{"id":1400,"date":"2025-07-25T01:00:16","date_gmt":"2025-07-25T01:00:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1400"},"modified":"2026-03-18T05:23:26","modified_gmt":"2026-03-18T05:23:26","slug":"dividing-polynomials-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/dividing-polynomials-fresh-take\/","title":{"raw":"Dividing Polynomials: Fresh Take","rendered":"Dividing Polynomials: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\r\n<h2>Polynomial Long Division<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nPolynomial long division might seem daunting at first, but it's quite similar to the long division we learned in elementary school. Just like we divide numbers, we can divide polynomials, breaking them down into simpler parts. The process involves dividing the highest degree terms and then subtracting to find the remainder, which we then bring down the next term to continue the process.\r\n\r\n<strong>Quick Tips: Steps to Polynomial Long Division<\/strong>\r\n<ol>\r\n \t<li><strong>Set Up<\/strong>: Write the polynomial (dividend) and the binomial (divisor) in long division format.<\/li>\r\n \t<li><strong>Divide the Leading Terms<\/strong>: Take the leading term of the dividend and divide it by the leading term of the divisor to find the first term of the quotient.<\/li>\r\n \t<li><strong>Multiply and Subtract<\/strong>: Multiply the entire divisor by the term just found and subtract it from the dividend.<\/li>\r\n \t<li><strong>Repeat<\/strong>: Bring down the next term of the dividend and repeat the process until you've worked through each term.<\/li>\r\n \t<li><strong>Remainder<\/strong>: If there's a remainder, express it as a fraction with the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<strong>The Division Algorithm: A Formal Approach<\/strong>The Division Algorithm is a fancy way of saying that any polynomial divided by another can be expressed as a multiplication (the quotient) plus something left over (the remainder). It's a structured method to ensure that every polynomial division can be broken down neatly.\r\n\r\n<\/div>\r\n<section class=\"textbox example\">Divide [latex]5{x}^{2}+3x - 2[\/latex] by [latex]x+1[\/latex].[reveal-answer q=\"996959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996959\"]<center>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"426\"]<img 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\" \/> Steps of division[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is [latex]0[\/latex]. 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 [latex]0[\/latex]. 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<\/section><section class=\"textbox example\">Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].[reveal-answer q=\"198989\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"198989\"][latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex][\/hidden-answer]<\/section>Watch the following video see an example of dividing polynomials with long division.\r\n\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgfbabgc-U2f5VzKGNPk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/U2f5VzKGNPk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hgfbabgc-U2f5VzKGNPk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328532&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgfbabgc-U2f5VzKGNPk&amp;vembed=0&amp;video_id=U2f5VzKGNPk&amp;video_target=tpm-plugin-hgfbabgc-U2f5VzKGNPk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dividing+polynomials+using+long+division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing polynomials using long division\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdfdchha-chyi4APQJi0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/chyi4APQJi0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gdfdchha-chyi4APQJi0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846614&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gdfdchha-chyi4APQJi0&amp;vembed=0&amp;video_id=chyi4APQJi0&amp;video_target=tpm-plugin-gdfdchha-chyi4APQJi0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/(New+Version+Available)+Ex+3+-++Divide+a+Polynomial+by+a+Binomial+Using+Long+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Ex 3: Divide a Polynomial by a Binomial Using Long Division\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Synthetic Division<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nSynthetic division is a quicker way to divide polynomials when the divisor is a binomial with a leading coefficient of [latex]1[\/latex]. It uses only the coefficients of the polynomials, simplifying the process significantly.\r\n\r\n<strong>How to Use Synthetic Division<\/strong>\r\n<ol>\r\n \t<li><strong>Write the Opposite of k<\/strong>: If dividing by [latex]x-k[\/latex], write down [latex]k[\/latex].<\/li>\r\n \t<li><strong>List the Coefficients<\/strong>: Write the coefficients of the polynomial you're dividing.<\/li>\r\n \t<li><strong>Bring Down the Leading Coefficient<\/strong>: This starts off your synthetic division.