{"id":4708,"date":"2017-12-26T16:55:46","date_gmt":"2017-12-26T16:55:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/divide-polynomials\/"},"modified":"2023-11-08T13:20:10","modified_gmt":"2023-11-08T13:20:10","slug":"divide-polynomials","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/divide-polynomials\/","title":{"raw":"8.3 - Division of Polynomials","rendered":"8.3 &#8211; Division of Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(8.3.1) - Divide a polynomial by a monomial<\/li>\r\n \t<li>(8.3.2) - Perform long division of polynomials<\/li>\r\n \t<li>(8.3.3) - Applications of polynomial division<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.[footnote]National Park Service. \"Lincoln Memorial Building Statistics.\" <a href=\"http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014[\/footnote]\u00a0We can easily find the volume using elementary geometry.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}V=l\\cdot w\\cdot h\\hfill \\\\ \\text{ }=61.5\\cdot 40\\cdot 30\\hfill \\\\ \\text{ }=73,800\\hfill \\end{array}[\/latex]<\/p>\r\nSo the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex].\u00a0Suppose we knew the volume, length, and width. We could divide to find the height.\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{l}h=\\frac{V}{l\\cdot w}\\hfill \\\\ \\text{ }=\\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ \\text{ }=30\\hfill \\end{array}[\/latex]<\/p>\r\nAs we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any or all of the measurements contain variable expressions. For example, suppose 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 [latex]3x[\/latex];\u00a0the width is given by [latex]x - 2[\/latex].\u00a0To find the height of the solid, we can use polynomial division, which is the focus of this section.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165512\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/> Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)[\/caption]\r\n<h1>(8.3.1) - Divide a polynomial by a monomial<\/h1>\r\nDivision of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will 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]\\displaystyle \\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\r\nOr you can\u00a0first divide each term by 2, then simplify the result.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\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't combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex]\\displaystyle \\frac{9a^3+6a}{3a^2}[\/latex]\r\n[reveal-answer q=\"641821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641821\"]\r\n\r\nDistribute [latex]3a^2[\/latex]<i>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]<\/p>\r\nDivide each term, a monomial divided by another monomial.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]<\/p>\r\nRewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.\r\n<p style=\"text-align: center\">[latex]\\displaystyle 3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\displaystyle \\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\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]\\displaystyle \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]\r\n[reveal-answer q=\"324719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324719\"]Divide each term in the polynomial by the monomial.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\r\nNote how the term[latex]\\displaystyle -\\frac{18}{-6y}[\/latex] does not have a [latex]y[\/latex] in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, [latex]27[\/latex] doesn't 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]\\displaystyle -\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\displaystyle \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe same principle will apply for polynomials with more than one variable.\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLE<\/h3>\r\nDivide. [latex]\\displaystyle \\frac{18x^3y-36xy^2+12x}{-3xy}[\/latex]\r\n\r\n[reveal-answer q=\"604284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604284\"]Divide each term by the divisor. Be careful with the signs!\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{18x^3y}{-3xy}-\\frac{36xy^2}{-3xy}+\\frac{12x}{-3xy} [\/latex]<\/p>\r\nSimplify, using the rules of exponents.\r\n<p style=\"text-align: center\">[latex]\\displaystyle -6x^2+12y-\\frac{4}{y}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\displaystyle -6x^2+12y-\\frac{4}{y}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice, that in the examples above, the answer was a<em><strong> rational expression.<\/strong><\/em>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]38831[\/ohm_question]\r\n\r\n<\/div>\r\n<h1>(8.3.2) - Performing Long Division of Polynomials<\/h1>\r\nWe 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.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165514\/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\" \/>\r\n\r\nAnother way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{dividend = }\\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder}\\hfill \\\\ 178=\\left(3\\cdot 59\\right)+1\\hfill \\\\ =177+1\\hfill \\\\ =178\\hfill \\end{array}[\/latex]<\/p>\r\nWe call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.\r\n\r\nDivision 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:\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165517\/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=\"522\" height=\"462\" \/>\r\n\r\nWe have found\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/p>\r\nWe can identify the <strong>dividend<\/strong>, the <strong>divisor<\/strong>, the <strong>quotient<\/strong>, and the <strong>remainder<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165520\/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\r\nWriting the result in this manner illustrates the Division Algorithm.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Division Algorithm<\/h3>\r\nThe <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\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[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].\r\n\r\nIf [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].