>\r\n<td style=\"height: 15.2344px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Since [latex]9&gt;5[\/latex], there are more [latex]a[\/latex] 's in the denominator and so we will end up with factors in the denominator.<\/td>\r\n<td style=\"height: 30px;\">[latex]\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Use the Quotient Property for [latex]n&gt;m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 30px;\">[latex]\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{1}{{a}^{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">2.<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\r\n<td style=\"height: 30px;\">[latex]\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.28125px;\">\r\n<td style=\"height: 15.28125px;\">Use the Quotient Property for [latex]m&gt;n,\\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}[\/latex].<\/td>\r\n<td style=\"height: 15.28125px;\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]{x}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146889[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]\\frac{40x^2y^6}{8xy}[\/latex]\r\n\r\n[reveal-answer q=\"903500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903500\"]\r\n\r\nSolution\r\nTo divide two monomials, perform the indicated division by using the laws of exponents. We can rewrite the problem in the following way:\r\n[latex]\\frac{40x^2y^6}{8xy}=\\frac{40}{8}\\cdot \\frac{x^2}{x} \\cdot \\frac{y^6}{y}[\/latex]\r\n\r\nBegin by dividing the numbers: [latex]\\frac{40}{8}=5[\/latex]\r\n\r\nNow use the laws of exponents [latex]\\frac{a^m}{a^n}=a^{m-n}[\/latex]\r\n\r\n[latex]\\frac{x^2}{x}=x^{2-1}=x[\/latex]\r\n[latex]\\frac{y^6}{y}=y^{6-1}=y^5[\/latex]\r\n\r\nUsing the information gathered above, we have the following:\r\n[latex]\\frac{40}{8}\\cdot \\frac{x^2}{x} \\cdot \\frac{y^6}{y}=5 \\cdot x \\cdot y^5[\/latex]\r\n\r\nTherefore, [latex]\\frac{40x^2y^6}{8xy}=5xy^5[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]248400[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.\r\n\r\nhttps:\/\/youtu.be\/Jmf-CPhm3XM\r\n<h2>Divide a polynomial by a monomial<\/h2>\r\nWe will expand upon what has already been discussed. We will now add another layer to this idea by dividing polynomials by monomials, and by binomials.\r\n\r\nThe distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\r\nOr you can first divide each term by 2, then simplify the result.\r\n<p style=\"text-align: center;\">[latex] \\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\r\nEither way gives you the same result. The second way is helpful when you can\u2019t combine like terms in the numerator. Let\u2019s try something similar with a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q641821\">Show Solution<\/span>\r\n<div id=\"q641821\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nDistribute [latex]3a^2[\/latex]<i> <\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]\r\n\r\n[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]\r\n\r\nDivide each term, a monomial divided by another monomial.\r\n\r\n[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]\r\n\r\nRewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.\r\n\r\n[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q324719\">Show Solution<\/span>\r\n<div id=\"q324719\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nDivide each term in the polynomial by the monomial.\r\n<p style=\"text-align: center;\">[latex] \\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\r\nNote how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn\u2019t divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.\r\n\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] -\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]248406[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use long division to divide polynomials<\/h2>\r\n<p id=\"fs-id1165135191647\">We are familiar with the <strong>long division<\/strong> algorithm for ordinary arithmetic. 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.<span id=\"fs-id1165137564295\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division. Step 1, 5 times 3 equals 15 and 17 minus 15 equals 2. Step 2: Bring down the 8. Step 3: 9 times 3 equals 27 and 28 minus 27 equals 1. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/><\/span><\/p>\r\n<p id=\"fs-id1165134170235\">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>\r\n\r\n<div id=\"eip-474\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{dividend}&amp;= \\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder} \\\\ 178&amp;=\\left(3\\cdot 59\\right)+1 \\\\ &amp;=177+1 \\\\ &amp;=178 \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\r\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:<span id=\"eip-id1167404718588\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"574\" height=\"508\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135191694\">We have found<\/p>\r\n\r\n<div id=\"eip-334\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137823279\">or<\/p>\r\n\r\n<div id=\"eip-212\" class=\"equation unnumbered\" 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]<\/div>\r\n<p id=\"fs-id1165135181270\">We can identify the <strong>dividend<\/strong>, the <strong>divisor<\/strong>, the <strong>quotient<\/strong>, and the <strong>remainder<\/strong>.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/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\" \/>\r\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\r\n\r\n<div id=\"fs-id1165135508595\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Division Algorithm<\/h3>\r\n<p id=\"fs-id1165137854177\">The <strong>Division Algorithm<\/strong> states that, given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\r\n<p id=\"fs-id1165137664631\">[latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165134540113\">If [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that, in this case, both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135638531\" class=\"note precalculus howto textbox\">\r\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/h3>\r\n<ol id=\"eip-id1165134557348\">\r\n \t<li>Set up the division problem.