{"id":87,"date":"2017-08-23T07:44:43","date_gmt":"2017-08-23T07:44:43","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=87"},"modified":"2017-09-18T15:43:00","modified_gmt":"2017-09-18T15:43:00","slug":"simplifying-rational-polynomials","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/simplifying-rational-polynomials\/","title":{"rendered":"Simplifying Rational Polynomials"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/factoring-polynomials\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-and-graphing-quadratic-equations\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Simplifying Rational Polynomials<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to simplify, add, subtract, multiply, and divide rational polynomials.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>How to <em><strong>factor <\/strong><\/em>polynomials using\u00a0the GCF or in the form\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p1_clip_image002.gif\" width=\"64\" height=\"14\" name=\"graphics2\" align=\"BOTTOM\" border=\"0\" \/>;\u00a0in the form <strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p1_clip_image004.gif\" width=\"72\" height=\"14\" name=\"graphics3\" align=\"TEXTTOP\" border=\"0\" \/><\/strong>;<strong>\u00a0<\/strong>and in special cases, such as the difference of two squares and\u00a0perfect square trinomials.<\/li>\n<\/ul>\n<section>\n<h3><strong>What are rational polynomials?<\/strong><\/h3>\n<p>A <abbr title=\" a polynomial with rational coefficients\">rational\u00a0<\/abbr>polynomials\u00a0a polynomial with rational\u00a0coefficients. The word <em><span style=\"text-decoration: none;\">rational\u00a0<\/span><\/em><span style=\"text-decoration: none;\">here is connected to the term <\/span><em><span style=\"text-decoration: none;\">ratio<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0because a rational number can be expressed as a ratio of integers.\u00a0Likewise, a rational function or a rational polynomial can be\u00a0expressed as a ratio of two polynomials. <\/span><\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image002.gif\" width=\"66\" height=\"34\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The numerator consists of a linear equation, and the\u00a0denominator is a quadratic.<\/p>\n<p><strong>How do we simplify rational polynomials?<\/strong><\/p>\n<p>Simplifying rational polynomials is directly related to the\u00a0methods in the previous section\u2014factoring and finding the\u00a0greatest common factors of polynomials. Before a rational\u00a0polynomial can be completely simplified, the GCF has to be\u00a0extracted from the polynomial (if one exists) and the resulting\u00a0polynomial has to be factored (if possible).<\/p>\n<p>For example, simplify the rational polynomial\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image004.gif\" width=\"99\" height=\"40\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<p>First, we must determine whether a GCF exists and, if so,\u00a0factor it out of the polynomial. In this example, the denominator\u00a0has a GCF = 3. Rewrite the ratio, factoring 3 out of the\u00a0denominator.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image006.gif\" width=\"86\" height=\"40\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now try factoring each quadratic in the numerator and<br \/>\ndenominator to see if any common terms will cancel.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image008.gif\" width=\"201\" height=\"41\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The linear term\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image010.gif\" width=\"40\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is common in both numerator and denominator, so it can be\u00a0cancelled or divided out. The simplified final answer is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image012.gif\" width=\"49\" height=\"37\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which shows the correct simplification of the rational\u00a0polynomial\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image014.gif\" width=\"83\" height=\"37\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image016.gif\" width=\"42\" height=\"37\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image018.gif\" width=\"42\" height=\"37\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image020.gif\" width=\"42\" height=\"37\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image022.gif\" width=\"40\" height=\"35\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. The first step is to see if a GCF\u00a0exists in either polynomial. In this problem there is not a GCF,\u00a0so we proceed to factor each quadratic. Doing so results in the\u00a0ratio\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image024.gif\" width=\"184\" height=\"40\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.\u00a0The next step is to cancel any common terms. In this case the\u00a0linear term\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image026.gif\" width=\"39\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is present in both the numerator and denominator, so it can be\u00a0canceled or divided out. The final simplified answer is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p2_clip_image027.gif\" width=\"42\" height=\"37\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<\/div>\n<\/section>\n<h3><strong>How do we perform operations on rational polynomials?<\/strong><\/h3>\n<p><strong>Adding and Subtracting Rational Polynomials <\/strong><\/p>\n<p>The process of adding and subtracting rational polynomials\u00a0follows the procedure for combining simple fractions. The\u00a0difference is that the common denominator is a polynomial instead\u00a0of a number. Once a common denominator is established, the\u00a0numerators can be added or subtracted.<\/p>\n<p>For example, add the two rational polynomials\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image002.gif\" width=\"50\" height=\"34\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0by first finding the common denominator, in this case, <span style=\"text-decoration: none;\">6<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em>.\u00a0Multiply the first polynomial by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image004.gif\" width=\"9\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and the second polynomial by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image006.gif\" width=\"9\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0to get a common denominator. Then add the numerators and simplify.\u00a0You will not change the value of the expression when you find a\u00a0common denominator since you are multiplying each part by one.