{"id":3143,"date":"2025-08-15T22:15:44","date_gmt":"2025-08-15T22:15:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3143"},"modified":"2026-01-05T17:26:09","modified_gmt":"2026-01-05T17:26:09","slug":"zeros-of-polynomial-functions-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/zeros-of-polynomial-functions-apply-it-2\/","title":{"raw":"Zeros of Polynomial Functions: Apply It","rendered":"Zeros of Polynomial Functions: Apply It"},"content":{"raw":"<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\r\n \t<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\r\n \t<li>Find zeros of a polynomial function.<\/li>\r\n \t<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Finding Polynomial Zeros in Engineering Design<\/h2>\r\n<p class=\"whitespace-normal break-words\">Engineers frequently encounter polynomial models where finding zeros reveals critical design points\u2014moments when systems reach equilibrium, structures experience zero stress, or processes achieve optimal efficiency. The systematic approach of using the Rational Zero Theorem and Factor Theorem transforms complex engineering problems into manageable calculations.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Structural Resonance Analysis<\/h3>\r\nCivil engineers must identify resonant frequencies where structures experience dangerous vibrations. These frequencies correspond to zeros of damping polynomials, making zero-finding techniques essential for safe design.\r\n\r\nA suspension bridge's damping system follows the polynomial [latex]D(f) = 2f^3 - 9f^2 + 10f - 3[\/latex], where [latex]f[\/latex] represents frequency in Hz and [latex]D(f)[\/latex] represents damping effectiveness. Engineers need to find frequencies where damping drops to zero (dangerous resonance points).\r\n\r\n[reveal-answer q=\"95468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95468\"]\r\n<p class=\"whitespace-normal break-words\">Using the Rational Zero Theorem:<\/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\">Factors of constant term (-3): [latex]\\pm 1, \\pm 3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factors of leading coefficient (2): [latex]\\pm 1, \\pm 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Possible rational zeros: [latex]\\pm 1, \\pm 3, \\pm \\frac{1}{2}, \\pm \\frac{3}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Testing [latex]f = 1[\/latex] using synthetic division:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{array}{c|rrrr}<br aria-hidden=\"true\" \/>1 &amp; 2 &amp; -9 &amp; +10 &amp; -3 \\\\<br aria-hidden=\"true\" \/>&amp; &amp; 2 &amp; -7 &amp; 3 \\\\<br aria-hidden=\"true\" \/>&amp; 2 &amp; -7 &amp; 3 &amp; 0 \\\\<br aria-hidden=\"true\" \/>\\end{array}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Since the remainder is 0, [latex]f = 1[\/latex] Hz is a resonant frequency. The Factor Theorem tells us [latex](f - 1)[\/latex] is a factor, so: [latex]D(f) = (f - 1)(2f^2 - 7f + +3).[\/latex]<\/p>\r\nTo find the other resonant frequencies we will factor the quadratic: [latex]D(f) = (f - 1)(2f^2 - 7f + +3)[\/latex]\r\n<p class=\"whitespace-normal break-words\">[latex]2f^2 - 7f + 3 = (2f - 1)(f - 3)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Complete factorization: [latex]D(f) = (f - 1)(2f - 1)(f - 3)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Critical resonant frequencies: 0.5 Hz, 1 Hz, and 3 Hz<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">When a polynomial has integer coefficients, the Rational Zero Theorem narrows your search dramatically. Test the simplest candidates first, often [latex]\\pm 1[\/latex] are zeros.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]317791[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How to: Finding All Polynomial Zeros Systematically<\/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\">Apply the Rational Zero Theorem to list possible rational zeros<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use synthetic division to test candidates, starting with simple values like [latex]\\pm 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When you find a zero, use the Factor Theorem to write the polynomial as a product<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the same process to the quotient polynomial (reduced degree)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continue until you reach a quadratic, then use factoring or the quadratic formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify your factorization by expanding back to the original polynomial<\/li>\r\n<\/ol>\r\n<\/section><\/div>","rendered":"<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the Factor Theorem to solve a polynomial equation.<\/li>\n<li>Use the Rational Zero Theorem to find rational zeros.