{"id":3137,"date":"2025-08-15T22:12:26","date_gmt":"2025-08-15T22:12:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3137"},"modified":"2025-09-05T16:42:46","modified_gmt":"2025-09-05T16:42:46","slug":"zeros-of-polynomial-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/zeros-of-polynomial-functions-apply-it\/","title":{"raw":"Polynomial Equations and Inequalities: Apply It","rendered":"Polynomial Equations and Inequalities: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">Type your <em>Learning Goals<\/em> text here<\/section>\r\n<div class=\"h-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Polynomial Inequalities in Break-Even Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\">We can use polynomial inequalities to determine when companies will be profitable by comparing revenue and cost functions. While real business models involve many variables, simplified polynomial models help illustrate the mathematical relationship between production levels and profitability.<\/p>\r\n\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Break-Even Analysis<\/h3>\r\n<p class=\"whitespace-normal break-words\">When a company's revenue and cost functions are both polynomials, finding when the business is profitable requires solving a polynomial inequality.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A small manufacturing company has:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Revenue function: [latex]R(x) = 12x - 0.02x^2[\/latex] (in thousands of dollars)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cost function: [latex]C(x) = 2x + 100[\/latex] (in thousands of dollars)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] represents units produced (in hundreds).<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">To find when the company is profitable, we solve [latex]R(x) &gt; C(x)[\/latex]: [latex]12x - 0.02x^2 &gt; 2x + 100[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Rearranging: [latex]-0.02x^2 + 10x - 100 &gt; 0[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Multiplying by -50 (and flipping the inequality): [latex]x^2 - 500x + 5000 &lt; 0[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Using the quadratic formula: [latex]x = \\frac{500 \\pm \\sqrt{250000 - 20000}}{2} = \\frac{500 \\pm \\sqrt{230000}}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]x = \\frac{500 \\pm 479.6}{2}[\/latex], so [latex]x \u2248 10.2[\/latex] or [latex]x \u2248 489.8[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Since the coefficient of [latex]x^2[\/latex] is positive, the parabola opens upward, making the expression negative between the roots.<\/p>\r\n<p class=\"whitespace-normal break-words\">The company is profitable when producing between 1,020 and 48,980 units.<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">When multiplying an inequality by a negative number, remember to flip the inequality sign.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311149[\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Complex Cost Structures<\/h3>\r\n<p class=\"whitespace-normal break-words\">Some businesses have more complex cost structures that create polynomial models, requiring careful analysis of multiple break-even points.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How To: Solving Polynomial Inequalities for Profitability<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Set up the inequality [latex]R(x) &gt; C(x)[\/latex] for profitability<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move all terms to one side: [latex]R(x) - C(x) &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the resulting polynomial or find its zeros<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test intervals between zeros to determine where the expression is positive<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider practical constraints (production levels must be non-negative)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret the solution in business terms<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A tech startup has:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Revenue: [latex]R(x) = 15x[\/latex] (thousands of dollars)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Costs: [latex]C(x) = x^3 - 9x^2 + 26x + 24[\/latex] (thousands of dollars)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] represents months of operation.<\/p>\r\n<p class=\"whitespace-normal break-words\">For profitability: [latex]15x &gt; x^3 - 9x^2 + 26x + 24[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Rearranging: [latex]0 &gt; x^3 - 9x^2 + 11x + 24[\/latex] Or: [latex]x^3 - 9x^2 + 11x + 24 &lt; 0[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">To find zeros of [latex]f(x) = x^3 - 9x^2 + 11x + 24[\/latex], we test rational roots.<\/p>\r\n<p class=\"whitespace-normal break-words\">Testing [latex]x = -1[\/latex]: <a class=\"underline\" href=\"-1\">latex<\/a>^3 - 9(-1)^2 + 11(-1) + 24 = -1 - 9 - 11 + 24 = 3 \\neq 0[\/latex] Testing [latex]x = -2[\/latex]: <a class=\"underline\" href=\"-2\">latex<\/a>^3 - 9(-2)^2 + 11(-2) + 24 = -8 - 36 - 22 + 24 = -42 \\neq 0[\/latex] Testing [latex]x = 3[\/latex]: [latex]27 - 81 + 33 + 24 = 3 \\neq 0[\/latex] Testing [latex]x = 4[\/latex]: [latex]64 - 144 + 44 + 24 = -12 \\neq 0[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Let's try [latex]x = 8[\/latex]: [latex]512 - 576 + 88 + 24 = 48 \\neq 0[\/latex] Try [latex]x = 6[\/latex]: [latex]216 - 324 + 66 + 24 = -18 \\neq 0[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Using numerical methods, suppose we find the roots are approximately [latex]x = -1.5[\/latex], [latex]x = 2.8[\/latex], and [latex]x = 7.7[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Testing intervals for [latex]x \u2265 0[\/latex] (since negative months don't make sense):<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]0 &lt; x &lt; 2.8[\/latex]: Test [latex]x = 1[\/latex]: [latex]1 - 9 + 11 + 24 = 27 &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [latex]2.8 &lt; x &lt; 7.7[\/latex]: Test [latex]x = 5[\/latex]: [latex]125 - 225 + 55 + 24 = -21 &lt; 0[\/latex] \u2713<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [latex]x &gt; 7.7[\/latex]: Test [latex]x = 9[\/latex]: [latex]729 - 729 + 99 + 24 = 123 &gt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The startup is profitable approximately between month3 and month 8.<\/p>\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Higher-degree polynomials can have multiple intervals where they are positive or negative, leading to several profitable periods separated by unprofitable ones.