{"id":248140,"date":"2025-07-08T07:04:00","date_gmt":"2025-07-08T07:04:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248140"},"modified":"2025-07-08T07:04:02","modified_gmt":"2025-07-08T07:04:02","slug":"form-polynomial-whose-real-zeros-and-degree-are-given","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/form-polynomial-whose-real-zeros-and-degree-are-given\/","title":{"rendered":"Form polynomial whose real zeros and degree are given."},"content":{"rendered":"\n

Form polynomial whose real zeros and degree are given. Zeros: 4,0, 9; degree: 3 Type a polynomial with integer coefficients and leading coefficient of 1. flx) = (Simplify your answer:<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To form a polynomial given the real zeros and the degree, you can use the factor theorem<\/strong>, which states that if rrr is a zero of a polynomial, then x\u2212rx – rx\u2212r is a factor of the polynomial.<\/p>\n\n\n\n

Given:<\/h3>\n\n\n\n
    \n
  • Zeros: 4, 0, and 9<\/li>\n\n\n\n
  • Degree: 3 (since there are 3 zeros)<\/li>\n<\/ul>\n\n\n\n

    We know that the polynomial will have the following factors:<\/p>\n\n\n\n

      \n
    • x\u22124x – 4x\u22124 for zero 4,<\/li>\n\n\n\n
    • x\u22120=xx – 0 = xx\u22120=x for zero 0,<\/li>\n\n\n\n
    • x\u22129x – 9x\u22129 for zero 9.<\/li>\n<\/ul>\n\n\n\n

      Thus, the polynomial can be written as:f(x)=(x\u22124)(x)(x\u22129)f(x) = (x – 4)(x)(x – 9)f(x)=(x\u22124)(x)(x\u22129)<\/p>\n\n\n\n

      Step-by-Step Expansion:<\/h3>\n\n\n\n
        \n
      1. First, expand (x\u22124)(x)(x – 4)(x)(x\u22124)(x):<\/li>\n<\/ol>\n\n\n\n

        (x\u22124)(x)=x2\u22124x(x – 4)(x) = x^2 – 4x(x\u22124)(x)=x2\u22124x<\/p>\n\n\n\n

          \n
        1. Now, multiply the result by (x\u22129)(x – 9)(x\u22129):<\/li>\n<\/ol>\n\n\n\n

          (x2\u22124x)(x\u22129)(x^2 – 4x)(x – 9)(x2\u22124x)(x\u22129)<\/p>\n\n\n\n

          Use distributive property (also known as FOIL for binomials):x2(x\u22129)=x3\u22129x2x^2(x – 9) = x^3 – 9x^2×2(x\u22129)=x3\u22129×2\u22124x(x\u22129)=\u22124×2+36x-4x(x – 9) = -4x^2 + 36x\u22124x(x\u22129)=\u22124×2+36x<\/p>\n\n\n\n

            \n
          1. Combine all terms:<\/li>\n<\/ol>\n\n\n\n

            x3\u22129×2\u22124×2+36xx^3 – 9x^2 – 4x^2 + 36xx3\u22129×2\u22124×2+36x=x3\u221213×2+36x= x^3 – 13x^2 + 36x=x3\u221213×2+36x<\/p>\n\n\n\n

            Thus, the polynomial is:f(x)=x3\u221213×2+36xf(x) = x^3 – 13x^2 + 36xf(x)=x3\u221213×2+36x<\/p>\n\n\n\n

            Explanation:<\/h3>\n\n\n\n
              \n
            • The degree of the polynomial is 3, which matches the number of zeros.<\/li>\n\n\n\n
            • The leading coefficient is 1, as required (the coefficient of x3x^3×3 is 1).<\/li>\n\n\n\n
            • The polynomial has integer coefficients, and the factorization follows directly from the given zeros.<\/li>\n<\/ul>\n\n\n\n

              This polynomial satisfies the conditions: real zeros at 4, 0, and 9, and a degree of 3.<\/p>\n\n\n\n

              \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

              Form polynomial whose real zeros and degree are given. Zeros: 4,0, 9; degree: 3 Type a polynomial with integer coefficients and leading coefficient of 1. flx) = (Simplify your answer: The Correct Answer and Explanation is: To form a polynomial given the real zeros and the degree, you can use the factor theorem, which states […]<\/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-248140","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248140","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=248140"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248140\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248140"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248140"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}