{"id":1513,"date":"2022-04-12T19:14:16","date_gmt":"2022-04-12T19:14:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1513"},"modified":"2026-01-06T17:55:58","modified_gmt":"2026-01-06T17:55:58","slug":"3-4-2-the-division-of-polynomials-synthetic-division","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-4-2-the-division-of-polynomials-synthetic-division\/","title":{"raw":"3.4.2: Synthetic Division","rendered":"3.4.2: Synthetic Division"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-429\" class=\"standard post-429 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Use synthetic division to divide polynomials by a linear binomial.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAlthough long division of polynomials will always work, there is a shorthand method for the special case of\u00a0dividing a polynomial by a linear factor whose leading coefficient is 1. This shorthand method is called <em><strong>synthetic division<\/strong><\/em>.\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">Consider the example of dividing [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm. Work through the example on your own to make sure you know how to get the solution in figure 1.<\/span>\r\n\r\n[caption id=\"attachment_1540\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1540 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-300x272.png\" alt=\"An example of division of a polynomial by a linear binomial as presented in section 3.4.1\" width=\"300\" height=\"272\" \/> Figure 1. Division of a polynomial by a linear binomial[\/caption]\r\n\r\nThere is a lot of repetition when we use the division algorithm. If we don\u2019t write the variables but instead line up their coefficients in columns under the division sign and also eliminate some partial products, we already have a simpler version of the entire problem.\r\n<div class=\"wp-nocaption aligncenter wp-image-2919 size-full\">\r\n\r\n[caption id=\"attachment_2919\" align=\"aligncenter\" width=\"487\"]<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=\"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> Figure 2. Losing the variables[\/caption]\r\n\r\n<\/div>\r\nSynthetic division takes this simplification further. Collapse the algorithm by moving each of the rows up to fill any vacant spots. Also, instead of multiplying and subtracting the product, we change the sign of the \u201cdivisor\u201d to \u20132, so we can add rather than subtract. The process starts by bringing down the leading coefficient.\u00a0<span style=\"font-size: 1em;\">We then multiply it by the \u201cdivisor\u201d 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.\u00a0<\/span>\r\n\r\n[caption id=\"attachment_2920\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-2920 size-full\" 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=\"The division restructured into synthetic division form described above.\" width=\"487\" height=\"74\" \/> Figure 3. Synthetic division[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h2>synthetic division<\/h2>\r\nSynthetic division is a shortcut that can be used when the divisor is a binomial in the form [latex]x\u2013k[\/latex], where [latex]k[\/latex] is a constant.\u00a0In\u00a0<em><strong>synthetic division<\/strong><\/em>, only the coefficients are used in the division process.\r\n<ol>\r\n \t<li>Write [latex]k[\/latex]<em>\u00a0<\/em>for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend written in descending order.<\/li>\r\n \t<li>Bring the leading coefficient down in the first column.<\/li>\r\n \t<li>Multiply the leading coefficient by\u00a0[latex]k[\/latex].\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\u00a0[latex]k[\/latex].\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. The degree of the quotient will be one degree less than the degree of the dividend.<\/li>\r\n<\/ol>\r\n<\/div>\r\nThere are then two basic rules to synthetic division:\r\n<ol>\r\n \t<li>Add vertically<\/li>\r\n \t<li>Multiply diagonally by [latex]k[\/latex]<\/li>\r\n<\/ol>\r\nThe process will be made clearer in the examples that follow.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nUse synthetic division to divide [latex]5x^2-3x-36[\/latex]\u00a0by [latex]x - 3[\/latex].\r\n<h4>Solution<\/h4>\r\nBegin by setting up the synthetic division. Determine the value of [latex]k[\/latex]: [latex]x-3=x-k[\/latex], so [latex]k=3[\/latex].\r\n\r\nWrite\u00a0[latex]k=3[\/latex] and the coefficients of the dividend in descending order.\r\n<div class=\"wp-nocaption aligncenter size-full wp-image-2921\"><\/div>\r\nBring down the leading coefficient. Multiply the leading coefficient 5 by [latex]k=3[\/latex] and put the answer under the second column.\r\n<div class=\"wp-nocaption aligncenter size-full wp-image-2922\"><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\">\r\n<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\" \/>\r\n<\/a><\/div>\r\nAdd the numbers vertically in the second column.\r\n\r\nMultiply the resulting number by [latex]k=3[\/latex] and write the result in the next column: [latex]12\\cdot 3=36[\/latex].\r\n\r\nThen add the numbers vertically in the third column.\r\n<div class=\"wp-nocaption aligncenter size-full wp-image-2923\"><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\">\r\n<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\" \/>\r\n<\/a><\/div>\r\nThe quotient is [latex]5x+12[\/latex] with remainder is 0.\r\n\r\n[latex]\\dfrac{5x^2-3x-36}{x-3}=5x+12[\/latex]\r\n\r\nWith no remainder, [latex]x - 3[\/latex] and [latex]5x+12[\/latex] are factors of the original polynomial.\r\n\r\n[latex]5x^2-3x-36=(x-3)(5x+12)[\/latex]\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](x - 3)(5x+12)+0\\\\=x(5x+12)-3(5x+12)\\\\=5x^2+12x-15x-36\\\\=5{x}^{2}-3x - 36[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nUse synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].