<\/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>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t<li>Subtract the bottom <strong>binomial<\/strong> from the top binomial.<\/li>\r\n \t<li>Bring down the next term of the dividend.<\/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 id=\"Example_03_05_01\" class=\"example\">\r\n<div id=\"fs-id1165137817675\" class=\"exercise\">\r\n<div id=\"fs-id1165137817678\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165137817683\">Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\r\n[reveal-answer q=\"463081\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"463081\"]\r\n<h3 id=\"eip-id1170045615164\"><span id=\"eip-id1169254772605\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"460\" height=\"312\" \/><\/span><\/h3>\r\n<p id=\"fs-id1165137639118\">The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\r\n<p id=\"fs-id1165134058382\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135372071\">This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_02\" class=\"example\">\r\n<div id=\"fs-id1165135372082\" class=\"exercise\">\r\n<div id=\"fs-id1165135372084\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165134352552\">Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\r\n[reveal-answer q=\"458321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458321\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"874\" height=\"206\" \/><\/a>\r\n<p id=\"fs-id1165135639821\">There is a remainder of 1. We can express the result as:<\/p>\r\n\r\n<div id=\"eip-id1165134294806\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/div>\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135340597\">We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\r\n\r\n<div id=\"eip-id1165135428302\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/div>\r\n<p id=\"fs-id1165135152076\">Notice, as we write our result,<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165135152079\">\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135545763\">Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].<\/p>\r\n[reveal-answer q=\"504428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"504428\"]\r\n\r\n[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]100259[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use synthetic division to divide polynomials<\/h2>\r\n<p id=\"fs-id1165137932627\">As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.<\/p>\r\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\r\n<p id=\"fs-id1165137932639\">Divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\r\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<span id=\"eip-id1163740536072\">\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"292\" height=\"263\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\r\n<span id=\"fs-id1165134305375\">\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0042.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" width=\"522\" height=\"118\" \/><\/span>\r\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \"divisor\" to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\r\n<span id=\"fs-id1165137696374\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0112.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" width=\"553\" height=\"84\" \/><\/span>\r\n<p id=\"fs-id1165137696388\">We then multiply it by the \"divisor\" and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in Example 3.<\/p>\r\n\r\n<div id=\"fs-id1165135383640\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Synthetic Division<\/h3>\r\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form <em>x<\/em> \u2013\u00a0<em>k<\/em>.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\r\n<ol id=\"fs-id1165135393418\">\r\n \t<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\r\n \t<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_05_03\" class=\"example\">\r\n<div id=\"fs-id1165135383099\" class=\"exercise\">\r\n<div id=\"fs-id1165135383101\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135383107\">Use synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].<\/p>\r\n[reveal-answer q=\"125978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"125978\"]\r\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.<\/p>\r\n<span id=\"fs-id1165135177629\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span>\r\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by <em>k<\/em>.<\/p>\r\n<span id=\"fs-id1165135439966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" \/><\/span>\r\n<p id=\"fs-id1165135179942\">Continue by adding the numbers in the second column. Multiply the resulting number by <em>k<\/em>.\u00a0Write the result in the next column. Then add the numbers in the third column.<\/p>\r\n<span id=\"fs-id1165135179966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row. \" \/><\/span>\r\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].\u00a0The remainder is 0. So [latex]x - 3[\/latex]\u00a0is a factor of the original polynomial.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135463247\">Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\r\n<p id=\"fs-id1165135463251\" style=\"text-align: center;\">[latex]\\left(x - 3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x - 36[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_04\" class=\"example\">\r\n<div id=\"fs-id1165135549012\" class=\"exercise\">\r\n<div id=\"fs-id1165135549014\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135549019\">Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].