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311150[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">Type your <em>Learning Goals<\/em> text here<\/section>\n<div class=\"h-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Polynomial Inequalities in Break-Even Analysis<\/h2>\n<p class=\"whitespace-normal break-words\">We can use polynomial inequalities to determine when companies will be profitable by comparing revenue and cost functions. While real business models involve many variables, simplified polynomial models help illustrate the mathematical relationship between production levels and profitability.<\/p>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Break-Even Analysis<\/h3>\n<p class=\"whitespace-normal break-words\">When a company&#8217;s revenue and cost functions are both polynomials, finding when the business is profitable requires solving a polynomial inequality.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A small manufacturing company has:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Revenue function: [latex]R(x) = 12x - 0.02x^2[\/latex] (in thousands of dollars)<\/li>\n<li class=\"whitespace-normal break-words\">Cost function: [latex]C(x) = 2x + 100[\/latex] (in thousands of dollars)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] represents units produced (in hundreds).<\/p>\n<p class=\"whitespace-pre-wrap break-words\">To find when the company is profitable, we solve [latex]R(x) > C(x)[\/latex]: [latex]12x - 0.02x^2 > 2x + 100[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Rearranging: [latex]-0.02x^2 + 10x - 100 > 0[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Multiplying by -50 (and flipping the inequality): [latex]x^2 - 500x + 5000 < 0[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Using the quadratic formula: [latex]x = \\frac{500 \\pm \\sqrt{250000 - 20000}}{2} = \\frac{500 \\pm \\sqrt{230000}}{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]x = \\frac{500 \\pm 479.6}{2}[\/latex], so [latex]x \u2248 10.2[\/latex] or [latex]x \u2248 489.8[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Since the coefficient of [latex]x^2[\/latex] is positive, the parabola opens upward, making the expression negative between the roots.<\/p>\n<p class=\"whitespace-normal break-words\">The company is profitable when producing between 1,020 and 48,980 units.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">When multiplying an inequality by a negative number, remember to flip the inequality sign.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311149\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311149&theme=lumen&iframe_resize_id=ohm311149&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Complex Cost Structures<\/h3>\n<p class=\"whitespace-normal break-words\">Some businesses have more complex cost structures that create polynomial models, requiring careful analysis of multiple break-even points.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How To: Solving Polynomial Inequalities for Profitability<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Set up the inequality [latex]R(x) > C(x)[\/latex] for profitability<\/li>\n<li class=\"whitespace-normal break-words\">Move all terms to one side: [latex]R(x) - C(x) > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the resulting polynomial or find its zeros<\/li>\n<li class=\"whitespace-normal break-words\">Test intervals between zeros to determine where the expression is positive<\/li>\n<li class=\"whitespace-normal break-words\">Consider practical constraints (production levels must be non-negative)<\/li>\n<li class=\"whitespace-normal break-words\">Interpret the solution in business terms<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A tech startup has:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Revenue: [latex]R(x) = 15x[\/latex] (thousands of dollars)<\/li>\n<li class=\"whitespace-normal break-words\">Costs: [latex]C(x) = x^3 - 9x^2 + 26x + 24[\/latex] (thousands of dollars)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] represents months of operation.<\/p>\n<p class=\"whitespace-normal break-words\">For profitability: [latex]15x > x^3 - 9x^2 + 26x + 24[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Rearranging: [latex]0 > x^3 - 9x^2 + 11x + 24[\/latex] Or: [latex]x^3 - 9x^2 + 11x + 24 < 0[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">To find zeros of [latex]f(x) = x^3 - 9x^2 + 11x + 24[\/latex], we test rational roots.<\/p>\n<p class=\"whitespace-normal break-words\">Testing [latex]x = -1[\/latex]: <a class=\"underline\" href=\"-1\">latex<\/a>^3 &#8211; 9(-1)^2 + 11(-1) + 24 = -1 &#8211; 9 &#8211; 11 + 24 = 3 \\neq 0[\/latex] Testing [latex]x = -2[\/latex]: <a class=\"underline\" href=\"-2\">latex<\/a>^3 &#8211; 9(-2)^2 + 11(-2) + 24 = -8 &#8211; 36 &#8211; 22 + 24 = -42 \\neq 0[\/latex] Testing [latex]x = 3[\/latex]: [latex]27 - 81 + 33 + 24 = 3 \\neq 0[\/latex] Testing [latex]x = 4[\/latex]: [latex]64 - 144 + 44 + 24 = -12 \\neq 0[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Let&#8217;s try [latex]x = 8[\/latex]: [latex]512 - 576 + 88 + 24 = 48 \\neq 0[\/latex] Try [latex]x = 6[\/latex]: [latex]216 - 324 + 66 + 24 = -18 \\neq 0[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Using numerical methods, suppose we find the roots are approximately [latex]x = -1.5[\/latex], [latex]x = 2.8[\/latex], and [latex]x = 7.7[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Testing intervals for [latex]x \u2265 0[\/latex] (since negative months don&#8217;t make sense):<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">For [latex]0 < x < 2.8[\/latex]: Test [latex]x = 1[\/latex]: [latex]1 - 9 + 11 + 24 = 27 > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]2.8 < x < 7.7[\/latex]: Test [latex]x = 5[\/latex]: [latex]125 - 225 + 55 + 24 = -21 < 0[\/latex] \u2713<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]x > 7.7[\/latex]: Test [latex]x = 9[\/latex]: [latex]729 - 729 + 99 + 24 = 123 > 0[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The startup is profitable approximately between month3 and month 8.<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Higher-degree polynomials can have multiple intervals where they are positive or negative, leading to several profitable periods separated by unprofitable ones.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311150\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311150&theme=lumen&iframe_resize_id=ohm311150&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":506,"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\/3137"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3137\/revisions"}],"predecessor-version":[{"id":3732,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3137\/revisions\/3732"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}