\r\n<h4>Solution<\/h4>\r\nThe binomial divisor is [latex]x+2=x-(-2)=x-k[\/latex], so [latex]k=-2[\/latex].\r\n\r\nBring down the leading coefficient. Multiply the leading coefficient 4 by [latex]k=-2[\/latex] and put the answer under the second column. Add vertically: 10+(-8) =2. Multiply diagonally by \u20132: 2(\u20132) = \u20134. Repeat until the last column is reached.\r\n\r\n<img class=\"alignnone\" 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 described above.\" width=\"487\" height=\"74\" \/>\r\n\r\nThe degree of the quotient is always one degree less than the dividend, so we start with an [latex]x^2[\/latex] term. The result is [latex]\\dfrac{4x^3+10x^2-6x-20}{x+2}=4{x}^{2}+2x - 10[\/latex].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0and [latex]4x^2+2x-10[\/latex] are factors 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<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204350\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2 showing that (-2,0) is an x-intercept, so x=-2 is a zero.\" width=\"487\" height=\"742\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/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<h4>Solution<\/h4>\r\n[latex]k=1[\/latex]. Notice there is no [latex]x[\/latex] term so we must use a zero as the coefficient for that term.\r\n\r\nStarting from the first column, add vertically then multiply diagonally by [latex]k=1[\/latex].\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">\r\n<\/span><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=\"An example of synthetic division with a missing degree. The coefficients of the dividend are -9, 10, 7, 0, -6.\" width=\"230\" height=\"300\" \/>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">The result is [latex]\\dfrac{-9{x}^{4}+10{x}^{3}+7{x}^{2}-6}{x-1}=-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/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=\"644060\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"644060\"]\r\n\r\n[latex]\\dfrac{3{x}^{4}+18{x}^{3}-3x+40}{x+7}=3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 2<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><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&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse synthetic division to divide the polynomial [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex] by [latex]x+6[\/latex].\r\n\r\n[reveal-answer q=\"106755\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"106755\"]\r\n\r\n[latex]\\dfrac{3{x}^{3}+14{x}^{2}-23x+6}{x+6}=3{x}^{2}-4x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-429\" class=\"standard post-429 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Use synthetic division to divide polynomials by a linear binomial.<\/li>\n<\/ul>\n<\/div>\n<p>Although long division of polynomials will always work, there is a shorthand method for the special case of\u00a0dividing a polynomial by a linear factor whose leading coefficient is 1. This shorthand method is called <em><strong>synthetic division<\/strong><\/em>.<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Consider the example of dividing [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm. Work through the example on your own to make sure you know how to get the solution in figure 1.<\/span><\/p>\n<div id=\"attachment_1540\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1540\" class=\"wp-image-1540 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-300x272.png\" alt=\"An example of division of a polynomial by a linear binomial as presented in section 3.4.1\" width=\"300\" height=\"272\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-300x272.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-768x696.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-1024x929.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-65x59.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-225x204.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision-350x317.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/3-4-2-LongDivision.png 1223w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1540\" class=\"wp-caption-text\">Figure 1. Division of a polynomial by a linear binomial<\/p>\n<\/div>\n<p>There is a lot of repetition when we use the division algorithm. If we don\u2019t write the variables but instead line up their coefficients in columns under the division sign and also eliminate some partial products, we already have a simpler version of the entire problem.<\/p>\n<div class=\"wp-nocaption aligncenter wp-image-2919 size-full\">\n<div id=\"attachment_2919\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><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\" aria-describedby=\"caption-attachment-2919\" class=\"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 id=\"caption-attachment-2919\" class=\"wp-caption-text\">Figure 2. Losing the variables<\/p>\n<\/div>\n<\/div>\n<p>Synthetic division takes this simplification further. Collapse the algorithm by moving each of the rows up to fill any vacant spots. Also, instead of multiplying and subtracting the product, we change the sign of the \u201cdivisor\u201d to \u20132, so we can add rather than subtract. The process starts by bringing down the leading coefficient.\u00a0<span style=\"font-size: 1em;\">We then multiply it by the \u201cdivisor\u201d 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.\u00a0<\/span><\/p>\n<div id=\"attachment_2920\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2920\" class=\"wp-image-2920 size-full\" 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=\"The division restructured into synthetic division form described above.\" width=\"487\" height=\"74\" \/><\/p>\n<p id=\"caption-attachment-2920\" class=\"wp-caption-text\">Figure 3. Synthetic division<\/p>\n<\/div>\n<div class=\"textbox\">\n<h2>synthetic division<\/h2>\n<p>Synthetic division is a shortcut that can be used when the divisor is a binomial in the form [latex]x\u2013k[\/latex], where [latex]k[\/latex] is a constant.