<\/p>\r\n[reveal-answer q=\"388484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388484\"]\r\n<p id=\"fs-id1165135173367\">The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span><\/p>\r\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of the polynomial function [latex]f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0in Figure 2\u00a0shows a zero at [latex]x=k=-2[\/latex].\u00a0This confirms that [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_05\" class=\"example\">\r\n<div id=\"fs-id1165133260470\" class=\"exercise\">\r\n<div id=\"fs-id1165133260472\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135481144\">Use synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].<\/p>\r\n[reveal-answer q=\"565402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"565402\"]\r\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_revised_eq_52.png\" alt=\"Synthetic Division of -9x^4+10x^3+7x^2-6 by x-1\" width=\"230\" height=\"300\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134037584\">Use synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[\/latex]\u00a0by [latex]x+7[\/latex].<\/p>\r\n[reveal-answer q=\"620042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620042\"]\r\n\r\n[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]126107[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use polynomial division to solve application problems<\/h2>\r\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\r\n\r\n<div id=\"Example_03_05_06\" class=\"example\">\r\n<div id=\"fs-id1165135403427\" class=\"exercise\">\r\n<div id=\"fs-id1165135403429\" class=\"problem textbox shaded\">\r\n<h3>Example: Using Polynomial Division in an Application Problem<\/h3>\r\n<p id=\"fs-id1165135403434\">The volume of a rectangular solid is given by the polynomial [latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x[\/latex].\u00a0The length of the solid is given by 3<em>x<\/em>\u00a0and the width is given by <em>x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\r\n[reveal-answer q=\"423911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"423911\"]\r\n<p id=\"fs-id1165135685837\">There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0102.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\" \/> <b>Figure 3<\/b>[\/caption]\r\n<p id=\"fs-id1165137843229\">We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\r\n<p style=\"text-align: center;\">[latex]V=l\\cdot w\\cdot h[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h[\/latex]<\/p>\r\n<p id=\"fs-id1165135457104\">To solve for <em>h<\/em>, first divide both sides by 3<em>x<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\\\[1 mm] \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{gathered}[\/latex]<\/p>\r\n<p id=\"fs-id1165135528878\">Now solve for <em>h<\/em>\u00a0using synthetic division.<\/p>\r\n<p style=\"text-align: center;\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\"><img class=\"aligncenter wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\" alt=\"Synthetic division with 2 as the divisor and {1, -1, -11, 18} as the quotient. The result is {1, 1, -9, 0}\" width=\"247\" height=\"142\" \/><\/a>\r\n<p id=\"fs-id1165134152722\">The quotient is [latex]{x}^{2}+x - 9[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex].\u00a0The width of the rectangle is given by <em>x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\r\n[reveal-answer q=\"145892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145892\"]\r\n\r\n[latex]3{x}^{2}-4x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<section id=\"fs-id1165135487276\" class=\"key-equations\">\r\n<table id=\"eip-id1165133432926\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Division Algorithm<\/td>\r\n<td>[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135531548\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135531552\">\r\n \t<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\r\n \t<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\r\n \t<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form <em>x \u2013\u00a0k<\/em>.<\/li>\r\n \t<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135471190\" class=\"definition\">\r\n \t<dt><strong>Division Algorithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135471195\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134486770\" class=\"definition\">\r\n \t<dt><strong>synthetic division<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form <em>x<\/em> \u2013<em> k<\/em><\/dd>\r\n<\/dl>\r\n<\/section>&nbsp;\r\n<h2 style=\"text-align: center;\">Section R.3 Homework Exercises<\/h2>\r\n1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?\r\n\r\n2.\u00a0If a polynomial of degree <em>n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?\r\n\r\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\r\n\r\n3. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]\r\n\r\n6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]\r\n\r\n7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]\r\n\r\n8.\u00a0[latex]\\left(-{x}^{2}-1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\nFor the following exercises, use synthetic division to find the quotient.\r\n\r\n14. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]\r\n\r\n16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\n18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n22.\u00a0[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]\r\n\r\n24.\u00a0[latex]\\left({x}^{3}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]\r\n\r\n26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n29. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n31. [latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]\r\n\r\n35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]\r\n\r\n36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]\r\n\r\n37. [latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\nFor the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.\r\n\r\n38. Factor is [latex]{x}^{2}-x+3[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2012.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\" \/>\r\n\r\n39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2022.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\" \/>\r\n\r\n40. Factor is [latex]{x}^{2}+2x+5[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2032.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\" \/>\r\n\r\n41. Factor is [latex]{x}^{2}+x+1[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2042.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\" \/>\r\n\r\n42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010746\/CNX_PreCalc_Figure_03_05_2052.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\" \/>\r\n\r\nFor the following exercises, use synthetic division to find the quotient and remainder.\r\n\r\n43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]\r\n\r\n44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]\r\n\r\n45. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]\r\n\r\n46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]\r\n\r\n47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]\r\n\r\nFor the following exercises, use a calculator with CAS to answer the questions.\r\n\r\n48. Consider [latex]\\frac{{x}^{k}-1}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if <em>k<\/em> = 4?\r\n\r\n49. Consider [latex]\\frac{{x}^{k}+1}{x+1}[\/latex] for [latex]k=1, 3, 5[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 7?\r\n\r\n50.\u00a0Consider [latex]\\frac{{x}^{4}-{k}^{4}}{x-k}[\/latex] for [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?\r\n\r\n51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?\r\n\r\n52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?\r\n\r\nFor the following exercises, use synthetic division to determine the quotient involving a complex number.\r\n\r\n53. [latex]\\frac{x+1}{x-i}[\/latex]\r\n\r\n54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]\r\n\r\n55. [latex]\\frac{x+1}{x+i}[\/latex]\r\n\r\n56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]\r\n\r\n57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]\r\n\r\nFor the following exercises, use the given length and area of a rectangle to express the width algebraically.\r\n\r\n58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].\r\n\r\n59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]\r\n\r\n60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]\r\n\r\nFor the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.\r\n\r\n61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].\r\n\r\n62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].\r\n\r\n63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].\r\n\r\n64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].\r\n\r\nFor the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.\r\n\r\n65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].\r\n\r\n66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].\r\n\r\n67. Volume is [latex]\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x - 32\\right)[\/latex], radius is [latex]x+4[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify a polynomial expression using the quotient property of exponents<\/li>\n<li>Divide a polynomial by a monomial<\/li>\n<li>Use long division to divide polynomials.<\/li>\n<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\n<p>Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]\\frac{a}{b}=\\frac{a\\cdot c}{b\\cdot c}\\text{ and }\\frac{a\\cdot c}{b\\cdot c}=\\frac{a}{b}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>As before, we&#8217;ll try to discover a property by looking at some examples.<br \/>\n[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill & & & \\hfill \\frac{{x}^{5}}{{x}^{2}}\\hfill & & & \\text{and}\\hfill & & & \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ \\text{What do they mean?}\\hfill & & & \\hfill \\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}\\hfill & & & & & & \\hfill \\frac{x\\cdot x}{x\\cdot x\\cdot x}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill & & & & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill & & & \\hfill {x}^{3}\\hfill & & & & & & \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<br \/>\nNotice that in each case the bases were the same and we subtracted the exponents.<\/p>\n<ul id=\"fs-id916657\">\n<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\n<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\n<\/ul>\n<p>We write:<\/p>\n<p>[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill & & & & \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill & & & & \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill & & & & \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then<\/p>\n<p>[latex]\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\\text{ and }\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}},n>m[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>A couple of examples with numbers may help to verify this property.<\/p>\n<p>[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill & & & \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill & & & \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill & & & \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]<\/p>\n<p>When we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{{x}^{10}}{{x}^{8}}[\/latex]<br \/>\n2. [latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/p>\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\n<td>[latex]\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex] .