{"id":16287,"date":"2019-10-02T21:57:12","date_gmt":"2019-10-02T21:57:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-simplify-polynomials\/"},"modified":"2020-10-22T09:29:51","modified_gmt":"2020-10-22T09:29:51","slug":"read-simplify-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-simplify-polynomials\/","title":{"raw":"12.3.c - Dividing Polynomials Using Synthetic Division","rendered":"12.3.c &#8211; Dividing Polynomials Using Synthetic Division"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Divide polynomials by binomials using synthetic division<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we have seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a polynomial\u00a0whose leading coefficient is\u00a0[latex]1[\/latex].\r\n<div class=\"textbox\">\r\n<h3 class=\"title\">\u00a0Synthetic Division<\/h3>\r\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form\u00a0\u00a0[latex]x\u2013k[\/latex]<em>, <\/em>for a real number\u00a0[latex]k[\/latex].\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\r\n\r\n<\/div>\r\n<p class=\"title\"><span style=\"font-size: 16px;line-height: 1.5\">To illustrate the process, d<\/span><span style=\"font-size: 16px;line-height: 1.5\">ivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/span><\/p>\r\n<p id=\"fs-id1165137932636\"><span id=\"eip-id1163740536072\"><img class=\"alignnone size-medium wp-image-5090\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/17031903\/Long-Division-277x300.jpg\" alt=\"\" width=\"277\" height=\"300\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137932377\">There is a lot of repetition in this process.\u00a0If we do not write the variables but, instead, line up their coefficients in columns under the division sign, we already have a simpler version of the entire problem.<\/p>\r\n<span id=\"fs-id1165134305375\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201529\/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.\" \/><\/span>\r\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by\u00a0[latex]2[\/latex], as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \"divisor\" to\u00a0[latex]\u20132[\/latex], multiply and add. The process starts by bringing down the leading coefficient.<span id=\"fs-id1165137696374\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201531\/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.\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137696388\">We then multiply it by the \"divisor\" and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is\u00a0[latex]\u201331[\/latex].\u00a0The process will be made more clear in the following example.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].\r\n[reveal-answer q=\"152802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152802\"]\r\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write\u00a0[latex]3[\/latex]\u00a0and the coefficients of the polynomial.<\/p>\r\n<span id=\"fs-id1165135177629\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201532\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span>\r\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by\u00a0[latex]3[\/latex]\u00a0and place the result in the second column.<\/p>\r\n<span id=\"fs-id1165135439966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201533\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" \/><\/span>\r\n<p id=\"fs-id1165135179942\">Continue by adding [latex]-3+15[\/latex]\u00a0in the second column. Multiply the resulting number, [latex]12[\/latex], by\u00a0[latex]3[\/latex].\u00a0Write the result, [latex]36[\/latex], in the next column. Then add the numbers in the third column.<\/p>\r\n<span id=\"fs-id1165135179966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201535\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" \/><\/span>\r\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].<\/p>\r\nWe can check our work by multiplying the result by the original divisor [latex]x-3[\/latex]. If we get\u00a0[latex]5{x}^{2}-3x - 36[\/latex], then we have used the method correctly.\r\n\r\nCheck:\r\n<p style=\"text-align: left\">\u00a0[latex]\\begin{array}{l}(5x+12)(x-3) \\\\ =5x^2-15x+12x-36 \\\\ =5x^2-3x-36\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Because we got a result of\u00a0[latex]5{x}^{2}-3x - 36[\/latex] when we multiplied the divisor and our answer, we can be sure that we have used synthetic division correctly.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnalysis of the solution\r\n\r\nIt is important to note that the result, [latex]5x+12[\/latex], of [latex]5{x}^{2}-3x - 36\\div{x-3}[\/latex] is one degree less than [latex]5{x}^{2}-3x - 36[\/latex]. Why is that? Think about how you would have solved this using long division. The first thing you would ask yourself is how many x's are in [latex]5x^2[\/latex]?\r\n<p style=\"text-align: center\">[latex]x-3\\overline{)5{x}^{2}-3x - 36}[\/latex]<\/p>\r\n<p style=\"text-align: left\">To get a result of [latex]5x^2[\/latex], you need to multiply [latex]x[\/latex] by [latex]5x[\/latex]. The next step in long division is to subtract this result from [latex]5x^2[\/latex]. This leaves us with no [latex]x^2[\/latex] term in the result.<\/p>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nReflect on this idea - if you multiply two polynomials and get a result whose degree is\u00a0[latex]2[\/latex], what are the possible degrees of the two polynomials that were multiplied? Write your ideas in the box below before looking at the discussion.[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"962896\"]Show Discussion[\/reveal-answer]\r\n[hidden-answer a=\"962896\"]\r\n\r\nA degree two polynomial will have a leading term with [latex]x^2[\/latex]. \u00a0Let us use [latex]2x^2-2x-24[\/latex] as an example. We can write\u00a0two products\u00a0that will give this as a result of multiplication:\r\n\r\n[latex]2(x^2-x-12) =2x^2-2x-24[\/latex]\r\n\r\n[latex](2x+6)(x-4)=2x^2-2x-24[\/latex]\r\n\r\nIf we work backward, starting from [latex]2x^2-2x-24[\/latex] and we divide by a binomial with degree one, such as [latex](x-4)[\/latex], our result will also have degree one.