<\/li>\n<li>Find zeros of a polynomial function.<\/li>\n<li>Use the Linear Factorization Theorem to find polynomials with given zeros.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Finding Polynomial Zeros in Engineering Design<\/h2>\n<p class=\"whitespace-normal break-words\">Engineers frequently encounter polynomial models where finding zeros reveals critical design points\u2014moments when systems reach equilibrium, structures experience zero stress, or processes achieve optimal efficiency. The systematic approach of using the Rational Zero Theorem and Factor Theorem transforms complex engineering problems into manageable calculations.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Structural Resonance Analysis<\/h3>\n<p>Civil engineers must identify resonant frequencies where structures experience dangerous vibrations. These frequencies correspond to zeros of damping polynomials, making zero-finding techniques essential for safe design.<\/p>\n<p>A suspension bridge&#8217;s damping system follows the polynomial [latex]D(f) = 2f^3 - 9f^2 + 10f - 3[\/latex], where [latex]f[\/latex] represents frequency in Hz and [latex]D(f)[\/latex] represents damping effectiveness. Engineers need to find frequencies where damping drops to zero (dangerous resonance points).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q95468\">Show Solution<\/button><\/p>\n<div id=\"q95468\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-normal break-words\">Using the Rational Zero Theorem:<\/p>\n<ul class=\"&#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Factors of constant term (-3): [latex]\\pm 1, \\pm 3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factors of leading coefficient (2): [latex]\\pm 1, \\pm 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Possible rational zeros: [latex]\\pm 1, \\pm 3, \\pm \\frac{1}{2}, \\pm \\frac{3}{2}[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Testing [latex]f = 1[\/latex] using synthetic division:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{array}{c|rrrr}<br aria-hidden=\"true\" \/>1 & 2 & -9 & +10 & -3 \\\\<br aria-hidden=\"true\" \/>& & 2 & -7 & 3 \\\\<br aria-hidden=\"true\" \/>& 2 & -7 & 3 & 0 \\\\<br aria-hidden=\"true\" \/>\\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Since the remainder is 0, [latex]f = 1[\/latex] Hz is a resonant frequency. The Factor Theorem tells us [latex](f - 1)[\/latex] is a factor, so: [latex]D(f) = (f - 1)(2f^2 - 7f + +3).[\/latex]<\/p>\n<p>To find the other resonant frequencies we will factor the quadratic: [latex]D(f) = (f - 1)(2f^2 - 7f + +3)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]2f^2 - 7f + 3 = (2f - 1)(f - 3)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Complete factorization: [latex]D(f) = (f - 1)(2f - 1)(f - 3)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Critical resonant frequencies: 0.5 Hz, 1 Hz, and 3 Hz<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">When a polynomial has integer coefficients, the Rational Zero Theorem narrows your search dramatically. Test the simplest candidates first, often [latex]\\pm 1[\/latex] are zeros.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm317791\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317791&theme=lumen&iframe_resize_id=ohm317791&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How to: Finding All Polynomial Zeros Systematically<\/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\">Apply the Rational Zero Theorem to list possible rational zeros<\/li>\n<li class=\"whitespace-normal break-words\">Use synthetic division to test candidates, starting with simple values like [latex]\\pm 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">When you find a zero, use the Factor Theorem to write the polynomial as a product<\/li>\n<li class=\"whitespace-normal break-words\">Apply the same process to the quotient polynomial (reduced degree)<\/li>\n<li class=\"whitespace-normal break-words\">Continue until you reach a quadratic, then use factoring or the quadratic formula<\/li>\n<li class=\"whitespace-normal break-words\">Verify your factorization by expanding back to the original polynomial<\/li>\n<\/ol>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":21,"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\/3143"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3143\/revisions"}],"predecessor-version":[{"id":5185,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3143\/revisions\/5185"}],"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\/3143\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3143"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3143"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3143"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}