<\/td>\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\n<td>[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex].<\/td>\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146219\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146219&theme=oea&iframe_resize_id=ohm146219&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{{b}^{10}}{{b}^{15}}[\/latex]<br \/>\n2. [latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738923\">Show Solution<\/span><\/p>\n<div id=\"q738923\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">1.<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15.34375px;\">\n<td style=\"height: 15.34375px;\">Since [latex]15>10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\n<td style=\"height: 15.34375px;\">[latex]\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Use the quotient property with [latex]n>m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 15px;\">[latex]\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]\\frac{1}{{b}^{5}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since [latex]5>3[\/latex], there are more factors of [\/latex]3[\/latex] in the denominator.<\/td>\n<td>[latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]n>m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td>[latex]\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{1}{{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Apply the exponent.<\/td>\n<td>[latex]\\frac{1}{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146220&theme=oea&iframe_resize_id=ohm146220&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now let&#8217;s see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{{a}^{5}}{{a}^{9}}[\/latex]<br \/>\n2. [latex]\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903400\">Show Solution<\/span><\/p>\n<div id=\"q903400\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15.2344px;\">\n<td style=\"height: 15.2344px;\">1.<\/td>\n<td style=\"height: 15.2344px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Since [latex]9>5[\/latex], there are more [latex]a[\/latex] &#8216;s in the denominator and so we will end up with factors in the denominator.<\/td>\n<td style=\"height: 30px;\">[latex]\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Use the Quotient Property for [latex]n>m,\\frac{{a}^{m}}{{a}^{n}}=\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 30px;\">[latex]\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]\\frac{1}{{a}^{4}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">2.<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\n<td style=\"height: 30px;\">[latex]\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.28125px;\">\n<td style=\"height: 15.28125px;\">Use the Quotient Property for [latex]m>n,\\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}[\/latex].<\/td>\n<td style=\"height: 15.28125px;\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]{x}^{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146889\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146889&theme=oea&iframe_resize_id=ohm146889&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expression: [latex]\\frac{40x^2y^6}{8xy}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903500\">Show Solution<\/span><\/p>\n<div id=\"q903500\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo divide two monomials, perform the indicated division by using the laws of exponents. We can rewrite the problem in the following way:<br \/>\n[latex]\\frac{40x^2y^6}{8xy}=\\frac{40}{8}\\cdot \\frac{x^2}{x} \\cdot \\frac{y^6}{y}[\/latex]<\/p>\n<p>Begin by dividing the numbers: [latex]\\frac{40}{8}=5[\/latex]<\/p>\n<p>Now use the laws of exponents [latex]\\frac{a^m}{a^n}=a^{m-n}[\/latex]<\/p>\n<p>[latex]\\frac{x^2}{x}=x^{2-1}=x[\/latex]<br \/>\n[latex]\\frac{y^6}{y}=y^{6-1}=y^5[\/latex]<\/p>\n<p>Using the information gathered above, we have the following:<br \/>\n[latex]\\frac{40}{8}\\cdot \\frac{x^2}{x} \\cdot \\frac{y^6}{y}=5 \\cdot x \\cdot y^5[\/latex]<\/p>\n<p>Therefore, [latex]\\frac{40x^2y^6}{8xy}=5xy^5[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm248400\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=248400&theme=oea&iframe_resize_id=ohm248400&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Divide a polynomial by a monomial<\/h2>\n<p>We will expand upon what has already been discussed. We will now add another layer to this idea by dividing polynomials by monomials, and by binomials.<\/p>\n<p>The distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\n<p>Or you can first divide each term by 2, then simplify the result.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\n<p>Either way gives you the same result. The second way is helpful when you can\u2019t combine like terms in the numerator. Let\u2019s try something similar with a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q641821\">Show Solution<\/span><\/p>\n<div id=\"q641821\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Distribute [latex]3a^2[\/latex]<i> <\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]<\/p>\n<p>[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]<\/p>\n<p>Divide each term, a monomial divided by another monomial.<\/p>\n<p>[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.