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video example, you will see another example of using synthetic division for division of a degree two polynomial by a degree one binomial.\r\n\r\nhttps:\/\/youtu.be\/KeZ_zMOYu9o\r\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\r\n<h2 id=\"fs-id1165135393414\">How To: Given two polynomials, where the divisor is in the form [latex]x-k[\/latex] use synthetic division to divide<\/h2>\r\n<ol id=\"fs-id1165135393418\">\r\n \t<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Repeat steps\u00a0[latex]5[\/latex] and\u00a0[latex]6[\/latex] 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\u00a0[latex]0[\/latex]. The next number from the right has degree\u00a0[latex]1[\/latex], and the next number from the right has degree\u00a0[latex]2[\/latex], and so on.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the next example, we will use synthetic division to divide a third-degree polynomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].\r\n[reveal-answer q=\"153403\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"153403\"]\r\n\r\nThe binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201536\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span>\r\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex]. Again notice\u00a0the degree of the result is less than the degree of the quotient,\u00a0[latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\r\nWe can check that we are correct by multiplying the result with the divisor:\r\n\r\n[latex](x+2)(4{x}^{2}+2x - 10)=4x^3+2x^2-10x+8x^2+4x-20=4x^3+10x^2-6x-20[\/latex]\r\n\r\nWe just verified that the answer is [latex]4{x}^{2}+2x - 10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, we will show division of a fourth degree polynomial by a binomial. \u00a0Note how there is no x term in the fourth degree polynomial, so we need to use a placeholder of 0 to ensure proper alignment of terms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/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[reveal-answer q=\"76281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"76281\"]\r\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201539\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]93542[\/ohm_question]\r\n\r\n<\/div>\r\nIn our last video example, we show another example of how to use synthetic division to divide a degree three polynomial by a degree one binomial.\r\n\r\nhttps:\/\/youtu.be\/h1oSCNuA9i0","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Divide polynomials by binomials using synthetic division<\/li>\n<\/ul>\n<\/div>\n<p>As we have seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a polynomial\u00a0whose leading coefficient is\u00a0[latex]1[\/latex].<\/p>\n<div class=\"textbox\">\n<h3 class=\"title\">\u00a0Synthetic Division<\/h3>\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form\u00a0\u00a0[latex]x\u2013k[\/latex]<em>, <\/em>for a real number\u00a0[latex]k[\/latex].\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\n<\/div>\n<p class=\"title\"><span style=\"font-size: 16px;line-height: 1.5\">To illustrate the process, d<\/span><span style=\"font-size: 16px;line-height: 1.5\">ivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/span><\/p>\n<p id=\"fs-id1165137932636\"><span id=\"eip-id1163740536072\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5090\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/17031903\/Long-Division-277x300.jpg\" alt=\"\" width=\"277\" height=\"300\" \/><\/span><\/p>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in this process.\u00a0If we do not write the variables but, instead, line up their coefficients in columns under the division sign, we already have a simpler version of the entire problem.<\/p>\n<p><span id=\"fs-id1165134305375\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201529\/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.\" \/><\/span><\/p>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by\u00a0[latex]2[\/latex], as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the &#8220;divisor&#8221; to\u00a0[latex]\u20132[\/latex], multiply and add. The process starts by bringing down the leading coefficient.<span id=\"fs-id1165137696374\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201531\/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.\" \/><\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the &#8220;divisor&#8221; and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is\u00a0[latex]\u201331[\/latex].\u00a0The process will be made more clear in the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/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=\"q152802\">Show Solution<\/span><\/p>\n<div id=\"q152802\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write\u00a0[latex]3[\/latex]\u00a0and the coefficients of the polynomial.<\/p>\n<p><span id=\"fs-id1165135177629\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201532\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span><\/p>\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by\u00a0[latex]3[\/latex]\u00a0and place the result in the second column.<\/p>\n<p><span id=\"fs-id1165135439966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201533\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" \/><\/span><\/p>\n<p id=\"fs-id1165135179942\">Continue by adding [latex]-3+15[\/latex]\u00a0in the second column. Multiply the resulting number, [latex]12[\/latex], by\u00a0[latex]3[\/latex].\u00a0Write the result, [latex]36[\/latex], in the next column. Then add the numbers in the third column.