<\/li>\r\n \t<li><strong>Multiply and Add<\/strong>: Multiply the leading coefficient by [latex]k[\/latex], add it to the next coefficient, and continue this process.<\/li>\r\n \t<li><strong>Find the Quotient and Remainder<\/strong>: The last number is the remainder, and the others form the coefficients of the quotient polynomial.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Use synthetic division to divide [latex]5x^2 - 3x - 36[\/latex] by [latex]x - 3[\/latex]. [reveal-answer q=\"981343\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981343\"]Begin by setting up the synthetic division. Write [latex]k[\/latex] and the coefficients.<center>\r\n\r\n[caption id=\"attachment_9522\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-9522 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150002\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f.webp\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"55\" \/> Synthetic division[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>Bring down the lead coefficient. Multiply the lead coefficient by [latex]k[\/latex].<center>\r\n\r\n[caption id=\"attachment_9523\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-9523 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150022\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e.webp\" 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\" \/> Synthetic division[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>Continue by adding the numbers in the second column. Multiply the resulting number by [latex]k[\/latex]. Write the result in the next column. Then add the numbers in the third column.<center>\r\n\r\n[caption id=\"attachment_9524\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-9524 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150038\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f.webp\" 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\" \/> Synthetic division[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>The result is [latex]5x + 12[\/latex]. The remainder is [latex]0[\/latex]. So [latex]x - 3[\/latex] is a factor of the original polynomial.<strong>Analysis<\/strong>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<center>[latex](x\u22123)(5x+12)+0=5x^2\u22123x\u221236[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\">Use synthetic division to divide [latex]-9x^4 + 10x^3 + 7x^2 - 6[\/latex] by [latex]x - 1[\/latex]. [reveal-answer q=\"981344\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981344\"]Notice there is no [latex]x[\/latex]-term. We will use a zero as the coefficient for that term.<center>\r\n\r\n[caption id=\"attachment_9533\" align=\"aligncenter\" width=\"205\"]<img class=\"wp-image-9533 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/23151200\/21c33f8114c651702043fec1fafe25f629656a9a.webp\" alt=\"A synthetic division table showing the division of a polynomial by the binomial (x - 1). The divisor '1' is on the far left. The row at the top lists the coefficients of the polynomial: -9, 10, 7, 0, and -6. The second row represents the products of the divisor and the results of each step: -9, 1, 8, and 8. The bottom row sums the coefficients and the products: -9, 1, 8, 8, and 2.\" width=\"205\" height=\"76\" \/> Synthetic division[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>The result is [latex]-9x^3 + x^2 + 8x + 8 + \\frac{2}{x-1}[\/latex].[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24625[\/ohm2_question]<\/section>Watch the following video see an example of dividing polynomials using synthetic division.\r\n\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgecdbac-FxHWoUOq2iQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FxHWoUOq2iQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cgecdbac-FxHWoUOq2iQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328533&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cgecdbac-FxHWoUOq2iQ&amp;vembed=0&amp;video_id=FxHWoUOq2iQ&amp;video_target=tpm-plugin-cgecdbac-FxHWoUOq2iQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Synthetic+Division+of+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSynthetic Division of Polynomials\u201d here (opens in new window). <\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heeadhhf-_Y0fRbh1RY8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_Y0fRbh1RY8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-heeadhhf-_Y0fRbh1RY8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846615&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-heeadhhf-_Y0fRbh1RY8&amp;vembed=0&amp;video_id=_Y0fRbh1RY8&amp;video_target=tpm-plugin-heeadhhf-_Y0fRbh1RY8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Divide+a+Polynomial+by+a+Binomial+Using+Synthetic+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Divide a Polynomial by a Binomial Using Synthetic Division\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Remainder Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nThe Remainder Theorem is a quick way to evaluate polynomials. If you divide a polynomial [latex] f(x) [\/latex]by [latex] x - k [\/latex], the remainder is simply the value of [latex] f(k) [\/latex]. It's like a shortcut in a maze, allowing you to reach the end without going through all the paths.