{"id":1515,"date":"2022-04-12T19:15:15","date_gmt":"2022-04-12T19:15:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1515"},"modified":"2026-01-06T18:03:24","modified_gmt":"2026-01-06T18:03:24","slug":"3-4-3-the-remainder-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-4-3-the-remainder-theorem\/","title":{"raw":"3.4.3: The Remainder Theorem","rendered":"3.4.3: The Remainder Theorem"},"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-1378\" class=\"standard post-1378 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Use the remainder theorem to find the reminder of the division of a polynomial divided by a binomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3 id=\"fs-id1165135471230\">The Remainder Theorem<\/h3>\r\nLet's first consider the synthetic division of a polynomial [latex]p(x)=4x^3-x^2-3x+2[\/latex] by a linear function [latex]d(x)=x-2[\/latex]:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1839\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/26214155\/Synthetic-division-with-arrows-300x34.png\" alt=\"Worked example of synthetic division. 4x cubed -x squared -3x + 2 divided by x - 2, giving 4x squared + 7x + 11 with remainder 24.\" width=\"609\" height=\"69\" \/><\/p>\r\n[latex]\\dfrac{4x^3-x^2-3x+2}{x-2}=4x^2+7x+11+\\dfrac{24}{x-2}[\/latex]\r\n\r\nThe remainder when we divide by [latex]x-2[\/latex] is 24.\r\n\r\nNow, let's evaluate the function value [latex]p(2)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}p(x)&amp;=4x^3-x^2-3x+2\\\\p(2)&amp;=4(2)^3-(2)^2-3(2)+2\\\\&amp;=32-4-6+2\\\\&amp;=24\\end{aligned}[\/latex]<\/p>\r\nWe have just shown that the remainder when we divide by [latex]x-2[\/latex] is equal to the function value [latex]p(2)=24[\/latex].\r\n\r\nThis is not a coincidence. If a polynomial is divided by [latex]x\u2013k[\/latex], the remainder is always equal to the polynomial function at [latex]k[\/latex], that is,\u00a0[latex]p(k)[\/latex].\u00a0Let\u2019s walk through the proof of the theorem.\r\n<p id=\"fs-id1165134085965\">Recall that the\u00a0<strong><em>Division Algorithm<\/em>\u00a0<\/strong>states that, given a polynomial dividend\u00a0[latex]p(x)[\/latex]\u00a0and a non-zero polynomial divisor\u00a0[latex]d(x)[\/latex]\u00a0where the degree of\u00a0[latex]d(x)[\/latex]\u00a0is less than or equal to the degree of\u00a0[latex]p(x)[\/latex], there exist unique polynomials\u00a0[latex]q(x)[\/latex]\u00a0and [latex]r(x)[\/latex] such that<\/p>\r\n\r\n<div id=\"eip-753\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]p\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165134094600\">If the divisor,\u00a0[latex]d(x)=x-k[\/latex], this takes the form<\/p>\r\n\r\n<div id=\"eip-567\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]p\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r(x)[\/latex]<\/div>\r\n<p id=\"fs-id1165137447771\">Since the divisor\u00a0[latex]x-k[\/latex] has degree 1, the remainder will be a constant,\u00a0[latex]r[\/latex]. And, if we evaluate this for\u00a0[latex]x=k[\/latex], we have<\/p>\r\n\r\n<div id=\"eip-791\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\begin{aligned}p(k)&amp;=(k-k)q(k)+r \\\\ &amp;=0\\cdot q\\left(k\\right)+r \\\\ &amp;=r\\hfill \\end{aligned}\\end{cases}[\/latex]<\/div>\r\n<div data-type=\"equation\" data-label=\"\"><\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">In other words,\u00a0[latex]p(k)[\/latex]\u00a0is the remainder obtained by dividing\u00a0[latex]p(x)[\/latex] by\u00a0[latex]x-k[\/latex].<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>THE REMAINDER THEOREM<\/h3>\r\n<p style=\"text-align: center;\">For any polynomial [latex]p(x)[\/latex] of degree 1 or higher and any linear function [latex]d(x)=x-k[\/latex],<\/p>\r\n<p style=\"text-align: center;\">[latex]p(k)[\/latex] = the remainder when [latex]p(x)[\/latex] is divided by [latex]x-k[\/latex].<\/p>\r\nAlgebraically:\r\n<p style=\"text-align: center;\">If [latex]\\dfrac{p(x)}{x-k}=q(x)+\\dfrac{r}{x-k}[\/latex], then [latex]r=p(k)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine the remainder of the division of\u00a0[latex]f\\left(x\\right)={x}^{3}-15{x}^{2}+2x - 7[\/latex] and [latex]x-2[\/latex] without doing the division.