<\/p>\n<p><span id=\"fs-id1165135179966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201535\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" \/><\/span><\/p>\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].<\/p>\n<p>We can check our work by multiplying the result by the original divisor [latex]x-3[\/latex]. If we get\u00a0[latex]5{x}^{2}-3x - 36[\/latex], then we have used the method correctly.<\/p>\n<p>Check:<\/p>\n<p style=\"text-align: left\">\u00a0[latex]\\begin{array}{l}(5x+12)(x-3) \\\\ =5x^2-15x+12x-36 \\\\ =5x^2-3x-36\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Because we got a result of\u00a0[latex]5{x}^{2}-3x - 36[\/latex] when we multiplied the divisor and our answer, we can be sure that we have used synthetic division correctly.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Analysis of the solution<\/p>\n<p>It is important to note that the result, [latex]5x+12[\/latex], of [latex]5{x}^{2}-3x - 36\\div{x-3}[\/latex] is one degree less than [latex]5{x}^{2}-3x - 36[\/latex]. Why is that? Think about how you would have solved this using long division. The first thing you would ask yourself is how many x&#8217;s are in [latex]5x^2[\/latex]?<\/p>\n<p style=\"text-align: center\">[latex]x-3\\overline{)5{x}^{2}-3x - 36}[\/latex]<\/p>\n<p style=\"text-align: left\">To get a result of [latex]5x^2[\/latex], you need to multiply [latex]x[\/latex] by [latex]5x[\/latex]. The next step in long division is to subtract this result from [latex]5x^2[\/latex]. This leaves us with no [latex]x^2[\/latex] term in the result.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Reflect on this idea &#8211; if you multiply two polynomials and get a result whose degree is\u00a0[latex]2[\/latex], what are the possible degrees of the two polynomials that were multiplied? Write your ideas in the box below before looking at the discussion.<textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962896\">Show Discussion<\/span><\/p>\n<div id=\"q962896\" class=\"hidden-answer\" style=\"display: none\">\n<p>A degree two polynomial will have a leading term with [latex]x^2[\/latex]. \u00a0Let us use [latex]2x^2-2x-24[\/latex] as an example. We can write\u00a0two products\u00a0that will give this as a result of multiplication:<\/p>\n<p>[latex]2(x^2-x-12) =2x^2-2x-24[\/latex]<\/p>\n<p>[latex](2x+6)(x-4)=2x^2-2x-24[\/latex]<\/p>\n<p>If we work backward, starting from [latex]2x^2-2x-24[\/latex] and we divide by a binomial with degree one, such as [latex](x-4)[\/latex], our result will also have degree one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video example, you will see another example of using synthetic division for division of a degree two polynomial by a degree one binomial.<\/p>\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 id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\n<h2 id=\"fs-id1165135393414\">How To: Given two polynomials, where the divisor is in the form [latex]x-k[\/latex] use synthetic division to divide<\/h2>\n<ol id=\"fs-id1165135393418\">\n<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Repeat steps\u00a0[latex]5[\/latex] and\u00a0[latex]6[\/latex] 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\u00a0[latex]0[\/latex]. The next number from the right has degree\u00a0[latex]1[\/latex], and the next number from the right has degree\u00a0[latex]2[\/latex], and so on.<\/li>\n<\/ol>\n<\/div>\n<p>In the next example, we will use synthetic division to divide a third-degree polynomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/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=\"q153403\">Show Solution<\/span><\/p>\n<div id=\"q153403\" class=\"hidden-answer\" style=\"display: none\">\n<p>The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201536\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span><\/p>\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex]. Again notice\u00a0the degree of the result is less than the degree of the quotient,\u00a0[latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<p>We can check that we are correct by multiplying the result with the divisor:<\/p>\n<p>[latex](x+2)(4{x}^{2}+2x - 10)=4x^3+2x^2-10x+8x^2+4x-20=4x^3+10x^2-6x-20[\/latex]<\/p>\n<p>We just verified that the answer is [latex]4{x}^{2}+2x - 10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we will show division of a fourth degree polynomial by a binomial. \u00a0Note how there is no x term in the fourth degree polynomial, so we need to use a placeholder of 0 to ensure proper alignment of terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/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=\"q76281\">Show Solution<\/span><\/p>\n<div id=\"q76281\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201539\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" \/><\/span><\/p>\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm93542\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93542&theme=oea&iframe_resize_id=ohm93542&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In our last video example, we show another example of how to use synthetic division to divide a degree three polynomial by a degree one binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 3:  Divide a Polynomial by a Binomial Using Synthetic Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/h1oSCNuA9i0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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-16287\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Divide a Trinomial by a Binomial Using Synthetic Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KeZ_zMOYu9o\">https:\/\/youtu.be\/KeZ_zMOYu9o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 3: Divide a Polynomial by a Binomial Using Synthetic Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/h1oSCNuA9i0\">https:\/\/youtu.be\/h1oSCNuA9i0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t 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