\r\n\r\nFor example, to evaluate [latex] f(x) = 6x^4 - x^3 - 15x^2 + 2x - 7 [\/latex] at [latex] x = 2 [\/latex], you can use synthetic division or directly substitute [latex] x [\/latex] with [latex]2[\/latex] to find the remainder.\r\n\r\n<\/div>\r\n<section class=\"textbox example\">Use the remainder theorem to evaluate [latex]f(x)=2x^5\u22123x^4\u22129x^3+8x^2+2 [\/latex] at [latex]x = -3[\/latex].[reveal-answer q=\"981346\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981346\"]To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x + 3[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{c|cccccc} -3 &amp; 2 &amp; -3 &amp; -9 &amp; 8 &amp; 0 &amp; 2 \\\\ &amp; &amp; -6 &amp; 27 &amp; -54 &amp; 138 &amp; -414 \\\\ \\hline &amp; 2 &amp; -9 &amp; 18 &amp; -46 &amp; 138 &amp; -412 \\\\ \\end{array} [\/latex]<\/p>\r\nThe remainder is [latex]-412[\/latex]. Therefore, [latex]f(-3) = -412[\/latex].\r\n\r\n<strong>Analysis<\/strong>We can check our answer by evaluating [latex]f(-3)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) &amp; = &amp; 2x^5 - 3x^4 - 9x^3 + 8x^2 + 2 \\\\ f(-3) &amp; =\u00a0 &amp;2(-3)^5 - 3(-3)^4 - 9(-3)^3 + 8(-3)^2 + 2 \\\\ &amp; = &amp; -412 \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video for more on the remainder theorem.\r\n\r\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fadfaeca-p1lSRAeEMR0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/p1lSRAeEMR0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fadfaeca-p1lSRAeEMR0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328534&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fadfaeca-p1lSRAeEMR0&amp;vembed=0&amp;video_id=p1lSRAeEMR0&amp;video_target=tpm-plugin-fadfaeca-p1lSRAeEMR0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Remainder+Theorem+and+Synthetic+Division+of+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRemainder Theorem and Synthetic Division of Polynomials\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use long division to divide polynomials.<\/li>\n<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/section>\n<h2>Polynomial Long Division<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Polynomial long division might seem daunting at first, but it&#8217;s quite similar to the long division we learned in elementary school. Just like we divide numbers, we can divide polynomials, breaking them down into simpler parts. The process involves dividing the highest degree terms and then subtracting to find the remainder, which we then bring down the next term to continue the process.<\/p>\n<p><strong>Quick Tips: Steps to Polynomial Long Division<\/strong><\/p>\n<ol>\n<li><strong>Set Up<\/strong>: Write the polynomial (dividend) and the binomial (divisor) in long division format.<\/li>\n<li><strong>Divide the Leading Terms<\/strong>: Take the leading term of the dividend and divide it by the leading term of the divisor to find the first term of the quotient.<\/li>\n<li><strong>Multiply and Subtract<\/strong>: Multiply the entire divisor by the term just found and subtract it from the dividend.<\/li>\n<li><strong>Repeat<\/strong>: Bring down the next term of the dividend and repeat the process until you&#8217;ve worked through each term.<\/li>\n<li><strong>Remainder<\/strong>: If there&#8217;s a remainder, express it as a fraction with the divisor as the denominator.<\/li>\n<\/ol>\n<p><strong>The Division Algorithm: A Formal Approach<\/strong>The Division Algorithm is a fancy way of saying that any polynomial divided by another can be expressed as a multiplication (the quotient) plus something left over (the remainder). It&#8217;s a structured method to ensure that every polynomial division can be broken down neatly.<\/p>\n<\/div>\n<section class=\"textbox example\">Divide [latex]5{x}^{2}+3x - 2[\/latex] by [latex]x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q996959\">Show Solution<\/button><\/p>\n<div id=\"q996959\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">\n<figure style=\"width: 426px\" 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\/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\" \/><figcaption class=\"wp-caption-text\">Steps of division<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is [latex]0[\/latex]. 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 [latex]0[\/latex]. 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<\/section>\n<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q198989\">Show Solution<\/button><\/p>\n<div id=\"q198989\" class=\"hidden-answer\" style=\"display: none\">[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video see an example of dividing polynomials with long division.