{"id":4987,"date":"2020-11-13T20:31:23","date_gmt":"2020-11-13T20:31:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebra\/?post_type=chapter&#038;p=4987"},"modified":"2020-11-17T23:53:17","modified_gmt":"2020-11-17T23:53:17","slug":"dividing-polynomials-custom-edit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/chapter\/dividing-polynomials-custom-edit\/","title":{"raw":"Dividing Polynomials","rendered":"Dividing Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\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\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{}V=61.5\\cdot 40\\cdot 30\\hfill \\\\ \\text{}V=73,800\\hfill \\end{array}[\/latex]<\/p>\r\nSo the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex]. Suppose we knew the volume, length, and width. We could divide to find the height.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}h=\\frac{V}{l\\cdot w}\\hfill \\\\ \\text{}h=\\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ \\text{}h=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 3<em>x<\/em>;\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\/896\/2016\/11\/02204315\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/> The Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)[\/caption]\r\n<h2>Polynomial Long Division<\/h2>\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\/896\/2016\/11\/02204318\/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}\\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder}\\text{ = dividend}\\hfill \\\\ \\left(3\\cdot 59\\right)+1 = 177+1 = 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\/896\/2016\/11\/02204321\/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]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/p>\r\nWe can identify the <strong>dividend<\/strong>,\u00a0<strong>divisor<\/strong>,\u00a0<strong>quotient<\/strong>, and\u00a0<strong>remainder<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204324\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/>\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 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 terms above it.<\/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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996959\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"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]\\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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"850001\"]\r\n\r\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"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]\\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 multiplying.\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>Try It<\/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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"198989\"]\r\n\r\n[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=210608&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n<h2>Synthetic Division<\/h2>\r\nAs we\u2019ve seen, long division with 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.\r\n\r\nTo illustrate the process, recall the example at the beginning of the section.\r\n\r\nDivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.\r\n\r\nThe final form of the process looked like this:\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182500\/CNX_Precalc_revised_eq_42.png\"><img class=\"aligncenter wp-image-2918 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182500\/CNX_Precalc_revised_eq_42-300x271.png\" alt=\"cnx_precalc_revised_eq_42\" width=\"300\" height=\"271\" \/><\/a>\r\n\r\nThere 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.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182617\/CNX_Precalc_Figure_03_05_0042.jpg\"><img class=\"aligncenter wp-image-2919 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182617\/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=\"487\" height=\"110\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\nSynthetic 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, and 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.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182738\/CNX_Precalc_Figure_03_05_0112.jpg\"><img class=\"aligncenter size-full wp-image-2920\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182738\/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=\"487\" height=\"74\" \/><\/a>\r\n\r\nWe 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] and the remainder is \u201331.\u00a0The process will be made more clear in the examples that follow.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Synthetic Division<\/h3>\r\nSynthetic 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.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two polynomials, use synthetic division to divide<\/h3>\r\n<ol>\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 leading coefficient down.<\/li>\r\n \t<li>Multiply the leading 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[embed]http:\/\/www.youtube.com\/watch?v=KeZ_zMOYu9o[\/embed]\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\r\nUse synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].\r\n\r\n[reveal-answer q=\"810134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"810134\"]\r\n\r\nBegin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.\r\n\r\n&nbsp;\r\n\r\n<img class=\" wp-image-5003 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5479\/2020\/11\/17161037\/synthetic-division-image.jpg\" alt=\"\" width=\"112\" height=\"59\" \/>\r\n\r\nBring down the leading coefficient. Multiply the leading coefficient by <em>k<\/em>.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183009\/CNX_Precalc_Figure_03_05_0062.jpg\"><img class=\"aligncenter size-full wp-image-2922\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183009\/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.\" width=\"487\" height=\"74\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\nContinue 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.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183049\/CNX_Precalc_Figure_03_05_0072.jpg\"><img class=\"aligncenter size-full wp-image-2923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183049\/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.\" width=\"487\" height=\"74\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\nThe result is [latex]5x+12[\/latex].\u00a0The remainder is 0. So [latex]x - 3[\/latex]\u00a0is a factor of the original polynomial.\r\n<h4>Analysis of the Solution<\/h4>\r\nJust as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.\r\n\r\n[latex]\\left(x - 3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x - 36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\r\nUse synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].\r\n\r\n[reveal-answer q=\"752360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752360\"]\r\n\r\nThe binomial divisor is [latex]x+2[\/latex], so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204348\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/>\r\n\r\nThe 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].\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] shows a zero at [latex]x=-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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204350\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\r\nUse synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].\r\n\r\n[reveal-answer q=\"371817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371817\"]\r\n\r\nNotice there is no <em>x\u00a0<\/em>term. We will use a zero as the coefficient for that term.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204352\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" \/>\r\n\r\nThe result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/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\nUse synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[\/latex]\u00a0by [latex]x+7[\/latex].\r\n\r\n[reveal-answer q=\"959022\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"959022\"]\r\n\r\n[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\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 3<em>x<\/em>\u00a0and the width is given by <em>x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.\r\n\r\n[reveal-answer q=\"426222\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426222\"]\r\n\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\/896\/2016\/11\/02204354\/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 <em>h<\/em>, first divide both sides by 3<em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\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 <em>h<\/em>\u00a0using synthetic division.\r\n<p style=\"text-align: center;\">[latex]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\/896\/2016\/11\/02204357\/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>Try It<\/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 <em>x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.\r\n\r\n[reveal-answer q=\"4034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"4034\"]\r\n\r\n[latex]3{x}^{2}-4x+1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<table>\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<h2>Key Concepts<\/h2>\r\n<ul>\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 of 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-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>Division Algorithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">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-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>synthetic division<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use long division to divide polynomials.