\r\n\r\n<\/div>\r\n<div class=\"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>\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 class=\"textbox exercises\">\r\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\r\nDivide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].\r\n\r\n[reveal-answer q=\"996959\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"996959\"]\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165522\/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=\"426\" height=\"288\" \/>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\r\nor\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\nThis 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\r\nDivide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].\r\n\r\n[reveal-answer q=\"850001\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"850001\"]\r\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165525\/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=\"621\" height=\"153\" \/><\/a>\r\n\r\nThere is a remainder of 1. We can express the result as:\r\n\r\n[latex]\\displaystyle \\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can check our work by using the Division Algorithm to rewrite the solution. Then multiply.\r\n\r\n[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]\r\n\r\nNotice, as we write our result,\r\n<ul id=\"fs-id1165135152079\">\r\n \t<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n \t<li>the divisor is [latex]3x - 2[\/latex]<\/li>\r\n \t<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\r\n \t<li>the remainder is\u00a01<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>EXAMPLE<\/h3>\r\nDivide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].\r\n\r\n[reveal-answer q=\"198989\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"198989\"][latex]\\displaystyle 4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide: [latex]\\displaystyle \\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]\r\n[reveal-answer q=\"523374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523374\"]In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152934\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/>\r\n\r\nFocus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex]\\displaystyle \\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.\r\n\r\nMultiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152935\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/>\r\n\r\nRewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152936\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nBring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152938\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nNow, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152939\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/>\r\n\r\nRemember to watch the signs!\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152940\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/>\r\n\r\nHow many [latex]x[\/latex]\u2019s are there in [latex]3x[\/latex]? Since there are 3, multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152941\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/>\r\n\r\nContinue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is 0, which is less than the degree of [latex]x-3[\/latex], which is [latex]1[\/latex].\r\n\r\nAlso notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152942\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/>\r\nYou can write the remainder using the symbol [latex]R[\/latex], or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.\r\n<h4>Answer<\/h4>\r\n[latex]\\large \\begin{array}{r}(x^{3}\u20136x\u201310)(x\u20133)=x^2+3x+3+R-1,\\\\ x^2+3x+3+ \\frac{-1}{x-3}, \\text{ or }\\\\ x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]\r\n\r\n<strong>Check the result:<\/strong>\r\n<p style=\"text-align: center\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we show another example of dividing a degree three trinomial by a binomial, also with a \"missing\" term..\r\n\r\nhttps:\/\/youtu.be\/Rxds7Q_UTeo\r\n<h1>(8.3.3) - Application of Polynomial Division<\/h1>\r\nPolynomial 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Polynomial Division in an Application Problem<\/h3>\r\nThe 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 [latex]3x[\/latex]\u00a0and the width is given by [latex]x-2[\/latex].\u00a0Find the height of the solid.\r\n\r\n[reveal-answer q=\"426222\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"426222\"]\r\nThere 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165543\/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\" \/>\r\n\r\nWe can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}V=l\\cdot w\\cdot h\\\\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h\\end{array}[\/latex]<\/p>\r\nTo solve for [latex]h[\/latex], first divide both sides by [latex]3x[\/latex].\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{l}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\\\ \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{array}[\/latex]<\/p>\r\nNow solve for [latex]h[\/latex]\u00a0using synthetic division.\r\n<p style=\"text-align: center\">[latex]\\displaystyle h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\r\n<img class=\"aligncenter size-full wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165546\/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=\"204\" height=\"118\" \/>\r\n\r\nThe 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>ExAMPLE<\/h3>\r\nThe 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 [latex]x+6[\/latex].\u00a0Find an expression for the length of the rectangle.\r\n\r\n[reveal-answer q=\"4034\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"4034\"][latex]3{x}^{2}-4x+1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(8.3.1) &#8211; Divide a polynomial by a monomial<\/li>\n<li>(8.3.2) &#8211; Perform long division of polynomials<\/li>\n<li>(8.3.3) &#8211; Applications of polynomial division<\/li>\n<\/ul>\n<\/div>\n<p>The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.<a class=\"footnote\" title=\"National Park Service. &quot;Lincoln Memorial Building Statistics.