{"id":1781,"date":"2015-11-12T18:30:45","date_gmt":"2015-11-12T18:30:45","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1781"},"modified":"2015-11-12T18:30:45","modified_gmt":"2015-11-12T18:30:45","slug":"key-concepts-glossary-30","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-30\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<h2>Key Concepts<\/h2>\n<ul><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><h2>Glossary<\/h2>\n<dl id=\"fs-id1165135436446\" class=\"definition\"><dt>partial fractions<\/dt><dd id=\"fs-id1165133203619\">the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression<\/dd><\/dl><dl id=\"fs-id1165131962224\" class=\"definition\"><dt>partial fraction decomposition<\/dt><dd id=\"fs-id1165135177574\">the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions<\/dd><\/dl>","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<dl id=\"fs-id1165135436446\" class=\"definition\">\n<dt>partial fractions<\/dt>\n<dd id=\"fs-id1165133203619\">the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131962224\" class=\"definition\">\n<dt>partial fraction decomposition<\/dt>\n<dd id=\"fs-id1165135177574\">the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions<\/dd>\n<\/dl>\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-1781\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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":276,"menu_order":6,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1781","chapter","type-chapter","status-publish","hentry"],"part":1775,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1781","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1781\/revisions"}],"predecessor-version":[{"id":2253,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1781\/revisions\/2253"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1775"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1781\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1781"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1781"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1781"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1781"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}