{"id":393,"date":"2019-07-15T22:44:43","date_gmt":"2019-07-15T22:44:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/summary-partial-fractions-an-application-of-systems\/"},"modified":"2019-07-15T22:44:43","modified_gmt":"2019-07-15T22:44:43","slug":"summary-partial-fractions-an-application-of-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/summary-partial-fractions-an-application-of-systems\/","title":{"raw":"Summary: Partial Fractions: an Application of Systems","rendered":"Summary: Partial Fractions: an Application of Systems"},"content":{"raw":"\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Decompose [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] by writing the partial fractions as [latex]\\frac{A}{{a}_{1}x+{b}_{1}}+\\frac{B}{{a}_{2}x+{b}_{2}}[\/latex]. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations.<\/li>\n \t<li>The decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] with repeated linear factors must account for the factors of the denominator in increasing powers.<\/li>\n \t<li>The decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in [latex]\\frac{A}{x}+\\frac{Bx+C}{\\left(a{x}^{2}+bx+c\\right)}[\/latex].<\/li>\n \t<li>In the decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex], where [latex]Q\\left(x\\right)[\/latex] has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as\n<p style=\"text-align: center;\">[latex]\\frac{Ax+B}{\\left(a{x}^{2}+bx+c\\right)}+\\frac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\cdots \\text{+}\\frac{{A}_{n}x+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>partial fractions<\/strong> the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression\n\n<strong>partial fraction decomposition<\/strong> the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>Decompose [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] by writing the partial fractions as [latex]\\frac{A}{{a}_{1}x+{b}_{1}}+\\frac{B}{{a}_{2}x+{b}_{2}}[\/latex]. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations.<\/li>\n<li>The decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] with repeated linear factors must account for the factors of the denominator in increasing powers.<\/li>\n<li>The decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex] with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in [latex]\\frac{A}{x}+\\frac{Bx+C}{\\left(a{x}^{2}+bx+c\\right)}[\/latex].<\/li>\n<li>In the decomposition of [latex]\\frac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex], where [latex]Q\\left(x\\right)[\/latex] has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as\n<p style=\"text-align: center;\">[latex]\\frac{Ax+B}{\\left(a{x}^{2}+bx+c\\right)}+\\frac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\cdots \\text{+}\\frac{{A}_{n}x+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>partial fractions<\/strong> the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression<\/p>\n<p><strong>partial fraction decomposition<\/strong> the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions<\/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-393\">\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":17533,"menu_order":25,"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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