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image008.gif\" width=\"262\" height=\"39\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the following correctly combines the two rational\u00a0polynomials\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image010.gif\" width=\"55\" height=\"34\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image012.gif\" width=\"93\" height=\"37\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image014.gif\" width=\"83\" height=\"37\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image016.gif\" width=\"83\" height=\"37\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image018.gif\" width=\"54\" height=\"37\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. To subtract, use the same method as\u00a0you would when adding. First, establish a common denominator. For\u00a0this problem, the common denominator is 10<em>x<\/em>. We then\u00a0multiply each numerator by the opposite denominator to &#8220;balance&#8221;\u00a0each ratio, giving us the following equivalent expression:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image020.gif\" width=\"432\" height=\"40\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p><strong>Multiplying Rational Polynomials<\/strong><\/p>\n<p>Once again, multiplying rational polynomials follows the same\u00a0rules as multiplying two fractions\u2014numerators are multiplied\u00a0together, and denominators are multiplied together.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image022.gif\" width=\"98\" height=\"37\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">The polynomial 2<\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">(which is actually\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image024.gif\" width=\"18\" height=\"34\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>)\u00a0is multiplied by the numerator. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image026.gif\" width=\"307\" height=\"41\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>After multiplying, check for any common terms in the numerator\u00a0and denominator to further simplify the ratio. To do this, factor\u00a0the denominator and see if a common term exists.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image028.gif\" width=\"173\" height=\"43\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image030.gif\" width=\"39\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0can be canceled or divided out, leaving\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image032.gif\" width=\"43\" height=\"41\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>After performing the operation, remember to simplify by canceling any\u00a0common GCF terms or common polynomial expressions.<\/p>\n<\/div>\n<h4>Dividing Rational Polynomials<\/h4>\n<p>Division consists of simply multiplying by the reciprocal, or,\u00a0in common terms, &#8220;flipping&#8221; the second ratio, and\u00a0changing the operation from division to multiplication.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image034.gif\" width=\"114\" height=\"39\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This is the exact procedure for rational polynomial division.<\/p>\n<p>For example, simplify the rational expression\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image036.gif\" width=\"136\" height=\"48\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The first step is to multiply by the reciprocal of the second\u00a0polynomial.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image038.gif\" width=\"286\" height=\"43\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Notice that there is a quadratic is in the numerator of the\u00a0second ratio. Factor this, with the hope of finding a common term\u00a0to cancel.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image040.gif\" width=\"292\" height=\"52\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The common term\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image042.gif\" width=\"38\" height=\"14\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0cancels from the numer<span style=\"text-decoration: none;\">ator and\u00a0denominator. 3<\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">will also cancel out of the numerator<\/span> and denominator\u00a0leaving\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p3_clip_image046.gif\" width=\"31\" height=\"34\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<h3><strong>Graphing Rational Polynomials <\/strong><\/h3>\n<p>If there are variables in the denominator after simplification,\u00a0then we can expect to find <abbr title=\"the point at which a rational function is undefined, or at which the denominator is equal to zero.\">asymptotes<\/abbr> as part of our graph since there will be values for the\u00a0denominator equal to zero.\u00a0For example, draw the graph of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s3_p4_clip_image002.gif\" width=\"43\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203%5B1%5D.1%20Art%20002.JPG\" alt=\"Graph of 6x (x+3)\" width=\"433\" height=\"325\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/center><span style=\"text-decoration: none;\">Because of the asymptotes,\u00a0the graph will approach <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em> = \u20133, but never reach this point. The function is <abbr title=\" points where, for a given domain, there is not a valid range\">undefined<\/abbr> at this point.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/factoring-polynomials\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-and-graphing-quadratic-equations\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<div><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Simplifying Rational Polynomials Objective In this lesson, you will study how to simplify, add, subtract, multiply, and divide rational polynomials. Previously Covered: How to factor polynomials using\u00a0the GCF or in the form\u00a0;\u00a0in the form ;\u00a0and in special cases, such as the difference of two squares and\u00a0perfect square trinomials. What [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-87","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/87","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=87"}],"version-history":[{"count":10,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/87\/revisions"}],"predecessor-version":[{"id":742,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/87\/revisions\/742"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=87"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}