<\/p>\n<p>[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q324719\">Show Solution<\/span><\/p>\n<div id=\"q324719\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Divide each term in the polynomial by the monomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\n<p>Note how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn\u2019t divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm248406\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=248406&theme=oea&iframe_resize_id=ohm248406&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use long division to divide polynomials<\/h2>\n<p id=\"fs-id1165135191647\">We are familiar with the <strong>long division<\/strong> algorithm for ordinary arithmetic. 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.<span id=\"fs-id1165137564295\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division. Step 1, 5 times 3 equals 15 and 17 minus 15 equals 2. Step 2: Bring down the 8. Step 3: 9 times 3 equals 27 and 28 minus 27 equals 1. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/><\/span><\/p>\n<p id=\"fs-id1165134170235\">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<div id=\"eip-474\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{dividend}&= \\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder} \\\\ 178&=\\left(3\\cdot 59\\right)+1 \\\\ &=177+1 \\\\ &=178 \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:<span id=\"eip-id1167404718588\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"574\" height=\"508\" \/><\/span><\/p>\n<p id=\"fs-id1165135191694\">We have found<\/p>\n<div id=\"eip-334\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/div>\n<p id=\"fs-id1165137823279\">or<\/p>\n<div id=\"eip-212\" class=\"equation unnumbered\" 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]<\/div>\n<p id=\"fs-id1165135181270\">We can identify the <strong>dividend<\/strong>, the <strong>divisor<\/strong>, the <strong>quotient<\/strong>, and the <strong>remainder<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/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\" \/><\/p>\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\n<div id=\"fs-id1165135508595\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Division Algorithm<\/h3>\n<p id=\"fs-id1165137854177\">The <strong>Division Algorithm<\/strong> states that, given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\n<p id=\"fs-id1165137664631\">[latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\n<p id=\"fs-id1165134540113\">If [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that, in this case, both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135638531\" class=\"note precalculus howto textbox\">\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/h3>\n<ol id=\"eip-id1165134557348\">\n<li>Set up the division problem.<\/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>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n<li>Subtract the bottom <strong>binomial<\/strong> from the top binomial.<\/li>\n<li>Bring down the next term of the dividend.<\/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 id=\"Example_03_05_01\" class=\"example\">\n<div id=\"fs-id1165137817675\" class=\"exercise\">\n<div id=\"fs-id1165137817678\" class=\"problem textbox shaded\">\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165137817683\">Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q463081\">Show Solution<\/span><\/p>\n<div id=\"q463081\" class=\"hidden-answer\" style=\"display: none\">\n<h3 id=\"eip-id1170045615164\"><span id=\"eip-id1169254772605\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"460\" height=\"312\" \/><\/span><\/h3>\n<p id=\"fs-id1165137639118\">The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p id=\"fs-id1165134058382\">or<\/p>\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135372071\">This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_02\" class=\"example\">\n<div id=\"fs-id1165135372082\" class=\"exercise\">\n<div id=\"fs-id1165135372084\" class=\"problem textbox shaded\">\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165134352552\">Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q458321\">Show Solution<\/span><\/p>\n<div id=\"q458321\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"874\" height=\"206\" \/><\/a><\/p>\n<p id=\"fs-id1165135639821\">There is a remainder of 1. We can express the result as:<\/p>\n<div id=\"eip-id1165134294806\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135340597\">We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\n<div id=\"eip-id1165135428302\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/div>\n<p id=\"fs-id1165135152076\">Notice, as we write our result,<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul id=\"fs-id1165135152079\">\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>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135545763\">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=\"q504428\">Show Solution<\/span><\/p>\n<div id=\"q504428\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm100259\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=100259&theme=oea&iframe_resize_id=ohm100259\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use synthetic division to divide polynomials<\/h2>\n<p id=\"fs-id1165137932627\">As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.<\/p>\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\n<p id=\"fs-id1165137932639\">Divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<span id=\"eip-id1163740536072\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"292\" height=\"263\" \/><\/span><\/p>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\n<p><span id=\"fs-id1165134305375\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0042.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" width=\"522\" height=\"118\" \/><\/span><\/p>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the &#8220;divisor&#8221; to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\n<p><span id=\"fs-id1165137696374\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0112.