{"id":248144,"date":"2025-07-08T07:05:44","date_gmt":"2025-07-08T07:05:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248144"},"modified":"2025-07-08T07:05:46","modified_gmt":"2025-07-08T07:05:46","slug":"form-a-polynomial-whose-real-zeros-and-degree-are-given","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/form-a-polynomial-whose-real-zeros-and-degree-are-given\/","title":{"rendered":"Form a polynomial whose real zeros and degree are given"},"content":{"rendered":"\n<p>Form a polynomial whose real zeros and degree are given. Zeros: -3, 0, 8; degree: 3 Type a polynomial with integer coefficients and a leading coefficient of 1. f(x) = \\boxed{} (Simplify your answer.)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To form a polynomial given the real zeros and degree, we use the fact that a polynomial can be factored based on its zeros. The degree of the polynomial tells us how many factors the polynomial will have.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Write the factors based on the zeros<\/h3>\n\n\n\n<p>The zeros given are: -3, 0, and 8. Each zero corresponds to a factor in the polynomial:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A zero of \u22123-3\u22123 corresponds to the factor (x+3)(x + 3)(x+3).<\/li>\n\n\n\n<li>A zero of 000 corresponds to the factor xxx.<\/li>\n\n\n\n<li>A zero of 888 corresponds to the factor (x\u22128)(x &#8211; 8)(x\u22128).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Multiply the factors to form the polynomial<\/h3>\n\n\n\n<p>Now, we multiply the factors to get the polynomial:f(x)=(x+3)(x)(x\u22128)f(x) = (x + 3)(x)(x &#8211; 8)f(x)=(x+3)(x)(x\u22128)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Expand the expression<\/h3>\n\n\n\n<p>We now expand the factors step by step.<\/p>\n\n\n\n<p>First, multiply (x+3)(x + 3)(x+3) and xxx:(x+3)(x)=x2+3x(x + 3)(x) = x^2 + 3x(x+3)(x)=x2+3x<\/p>\n\n\n\n<p>Now, multiply (x2+3x)(x^2 + 3x)(x2+3x) by (x\u22128)(x &#8211; 8)(x\u22128):(x2+3x)(x\u22128)=x2(x\u22128)+3x(x\u22128)(x^2 + 3x)(x &#8211; 8) = x^2(x &#8211; 8) + 3x(x &#8211; 8)(x2+3x)(x\u22128)=x2(x\u22128)+3x(x\u22128)<\/p>\n\n\n\n<p>Distribute each term:x2(x\u22128)=x3\u22128x2x^2(x &#8211; 8) = x^3 &#8211; 8x^2&#215;2(x\u22128)=x3\u22128x23x(x\u22128)=3&#215;2\u221224x3x(x &#8211; 8) = 3x^2 &#8211; 24x3x(x\u22128)=3&#215;2\u221224x<\/p>\n\n\n\n<p>Now, combine all the terms:f(x)=x3\u22128&#215;2+3&#215;2\u221224xf(x) = x^3 &#8211; 8x^2 + 3x^2 &#8211; 24xf(x)=x3\u22128&#215;2+3&#215;2\u221224x<\/p>\n\n\n\n<p>Combine like terms:f(x)=x3\u22125&#215;2\u221224xf(x) = x^3 &#8211; 5x^2 &#8211; 24xf(x)=x3\u22125&#215;2\u221224x<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Final result<\/h3>\n\n\n\n<p>Thus, the polynomial is:f(x)=x3\u22125&#215;2\u221224xf(x) = x^3 &#8211; 5x^2 &#8211; 24xf(x)=x3\u22125&#215;2\u221224x<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The degree of the polynomial is 3, which matches the number of given zeros, and the leading coefficient is 1, as required.<\/li>\n\n\n\n<li>We started with the factored form using the given zeros and then expanded the expression to obtain the polynomial in standard form.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-160.jpeg\" alt=\"\" class=\"wp-image-248145\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Form a polynomial whose real zeros and degree are given. Zeros: -3, 0, 8; degree: 3 Type a polynomial with integer coefficients and a leading coefficient of 1. f(x) = \\boxed{} (Simplify your answer.) The Correct Answer and Explanation is: To form a polynomial given the real zeros and degree, we use the fact that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-248144","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=248144"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248144\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248144"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248144"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}