{"id":1817,"date":"2016-11-02T20:51:49","date_gmt":"2016-11-02T20:51:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1817"},"modified":"2025-10-13T19:03:48","modified_gmt":"2025-10-13T19:03:48","slug":"synthetic-division","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/synthetic-division\/","title":{"raw":"Synthetic Division","rendered":"Synthetic Division"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use synthetic division to divide polynomials.<\/li>\r\n<\/ul>\r\n<\/div>\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<img class=\"aligncenter wp-image-5638\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16231756\/Screen-Shot-2020-11-16-at-3.17.05-PM-300x222.png\" alt=\"Long division of [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0\" width=\"348\" height=\"259\" \/>\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<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<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182931\/CNX_Precalc_Figure_03_05_01121.jpg\"><img class=\"aligncenter size-full wp-image-2921\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182931\/CNX_Precalc_Figure_03_05_01121.jpg\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"74\" \/><\/a>\r\n\r\n&nbsp;\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\r\n[embed]https:\/\/www.myopenmath.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/div>\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><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5638\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/16231756\/Screen-Shot-2020-11-16-at-3.17.05-PM-300x222.png\" alt=\"Long division of [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0\" width=\"348\" height=\"259\" \/><\/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<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><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182931\/CNX_Precalc_Figure_03_05_01121.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2921\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/01182931\/CNX_Precalc_Figure_03_05_01121.jpg\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"74\" \/><\/a><\/p>\n<p>&nbsp;<\/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><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=29483&#38;theme=oea&#38;iframe_resize_id=mom1\">https:\/\/www.myopenmath.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1<\/a><\/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\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-1817\">\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>Question ID 29483. <strong>Authored by<\/strong>: McClure,Caren. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: 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