<\/p>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgfbabgc-U2f5VzKGNPk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/U2f5VzKGNPk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hgfbabgc-U2f5VzKGNPk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328532&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgfbabgc-U2f5VzKGNPk&amp;vembed=0&amp;video_id=U2f5VzKGNPk&amp;video_target=tpm-plugin-hgfbabgc-U2f5VzKGNPk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dividing+polynomials+using+long+division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDividing polynomials using long division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdfdchha-chyi4APQJi0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/chyi4APQJi0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gdfdchha-chyi4APQJi0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846614&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gdfdchha-chyi4APQJi0&amp;vembed=0&amp;video_id=chyi4APQJi0&amp;video_target=tpm-plugin-gdfdchha-chyi4APQJi0\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/(New+Version+Available)+Ex+3+-++Divide+a+Polynomial+by+a+Binomial+Using+Long+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Ex 3: Divide a Polynomial by a Binomial Using Long Division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Synthetic Division<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Synthetic division is a quicker way to divide polynomials when the divisor is a binomial with a leading coefficient of [latex]1[\/latex]. It uses only the coefficients of the polynomials, simplifying the process significantly.<\/p>\n<p><strong>How to Use Synthetic Division<\/strong><\/p>\n<ol>\n<li><strong>Write the Opposite of k<\/strong>: If dividing by [latex]x-k[\/latex], write down [latex]k[\/latex].<\/li>\n<li><strong>List the Coefficients<\/strong>: Write the coefficients of the polynomial you&#8217;re dividing.<\/li>\n<li><strong>Bring Down the Leading Coefficient<\/strong>: This starts off your synthetic division.<\/li>\n<li><strong>Multiply and Add<\/strong>: Multiply the leading coefficient by [latex]k[\/latex], add it to the next coefficient, and continue this process.<\/li>\n<li><strong>Find the Quotient and Remainder<\/strong>: The last number is the remainder, and the others form the coefficients of the quotient polynomial.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use synthetic division to divide [latex]5x^2 - 3x - 36[\/latex] by [latex]x - 3[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981343\">Show Solution<\/button> <\/p>\n<div id=\"q981343\" class=\"hidden-answer\" style=\"display: none\">Begin by setting up the synthetic division. Write [latex]k[\/latex] and the coefficients.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_9522\" aria-describedby=\"caption-attachment-9522\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9522 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150002\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f.webp\" alt=\"A collapsed version of the previous synthetic division.\" width=\"487\" height=\"55\" \/><figcaption id=\"caption-attachment-9522\" class=\"wp-caption-text\">Synthetic division<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Bring down the lead coefficient. Multiply the lead coefficient by [latex]k[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_9523\" aria-describedby=\"caption-attachment-9523\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9523 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150022\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e.webp\" 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\" \/><figcaption id=\"caption-attachment-9523\" class=\"wp-caption-text\">Synthetic division<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Continue by adding the numbers in the second column. Multiply the resulting number by [latex]k[\/latex]. Write the result in the next column. Then add the numbers in the third column.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_9524\" aria-describedby=\"caption-attachment-9524\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9524 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/23150038\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f.webp\" 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\" \/><figcaption id=\"caption-attachment-9524\" class=\"wp-caption-text\">Synthetic division<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The result is [latex]5x + 12[\/latex]. The remainder is [latex]0[\/latex]. So [latex]x - 3[\/latex] is a factor of the original polynomial.<strong>Analysis<\/strong>Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<div style=\"text-align: center;\">[latex](x\u22123)(5x+12)+0=5x^2\u22123x\u221236[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use synthetic division to divide [latex]-9x^4 + 10x^3 + 7x^2 - 6[\/latex] by [latex]x - 1[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981344\">Show Solution<\/button> <\/p>\n<div id=\"q981344\" class=\"hidden-answer\" style=\"display: none\">Notice there is no [latex]x[\/latex]-term. We will use a zero as the coefficient for that term.