<\/li>\n<li>Use synthetic division to divide polynomials.<\/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-4987-1\" href=\"#footnote-4987-1\" 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{}V=61.5\\cdot 40\\cdot 30\\hfill \\\\ \\text{}V=73,800\\hfill \\end{array}[\/latex]<\/p>\n<p>So the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex]. Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}h=\\frac{V}{l\\cdot w}\\hfill \\\\ \\text{}h=\\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ \\text{}h=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 3<em>x<\/em>;\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\/896\/2016\/11\/02204315\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\">The Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<\/p>\n<\/div>\n<h2>Polynomial Long Division<\/h2>\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\/896\/2016\/11\/02204318\/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}\\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder}\\text{ = dividend}\\hfill \\\\ \\left(3\\cdot 59\\right)+1 = 177+1 = 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\/896\/2016\/11\/02204321\/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]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\n<p style=\"text-align: center;\">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>,\u00a0<strong>divisor<\/strong>,\u00a0<strong>quotient<\/strong>, and\u00a0<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\/896\/2016\/11\/02204324\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/><\/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 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 terms above it.<\/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\">Show Solution<\/span><\/p>\n<div id=\"q996959\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"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]\\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\">Show Solution<\/span><\/p>\n<div id=\"q850001\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"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]\\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 multiplying.<\/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>Try It<\/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\">Show Solution<\/span><\/p>\n<div id=\"q198989\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=210608&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<h2>Synthetic Division<\/h2>\n<p>As we\u2019ve seen, long division with 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>To illustrate the process, recall the example at the beginning of the section.<\/p>\n<p>Divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\n<p>The final form of the process looked like this:<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182500\/CNX_Precalc_revised_eq_42.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2918 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182500\/CNX_Precalc_revised_eq_42-300x271.png\" alt=\"cnx_precalc_revised_eq_42\" width=\"300\" height=\"271\" \/><\/a><\/p>\n<p>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><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182617\/CNX_Precalc_Figure_03_05_0042.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2919 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182617\/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=\"487\" height=\"110\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>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, and 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><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182738\/CNX_Precalc_Figure_03_05_0112.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2920\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182738\/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=\"487\" height=\"74\" \/><\/a><\/p>\n<p>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] and the remainder is \u201331.\u00a0The process will be made more clear in the examples that follow.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Synthetic Division<\/h3>\n<p>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 class=\"textbox\">\n<h3>How To: Given two polynomials, use synthetic division to divide<\/h3>\n<ol>\n<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the leading coefficient down.<\/li>\n<li>Multiply the leading 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<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Divide a Trinomial by a Binomial Using Synthetic Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KeZ_zMOYu9o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\n<p>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=\"q810134\">Show Solution<\/span><\/p>\n<div id=\"q810134\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5003 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5479\/2020\/11\/17161037\/synthetic-division-image.jpg\" alt=\"\" width=\"112\" height=\"59\" \/><\/p>\n<p>Bring down the leading coefficient. Multiply the leading coefficient by <em>k<\/em>.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183009\/CNX_Precalc_Figure_03_05_0062.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2922\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183009\/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.\" width=\"487\" height=\"74\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>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><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183049\/CNX_Precalc_Figure_03_05_0072.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01183049\/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.\" width=\"487\" height=\"74\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>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>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<p>[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 class=\"textbox exercises\">\n<h3>Example: Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\n<p>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=\"q752360\">Show Solution<\/span><\/p>\n<div id=\"q752360\" class=\"hidden-answer\" style=\"display: none\">\n<p>The binomial divisor is [latex]x+2[\/latex], so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204348\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/p>\n<p>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] shows a zero at [latex]x=-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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204350\/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<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\n<p>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=\"q371817\">Show Solution<\/span><\/p>\n<div id=\"q371817\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice there is no <em>x\u00a0<\/em>term. We will use a zero as the coefficient for that term.<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204352\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" \/><\/p>\n<p>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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=\"q959022\">Show Solution<\/span><\/p>\n<div id=\"q959022\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\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 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=\"q426222\">Show Solution<\/span><\/p>\n<div id=\"q426222\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204354\/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 <em>h<\/em>, first divide both sides by 3<em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\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 <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<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\/896\/2016\/11\/02204357\/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>Try It<\/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 <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=\"q4034\">Show Solution<\/span><\/p>\n<div id=\"q4034\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{2}-4x+1[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\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<h2>Key Concepts<\/h2>\n<ul>\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 of 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-id1165132943522\" class=\"definition\">\n<dt><strong>Division Algorithm<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">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-id1165132943522\" class=\"definition\">\n<dt><strong>synthetic division<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">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<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-4987-1\">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-4987-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":167848,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4987","chapter","type-chapter","status-publish","hentry"],"part":764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4987","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4987\/revisions"}],"predecessor-version":[{"id":5042,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4987\/revisions\/5042"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/parts\/764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4987\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/media?parent=4987"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapter-type?post=4987"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/contributor?post=4987"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/license?post=4987"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}