{"id":3185,"date":"2025-08-15T22:30:35","date_gmt":"2025-08-15T22:30:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3185"},"modified":"2025-09-18T20:23:48","modified_gmt":"2025-09-18T20:23:48","slug":"partial-fractions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/partial-fractions-apply-it\/","title":{"raw":"Partial Fractions: Apply It","rendered":"Partial Fractions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Decompose [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] , where Q( x ) has only linear factors.<\/li>\r\n \t<li>Decompose [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] , where Q( x ) has an irreducible quadratic factor.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">A pharmaceutical research team is developing a new medication and needs to understand how the drug is processed in the human body. The concentration of the drug in the bloodstream over time follows complex mathematical models that describe how the medication is absorbed, distributed, and eliminated through different pathways. To optimize dosing schedules and ensure patient safety, the team must break down these complex expressions into simpler components.<\/p>\r\n\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">The Pharmacokinetics Challenge<\/h2>\r\n<p class=\"whitespace-normal break-words\">After a patient receives a single dose of the experimental medication, the drug concentration in the blood (measured in mg\/L) follows the function:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]C(t) = \\frac{90t + 300}{(t + 2)(t + 5)}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] represents time in hours after administration. This expression combines multiple biological processes: the drug's absorption into the bloodstream and its elimination through the liver and kidneys.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312328[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312329[\/ohm_question]\r\n\r\n<\/section><\/section>\r\n<p class=\"whitespace-normal break-words\">The research team studies a second medication with a more complex elimination pattern:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]D(t) = \\frac{2t^2 + 14t + 32}{t(t + 4)^2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This represents a drug that is eliminated through three processes: immediate clearance, primary organ elimination, and secondary metabolic pathway with the same organ showing saturation effects.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312331[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Decompose [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] , where Q( x ) has only linear factors.<\/li>\n<li>Decompose [latex]\\frac{{P( x )}}{{ Q( x )}}[\/latex] , where Q( x ) has an irreducible quadratic factor.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">A pharmaceutical research team is developing a new medication and needs to understand how the drug is processed in the human body. The concentration of the drug in the bloodstream over time follows complex mathematical models that describe how the medication is absorbed, distributed, and eliminated through different pathways. To optimize dosing schedules and ensure patient safety, the team must break down these complex expressions into simpler components.<\/p>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">The Pharmacokinetics Challenge<\/h2>\n<p class=\"whitespace-normal break-words\">After a patient receives a single dose of the experimental medication, the drug concentration in the blood (measured in mg\/L) follows the function:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]C(t) = \\frac{90t + 300}{(t + 2)(t + 5)}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] represents time in hours after administration. This expression combines multiple biological processes: the drug&#8217;s absorption into the bloodstream and its elimination through the liver and kidneys.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312328\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312328&theme=lumen&iframe_resize_id=ohm312328&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312329\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312329&theme=lumen&iframe_resize_id=ohm312329&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/section>\n<p class=\"whitespace-normal break-words\">The research team studies a second medication with a more complex elimination pattern:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]D(t) = \\frac{2t^2 + 14t + 32}{t(t + 4)^2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This represents a drug that is eliminated through three processes: immediate clearance, primary organ elimination, and secondary metabolic pathway with the same organ showing saturation effects.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312331\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312331&theme=lumen&iframe_resize_id=ohm312331&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3185"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3185\/revisions"}],"predecessor-version":[{"id":4141,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3185\/revisions\/4141"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3185\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3185"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3185"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3185"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}