{"id":2243,"date":"2016-11-03T19:20:56","date_gmt":"2016-11-03T19:20:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2243"},"modified":"2017-08-15T21:24:22","modified_gmt":"2017-08-15T21:24:22","slug":"introduction-partial-fractions-an-application-of-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/introduction-partial-fractions-an-application-of-systems\/","title":{"raw":"Introduction to Partial Fractions: an Application of Systems","rendered":"Introduction to Partial Fractions: an Application of Systems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Decompose \u2009 [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has only nonrepeated linear factors.<\/li>\r\n \t<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has repeated linear factors.<\/li>\r\n \t<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has a nonrepeated irreducible quadratic factor.<\/li>\r\n \t<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has a repeated irreducible quadratic factor.<\/li>\r\n<\/ul>\r\n<\/div>\r\nEarlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized\u2014the decomposition of rational expressions.\r\n\r\nFractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Decompose \u2009 [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has only nonrepeated linear factors.<\/li>\n<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has repeated linear factors.<\/li>\n<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has a nonrepeated irreducible quadratic factor.<\/li>\n<li>Decompose \u2009[latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] ,\u2009 where \u2009Q( x )\u2009 has a repeated irreducible quadratic factor.<\/li>\n<\/ul>\n<\/div>\n<p>Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized\u2014the decomposition of rational expressions.<\/p>\n<p>Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.<\/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-2243\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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 <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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