\r\n<h4>Solution<\/h4>\r\nThe remainder theorem tells us that\u00a0\u00a0[latex]\\dfrac{f(x)}{d(x)}=\\dfrac{{x}^{3}-15{x}^{2}+2x - 7}{x-2}=q(x)+r[\/latex] and that [latex]r=f(2)[\/latex].\r\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=(2)^{3}-15(2)^{2}+2(2) - 7 = -55[\/latex]<\/p>\r\nSo the remainder is \u201355.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nFind the remainder of the division of the polynomial\u00a0[latex]{x}^{2}-2x+3[\/latex] and [latex]x+1[\/latex].\r\n<h4>Solution<\/h4>\r\nLet [latex]p(x)={x}^{2}-2x+3[\/latex], then use the remainder theorem to find the remainder = [latex]p(-1)[\/latex].\r\n<p style=\"text-align: center;\">[latex]p(-1)=(-1)^{2}-2(-1)+3 = 1 + 2 + 3 = 6[\/latex]<\/p>\r\nSo the remainder is 6.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine the remainder when the polynomial\u00a0[latex]2x^{2}-3x+4[\/latex] is divided by [latex]x-3[\/latex].\r\n\r\n[reveal-answer q=\"hjm352\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm352\"]\r\n\r\nRemainder = 13\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine the remainder when the polynomial\u00a0[latex]-3x^3+2x^{2}-3x+4[\/latex] is divided by [latex]x+2[\/latex].\r\n\r\n[reveal-answer q=\"hjm700\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm700\"]\r\n\r\nRemainder = 42\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\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-1378\" class=\"standard post-1378 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Use the remainder theorem to find the reminder of the division of a polynomial divided by a binomial<\/li>\n<\/ul>\n<\/div>\n<h3 id=\"fs-id1165135471230\">The Remainder Theorem<\/h3>\n<p>Let&#8217;s first consider the synthetic division of a polynomial [latex]p(x)=4x^3-x^2-3x+2[\/latex] by a linear function [latex]d(x)=x-2[\/latex]:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1839\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/26214155\/Synthetic-division-with-arrows-300x34.png\" alt=\"Worked example of synthetic division. 4x cubed -x squared -3x + 2 divided by x - 2, giving 4x squared + 7x + 11 with remainder 24.\" width=\"609\" height=\"69\" \/><\/p>\n<p>[latex]\\dfrac{4x^3-x^2-3x+2}{x-2}=4x^2+7x+11+\\dfrac{24}{x-2}[\/latex]<\/p>\n<p>The remainder when we divide by [latex]x-2[\/latex] is 24.<\/p>\n<p>Now, let&#8217;s evaluate the function value [latex]p(2)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}p(x)&=4x^3-x^2-3x+2\\\\p(2)&=4(2)^3-(2)^2-3(2)+2\\\\&=32-4-6+2\\\\&=24\\end{aligned}[\/latex]<\/p>\n<p>We have just shown that the remainder when we divide by [latex]x-2[\/latex] is equal to the function value [latex]p(2)=24[\/latex].<\/p>\n<p>This is not a coincidence. If a polynomial is divided by [latex]x\u2013k[\/latex], the remainder is always equal to the polynomial function at [latex]k[\/latex], that is,\u00a0[latex]p(k)[\/latex].\u00a0Let\u2019s walk through the proof of the theorem.<\/p>\n<p id=\"fs-id1165134085965\">Recall that the\u00a0<strong><em>Division Algorithm<\/em>\u00a0<\/strong>states that, given a polynomial dividend\u00a0[latex]p(x)[\/latex]\u00a0and a non-zero polynomial divisor\u00a0[latex]d(x)[\/latex]\u00a0where the degree of\u00a0[latex]d(x)[\/latex]\u00a0is less than or equal to the degree of\u00a0[latex]p(x)[\/latex], there exist unique polynomials\u00a0[latex]q(x)[\/latex]\u00a0and [latex]r(x)[\/latex] such that<\/p>\n<div id=\"eip-753\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]p\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/div>\n<p id=\"fs-id1165134094600\">If the divisor,\u00a0[latex]d(x)=x-k[\/latex], this takes the form<\/p>\n<div id=\"eip-567\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]p\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r(x)[\/latex]<\/div>\n<p id=\"fs-id1165137447771\">Since the divisor\u00a0[latex]x-k[\/latex] has degree 1, the remainder will be a constant,\u00a0[latex]r[\/latex]. And, if we evaluate this for\u00a0[latex]x=k[\/latex], we have<\/p>\n<div id=\"eip-791\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\begin{aligned}p(k)&=(k-k)q(k)+r \\\\ &=0\\cdot q\\left(k\\right)+r \\\\ &=r\\hfill \\end{aligned}\\end{cases}[\/latex]<\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">In other words,\u00a0[latex]p(k)[\/latex]\u00a0is the remainder obtained by dividing\u00a0[latex]p(x)[\/latex] by\u00a0[latex]x-k[\/latex].