\u00a0In\u00a0<em><strong>synthetic division<\/strong><\/em>, only the coefficients are used in the division process.<\/p>\n<ol>\n<li>Write [latex]k[\/latex]<em>\u00a0<\/em>for the divisor.<\/li>\n<li>Write the coefficients of the dividend written in descending order.<\/li>\n<li>Bring the leading coefficient down in the first column.<\/li>\n<li>Multiply the leading coefficient by\u00a0[latex]k[\/latex].\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by\u00a0[latex]k[\/latex].\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. The degree of the quotient will be one degree less than the degree of the dividend.<\/li>\n<\/ol>\n<\/div>\n<p>There are then two basic rules to synthetic division:<\/p>\n<ol>\n<li>Add vertically<\/li>\n<li>Multiply diagonally by [latex]k[\/latex]<\/li>\n<\/ol>\n<p>The process will be made clearer in the examples that follow.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Use synthetic division to divide [latex]5x^2-3x-36[\/latex]\u00a0by [latex]x - 3[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Begin by setting up the synthetic division. Determine the value of [latex]k[\/latex]: [latex]x-3=x-k[\/latex], so [latex]k=3[\/latex].<\/p>\n<p>Write\u00a0[latex]k=3[\/latex] and the coefficients of the dividend in descending order.<\/p>\n<div class=\"wp-nocaption aligncenter size-full wp-image-2921\"><\/div>\n<p>Bring down the leading coefficient. Multiply the leading coefficient 5 by [latex]k=3[\/latex] and put the answer under the second column.<\/p>\n<div class=\"wp-nocaption aligncenter size-full wp-image-2922\"><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\"><br \/>\n<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\" \/><br \/>\n<\/a><\/div>\n<p>Add the numbers vertically in the second column.<\/p>\n<p>Multiply the resulting number by [latex]k=3[\/latex] and write the result in the next column: [latex]12\\cdot 3=36[\/latex].<\/p>\n<p>Then add the numbers vertically in the third column.<\/p>\n<div class=\"wp-nocaption aligncenter size-full wp-image-2923\"><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\"><br \/>\n<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\" \/><br \/>\n<\/a><\/div>\n<p>The quotient is [latex]5x+12[\/latex] with remainder is 0.<\/p>\n<p>[latex]\\dfrac{5x^2-3x-36}{x-3}=5x+12[\/latex]<\/p>\n<p>With no remainder, [latex]x - 3[\/latex] and [latex]5x+12[\/latex] are factors of the original polynomial.<\/p>\n<p>[latex]5x^2-3x-36=(x-3)(5x+12)[\/latex]<\/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](x - 3)(5x+12)+0\\\\=x(5x+12)-3(5x+12)\\\\=5x^2+12x-15x-36\\\\=5{x}^{2}-3x - 36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/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<h4>Solution<\/h4>\n<p>The binomial divisor is [latex]x+2=x-(-2)=x-k[\/latex], so [latex]k=-2[\/latex].<\/p>\n<p>Bring down the leading coefficient. Multiply the leading coefficient 4 by [latex]k=-2[\/latex] and put the answer under the second column. Add vertically: 10+(-8) =2. Multiply diagonally by \u20132: 2(\u20132) = \u20134. Repeat until the last column is reached.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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 described above.\" width=\"487\" height=\"74\" \/><\/p>\n<p>The degree of the quotient is always one degree less than the dividend, so we start with an [latex]x^2[\/latex] term. The result is [latex]\\dfrac{4x^3+10x^2-6x-20}{x+2}=4{x}^{2}+2x - 10[\/latex].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0and [latex]4x^2+2x-10[\/latex] are factors 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<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204350\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2 showing that (-2,0) is an x-intercept, so x=-2 is a zero.\" width=\"487\" height=\"742\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/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<h4>Solution<\/h4>\n<p>[latex]k=1[\/latex]. Notice there is no [latex]x[\/latex] term so we must use a zero as the coefficient for that term.<\/p>\n<p>Starting from the first column, add vertically then multiply diagonally by [latex]k=1[\/latex].<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><br \/>\n<\/span><img loading=\"lazy\" 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=\"An example of synthetic division with a missing degree. The coefficients of the dividend are -9, 10, 7, 0, -6.\" width=\"230\" height=\"300\" \/><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">The result is [latex]\\dfrac{-9{x}^{4}+10{x}^{3}+7{x}^{2}-6}{x-1}=-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/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=\"q644060\">Show Answer<\/span><\/p>\n<div id=\"q644060\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{3{x}^{4}+18{x}^{3}-3x+40}{x+7}=3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=29483&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><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<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use synthetic division to divide the polynomial [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex] by [latex]x+6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q106755\">Show Answer<\/span><\/p>\n<div id=\"q106755\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{3{x}^{3}+14{x}^{2}-23x+6}{x+6}=3{x}^{2}-4x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1513\">\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>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 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: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 29483\",\"author\":\"McClure,Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"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 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