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_9533\" aria-describedby=\"caption-attachment-9533\" style=\"width: 205px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9533 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/23151200\/21c33f8114c651702043fec1fafe25f629656a9a.webp\" alt=\"A synthetic division table showing the division of a polynomial by the binomial (x - 1). The divisor '1' is on the far left. The row at the top lists the coefficients of the polynomial: -9, 10, 7, 0, and -6. The second row represents the products of the divisor and the results of each step: -9, 1, 8, and 8. The bottom row sums the coefficients and the products: -9, 1, 8, 8, and 2.\" width=\"205\" height=\"76\" \/><figcaption id=\"caption-attachment-9533\" class=\"wp-caption-text\">Synthetic division<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The result is [latex]-9x^3 + x^2 + 8x + 8 + \\frac{2}{x-1}[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24625\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24625&theme=lumen&iframe_resize_id=ohm24625&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Watch the following video see an example of dividing polynomials using synthetic division.<\/p>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgecdbac-FxHWoUOq2iQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FxHWoUOq2iQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cgecdbac-FxHWoUOq2iQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328533&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cgecdbac-FxHWoUOq2iQ&amp;vembed=0&amp;video_id=FxHWoUOq2iQ&amp;video_target=tpm-plugin-cgecdbac-FxHWoUOq2iQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Synthetic+Division+of+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSynthetic Division of Polynomials\u201d here (opens in new window). <\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heeadhhf-_Y0fRbh1RY8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_Y0fRbh1RY8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-heeadhhf-_Y0fRbh1RY8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846615&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-heeadhhf-_Y0fRbh1RY8&amp;vembed=0&amp;video_id=_Y0fRbh1RY8&amp;video_target=tpm-plugin-heeadhhf-_Y0fRbh1RY8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Divide+a+Polynomial+by+a+Binomial+Using+Synthetic+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Divide a Polynomial by a Binomial Using Synthetic Division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Remainder Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The Remainder Theorem is a quick way to evaluate polynomials. If you divide a polynomial [latex]f(x)[\/latex]by [latex]x - k[\/latex], the remainder is simply the value of [latex]f(k)[\/latex]. It&#8217;s like a shortcut in a maze, allowing you to reach the end without going through all the paths.<\/p>\n<p>For example, to evaluate [latex]f(x) = 6x^4 - x^3 - 15x^2 + 2x - 7[\/latex] at [latex]x = 2[\/latex], you can use synthetic division or directly substitute [latex]x[\/latex] with [latex]2[\/latex] to find the remainder.<\/p>\n<\/div>\n<section class=\"textbox example\">Use the remainder theorem to evaluate [latex]f(x)=2x^5\u22123x^4\u22129x^3+8x^2+2[\/latex] at [latex]x = -3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981346\">Show Solution<\/button> <\/p>\n<div id=\"q981346\" class=\"hidden-answer\" style=\"display: none\">To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by [latex]x + 3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c|cccccc} -3 & 2 & -3 & -9 & 8 & 0 & 2 \\\\ & & -6 & 27 & -54 & 138 & -414 \\\\ \\hline & 2 & -9 & 18 & -46 & 138 & -412 \\\\ \\end{array}[\/latex]<\/p>\n<p>The remainder is [latex]-412[\/latex]. Therefore, [latex]f(-3) = -412[\/latex].<\/p>\n<p><strong>Analysis<\/strong>We can check our answer by evaluating [latex]f(-3)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) & = & 2x^5 - 3x^4 - 9x^3 + 8x^2 + 2 \\\\ f(-3) & =\u00a0 &2(-3)^5 - 3(-3)^4 - 9(-3)^3 + 8(-3)^2 + 2 \\\\ & = & -412 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on the remainder theorem.<\/p>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fadfaeca-p1lSRAeEMR0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/p1lSRAeEMR0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fadfaeca-p1lSRAeEMR0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328534&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fadfaeca-p1lSRAeEMR0&amp;vembed=0&amp;video_id=p1lSRAeEMR0&amp;video_target=tpm-plugin-fadfaeca-p1lSRAeEMR0\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Remainder+Theorem+and+Synthetic+Division+of+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRemainder Theorem and Synthetic Division of Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Dividing polynomials using long 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