&quot; http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm. Accessed 4\/3\/2014\" id=\"return-footnote-4708-1\" href=\"#footnote-4708-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0We can easily find the volume using elementary geometry.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}V=l\\cdot w\\cdot h\\hfill \\\\ \\text{ }=61.5\\cdot 40\\cdot 30\\hfill \\\\ \\text{ }=73,800\\hfill \\end{array}[\/latex]<\/p>\n<p>So the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex].\u00a0Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{l}h=\\frac{V}{l\\cdot w}\\hfill \\\\ \\text{ }=\\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ \\text{ }=30\\hfill \\end{array}[\/latex]<\/p>\n<p>As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any or all of the measurements contain variable expressions. For example, suppose 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 [latex]3x[\/latex];\u00a0the width is given by [latex]x - 2[\/latex].\u00a0To find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165512\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\">Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<\/p>\n<\/div>\n<h1>(8.3.1) &#8211; Divide a polynomial by a monomial<\/h1>\n<p>Division of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will 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]\\displaystyle \\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\n<p>Or you can\u00a0first divide each term by 2, then simplify the result.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\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&#8217;t combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\displaystyle \\frac{9a^3+6a}{3a^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\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 style=\"text-align: center\">[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 style=\"text-align: center\">[latex]\\displaystyle 3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\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]\\displaystyle \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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\">Divide each term in the polynomial by the monomial.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\n<p>Note how the term[latex]\\displaystyle -\\frac{18}{-6y}[\/latex] does not have a [latex]y[\/latex] in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, [latex]27[\/latex] doesn&#8217;t 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]\\displaystyle -\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\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<p>The same principle will apply for polynomials with more than one variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>Divide. [latex]\\displaystyle \\frac{18x^3y-36xy^2+12x}{-3xy}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604284\">Show Solution<\/span><\/p>\n<div id=\"q604284\" class=\"hidden-answer\" style=\"display: none\">Divide each term by the divisor. Be careful with the signs!<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{18x^3y}{-3xy}-\\frac{36xy^2}{-3xy}+\\frac{12x}{-3xy}[\/latex]<\/p>\n<p>Simplify, using the rules of exponents.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle -6x^2+12y-\\frac{4}{y}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle -6x^2+12y-\\frac{4}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice, that in the examples above, the answer was a<em><strong> rational expression.<\/strong><\/em><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm38831\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=38831&theme=oea&iframe_resize_id=ohm38831&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h1>(8.3.2) &#8211; Performing Long Division of Polynomials<\/h1>\n<p>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.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165514\/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\" \/><\/p>\n<p>Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{dividend = }\\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder}\\hfill \\\\ 178=\\left(3\\cdot 59\\right)+1\\hfill \\\\ =177+1\\hfill \\\\ =178\\hfill \\end{array}[\/latex]<\/p>\n<p>We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\n<p>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:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165517\/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=\"522\" height=\"462\" \/><\/p>\n<p>We have found<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/p>\n<p>We can identify the <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\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165520\/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>Writing the result in this manner illustrates the Division Algorithm.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Division Algorithm<\/h3>\n<p>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>[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>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 class=\"textbox\">\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p>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=\"q996959\">Solution<\/span><\/p>\n<div id=\"q996959\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165522\/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=\"426\" height=\"288\" \/>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p>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>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 class=\"textbox exercises\">\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\n<p>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=\"q850001\">Solution<\/span><\/p>\n<div id=\"q850001\" class=\"hidden-answer\" style=\"display: none\">\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165525\/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=\"621\" height=\"153\" \/><\/a><\/p>\n<p>There is a remainder of 1. We can express the result as:<\/p>\n<p>[latex]\\displaystyle \\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\n<p>[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/p>\n<p>Notice, as we write our result,<\/p>\n<ul