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" width=\"553\" height=\"84\" \/><\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the &#8220;divisor&#8221; and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in Example 3.<\/p>\n<div id=\"fs-id1165135383640\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Synthetic Division<\/h3>\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form <em>x<\/em> \u2013\u00a0<em>k<\/em>.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\n<\/div>\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\n<ol id=\"fs-id1165135393418\">\n<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\n<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_05_03\" class=\"example\">\n<div id=\"fs-id1165135383099\" class=\"exercise\">\n<div id=\"fs-id1165135383101\" class=\"problem textbox shaded\">\n<h3>Example: Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135383107\">Use synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125978\">Show Solution<\/span><\/p>\n<div id=\"q125978\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.<\/p>\n<p><span id=\"fs-id1165135177629\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span><\/p>\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by <em>k<\/em>.<\/p>\n<p><span id=\"fs-id1165135439966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" \/><\/span><\/p>\n<p id=\"fs-id1165135179942\">Continue by adding the numbers in the second column. Multiply the resulting number by <em>k<\/em>.\u00a0Write the result in the next column. Then add the numbers in the third column.<\/p>\n<p><span id=\"fs-id1165135179966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" \/><\/span><\/p>\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].\u00a0The remainder is 0. So [latex]x - 3[\/latex]\u00a0is a factor of the original polynomial.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135463247\">Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<p id=\"fs-id1165135463251\" style=\"text-align: center;\">[latex]\\left(x - 3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x - 36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_04\" class=\"example\">\n<div id=\"fs-id1165135549012\" class=\"exercise\">\n<div id=\"fs-id1165135549014\" class=\"problem textbox shaded\">\n<h3>Example: Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135549019\">Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388484\">Show Solution<\/span><\/p>\n<div id=\"q388484\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173367\">The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span><\/p>\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of the polynomial function [latex]f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0in Figure 2\u00a0shows a zero at [latex]x=k=-2[\/latex].\u00a0This confirms that [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_05\" class=\"example\">\n<div id=\"fs-id1165133260470\" class=\"exercise\">\n<div id=\"fs-id1165133260472\" class=\"problem textbox shaded\">\n<h3>Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135481144\">Use synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q565402\">Show Solution<\/span><\/p>\n<div id=\"q565402\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_revised_eq_52.png\" alt=\"Synthetic Division of -9x^4+10x^3+7x^2-6 by x-1\" width=\"230\" height=\"300\" \/><\/span><\/p>\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134037584\">Use synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[\/latex]\u00a0by [latex]x+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q620042\">Show Solution<\/span><\/p>\n<div id=\"q620042\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm126107\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=126107&theme=oea&iframe_resize_id=ohm126107\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use polynomial division to solve application problems<\/h2>\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n<div id=\"Example_03_05_06\" class=\"example\">\n<div id=\"fs-id1165135403427\" class=\"exercise\">\n<div id=\"fs-id1165135403429\" class=\"problem textbox shaded\">\n<h3>Example: Using Polynomial Division in an Application Problem<\/h3>\n<p id=\"fs-id1165135403434\">The volume of a rectangular solid is given by the polynomial [latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x[\/latex].\u00a0The length of the solid is given by 3<em>x<\/em>\u00a0and the width is given by <em>x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q423911\">Show Solution<\/span><\/p>\n<div id=\"q423911\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135685837\">There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0102.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137843229\">We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n<p style=\"text-align: center;\">[latex]V=l\\cdot w\\cdot h[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h[\/latex]<\/p>\n<p id=\"fs-id1165135457104\">To solve for <em>h<\/em>, first divide both sides by 3<em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\\\[1 mm] \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{gathered}[\/latex]<\/p>\n<p id=\"fs-id1165135528878\">Now solve for <em>h<\/em>\u00a0using synthetic division.<\/p>\n<p style=\"text-align: center;\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\" alt=\"Synthetic division with 2 as the divisor and {1, -1, -11, 18} as the quotient. The result is {1, 1, -9, 0}\" width=\"247\" height=\"142\" \/><\/a><\/p>\n<p id=\"fs-id1165134152722\">The quotient is [latex]{x}^{2}+x - 9[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex].\u00a0The width of the rectangle is given by <em>x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q145892\">Show Solution<\/span><\/p>\n<div id=\"q145892\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{2}-4x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<section id=\"fs-id1165135487276\" class=\"key-equations\">\n<table id=\"eip-id1165133432926\" summary=\"..\">\n<tbody>\n<tr>\n<td>Division Algorithm<\/td>\n<td>[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135531548\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135531552\">\n<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\n<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\n<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form <em>x \u2013\u00a0k<\/em>.<\/li>\n<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135471190\" class=\"definition\">\n<dt><strong>Division Algorithm<\/strong><\/dt>\n<dd id=\"fs-id1165135471195\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134486770\" class=\"definition\">\n<dt><strong>synthetic division<\/strong><\/dt>\n<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form <em>x<\/em> \u2013<em> k<\/em><\/dd>\n<\/dl>\n<\/section>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center;\">Section R.3 Homework Exercises<\/h2>\n<p>1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n<p>2.\u00a0If a polynomial of degree <em>n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n<p>For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n<p>3. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n<p>4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]<\/p>\n<p>6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]<\/p>\n<p>7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]<\/p>\n<p>8.\u00a0[latex]\\left(-{x}^{2}-1\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n<p>12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>For the following exercises, use synthetic division to find the quotient.<\/p>\n<p>14. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]<\/p>\n<p>16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]<\/p>\n<p>18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n<p>19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n<p>21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n<p>22.\u00a0[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]<\/p>\n<p>24.\u00a0[latex]\\left({x}^{3}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]<\/p>\n<p>26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n<p>28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n<p>29. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n<p>31. [latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]<\/p>\n<p>35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]<\/p>\n<p>36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]<\/p>\n<p>37. [latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\\div \\left(2x+1\\right)[\/latex]<\/p>\n<p>For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\n<p>38. Factor is [latex]{x}^{2}-x+3[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2012.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\" \/><\/p>\n<p>39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2022.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\" \/><\/p>\n<p>40. Factor is [latex]{x}^{2}+2x+5[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2032.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\" \/><\/p>\n<p>41. Factor is [latex]{x}^{2}+x+1[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2042.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\" \/><\/p>\n<p>42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010746\/CNX_PreCalc_Figure_03_05_2052.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\" \/><\/p>\n<p>For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n<p>43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]<\/p>\n<p>44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]<\/p>\n<p>45. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]<\/p>\n<p>46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]<\/p>\n<p>47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]<\/p>\n<p>For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n<p>48. Consider [latex]\\frac{{x}^{k}-1}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if <em>k<\/em> = 4?<\/p>\n<p>49. Consider [latex]\\frac{{x}^{k}+1}{x+1}[\/latex] for [latex]k=1, 3, 5[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 7?<\/p>\n<p>50.\u00a0Consider [latex]\\frac{{x}^{4}-{k}^{4}}{x-k}[\/latex] for [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?<\/p>\n<p>51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?<\/p>\n<p>52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?<\/p>\n<p>For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n<p>53. [latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n<p>54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]<\/p>\n<p>55. [latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n<p>56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]<\/p>\n<p>57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]<\/p>\n<p>For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n<p>58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].<\/p>\n<p>59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]<\/p>\n<p>60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]<\/p>\n<p>For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n<p>61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].<\/p>\n<p>62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].<\/p>\n<p>63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].<\/p>\n<p>64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].<\/p>\n<p>For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n<p>65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].<\/p>\n<p>66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].<\/p>\n<p>67. 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