<\/div>\n<div class=\"textbox shaded\">\n<h3>THE REMAINDER THEOREM<\/h3>\n<p style=\"text-align: center;\">For any polynomial [latex]p(x)[\/latex] of degree 1 or higher and any linear function [latex]d(x)=x-k[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]p(k)[\/latex] = the remainder when [latex]p(x)[\/latex] is divided by [latex]x-k[\/latex].<\/p>\n<p>Algebraically:<\/p>\n<p style=\"text-align: center;\">If [latex]\\dfrac{p(x)}{x-k}=q(x)+\\dfrac{r}{x-k}[\/latex], then [latex]r=p(k)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine the remainder of the division of\u00a0[latex]f\\left(x\\right)={x}^{3}-15{x}^{2}+2x - 7[\/latex] and [latex]x-2[\/latex] without doing the division.<\/p>\n<h4>Solution<\/h4>\n<p>The remainder theorem tells us that\u00a0\u00a0[latex]\\dfrac{f(x)}{d(x)}=\\dfrac{{x}^{3}-15{x}^{2}+2x - 7}{x-2}=q(x)+r[\/latex] and that [latex]r=f(2)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=(2)^{3}-15(2)^{2}+2(2) - 7 = -55[\/latex]<\/p>\n<p>So the remainder is \u201355.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Find the remainder of the division of the polynomial\u00a0[latex]{x}^{2}-2x+3[\/latex] and [latex]x+1[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Let [latex]p(x)={x}^{2}-2x+3[\/latex], then use the remainder theorem to find the remainder = [latex]p(-1)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]p(-1)=(-1)^{2}-2(-1)+3 = 1 + 2 + 3 = 6[\/latex]<\/p>\n<p>So the remainder is 6.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine the remainder when the polynomial\u00a0[latex]2x^{2}-3x+4[\/latex] is divided by [latex]x-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm352\">Show Answer<\/span><\/p>\n<div id=\"qhjm352\" class=\"hidden-answer\" style=\"display: none\">\n<p>Remainder = 13<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine the remainder when the polynomial\u00a0[latex]-3x^3+2x^{2}-3x+4[\/latex] is divided by [latex]x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm700\">Show Answer<\/span><\/p>\n<div id=\"qhjm700\" class=\"hidden-answer\" style=\"display: none\">\n<p>Remainder = 42<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-1515\">\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>Example 1 and Example 2. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <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><li>Try it hjm504; hjm700; hjm352. Introduction.. <strong>Authored by<\/strong>: Hazel McKenna. <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>Adapted and revised: Precalculus. <strong>Authored by<\/strong>: Jay Abramson, 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>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Example 1 and Example 2\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Adapted and revised: Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"},{\"type\":\"original\",\"description\":\"Try it hjm504; hjm700; hjm352. Introduction.\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1515","chapter","type-chapter","status-publish","hentry"],"part":1176,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1515","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1515\/revisions"}],"predecessor-version":[{"id":4782,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1515\/revisions\/4782"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/1176"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1515\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1515"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1515"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1515"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1515"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}