id=\"fs-id1165135152079\">\n<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\n<li>the divisor is [latex]3x - 2[\/latex]<\/li>\n<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\n<li>the remainder is\u00a01<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>EXAMPLE<\/h3>\n<p>Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q198989\">Solution<\/span><\/p>\n<div id=\"q198989\" class=\"hidden-answer\" style=\"display: none\">[latex]\\displaystyle 4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide: [latex]\\displaystyle \\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523374\">Show Solution<\/span><\/p>\n<div id=\"q523374\" class=\"hidden-answer\" style=\"display: none\">In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152934\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/><\/p>\n<p>Focus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex]\\displaystyle \\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.<\/p>\n<p>Multiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152935\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/><\/p>\n<p>Rewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152936\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Bring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152938\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Now, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152939\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/><\/p>\n<p>Remember to watch the signs!<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152940\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/><\/p>\n<p>How many [latex]x[\/latex]\u2019s are there in [latex]3x[\/latex]? Since there are 3, multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152941\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/><\/p>\n<p>Continue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is 0, which is less than the degree of [latex]x-3[\/latex], which is [latex]1[\/latex].<\/p>\n<p>Also notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152942\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/><br \/>\nYou can write the remainder using the symbol [latex]R[\/latex], or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\large \\begin{array}{r}(x^{3}\u20136x\u201310)(x\u20133)=x^2+3x+3+R-1,\\\\ x^2+3x+3+ \\frac{-1}{x-3}, \\text{ or }\\\\ x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]<\/p>\n<p><strong>Check the result:<\/strong><\/p>\n<p style=\"text-align: center\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show another example of dividing a degree three trinomial by a binomial, also with a &#8220;missing&#8221; term..<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Rxds7Q_UTeo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>(8.3.3) &#8211; Application of Polynomial Division<\/h1>\n<p>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 class=\"textbox exercises\">\n<h3>Example: Using Polynomial Division in an Application Problem<\/h3>\n<p>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 [latex]3x[\/latex]\u00a0and the width is given by [latex]x-2[\/latex].\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=\"q426222\">Solution<\/span><\/p>\n<div id=\"q426222\" class=\"hidden-answer\" style=\"display: none\">\nThere 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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165543\/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>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]\\begin{array}{l}V=l\\cdot w\\cdot h\\\\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h\\end{array}[\/latex]<\/p>\n<p>To solve for [latex]h[\/latex], first divide both sides by [latex]3x[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{l}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\\\ \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{array}[\/latex]<\/p>\n<p>Now solve for [latex]h[\/latex]\u00a0using synthetic division.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165546\/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=\"204\" height=\"118\" \/><\/p>\n<p>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 class=\"textbox key-takeaways\">\n<h3>ExAMPLE<\/h3>\n<p>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 [latex]x+6[\/latex].\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=\"q4034\">Solution<\/span><\/p>\n<div id=\"q4034\" class=\"hidden-answer\" style=\"display: none\">[latex]3{x}^{2}-4x+1[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4708\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 29482, 29483. <strong>Authored by<\/strong>: McClure, Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Rxds7Q_UTeo\">https:\/\/youtu.be\/Rxds7Q_UTeo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID# 38831. <strong>Authored by<\/strong>: Meacham,William, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-4708-1\">National Park Service. \"Lincoln Memorial Building Statistics.\" <a href=\"http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014 <a href=\"#return-footnote-4708-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23485,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 29482, 29483\",\"author\":\"McClure, Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Rxds7Q_UTeo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID# 38831\",\"author\":\"Meacham,William, mb Sousa,James\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4708","chapter","type-chapter","status-web-only","hentry"],"part":876,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":32,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4708\/revisions"}],"predecessor-version":[{"id":5500,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4708\/revisions\/5500"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/876"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4708\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/wp\/v2\/media?parent=4708"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4708"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/wp\/v2\/contributor?post=4708"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/wp-json\/wp\/v2\/license?post=4708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}