{"id":191492,"date":"2025-02-15T07:16:16","date_gmt":"2025-02-15T07:16:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191492"},"modified":"2025-02-15T07:16:19","modified_gmt":"2025-02-15T07:16:19","slug":"form-a-polynomial-whose-zeros-and-degree-are-given","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/15\/form-a-polynomial-whose-zeros-and-degree-are-given\/","title":{"rendered":"Form a polynomial whose zeros and degree are given"},"content":{"rendered":"\n
    \n
  1. Form a polynomial whose zeros and degree are given.
    Zeros: -3,3.4; degree: 3
    Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below.
    f(x) (Simplify your answer.)<\/li>\n\n\n\n
  2. Form a polynomial whose real zeros and degree are given. Zeros: -2,0,6; degree: 3
    Type a polynomial with integer coefficients and a leading coefficient of 1.
    f(x)= (Simplify your answer.)<\/li>\n<\/ol>\n\n\n\n

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

    Solution to Problem 1: Zeros: -3, 3.4; Degree: 3<\/h3>\n\n\n\n

    To form the polynomial with integer coefficients and a leading coefficient of 1, we follow these steps:<\/p>\n\n\n\n

      \n
    1. Zeros Given<\/strong>: -3 and 3.4<\/li>\n<\/ol>\n\n\n\n
        \n
      • The polynomial must have these values as its zeros.<\/li>\n<\/ul>\n\n\n\n
          \n
        1. Degree<\/strong>: 3<\/li>\n<\/ol>\n\n\n\n
            \n
          • The degree of the polynomial is 3, meaning there will be three factors in the polynomial.<\/li>\n\n\n\n
          • For the zero -3, the factor will be:
            [
            (x + 3)
            ]<\/li>\n\n\n\n
          • For the zero 3.4, the factor will be:
            [
            (x – 3.4)
            ]
            However, since we need integer coefficients, we’ll multiply this factor by 10 to eliminate the decimal, giving us:
            [
            10(x – 3.4) = 10(x – \\frac{17}{5}) = 5x – 17
            ]
            So, the factor corresponding to the zero 3.4 becomes (5x – 17).<\/li>\n<\/ul>\n\n\n\n
              \n
            1. Third Zero<\/strong>: We need one more zero to make the degree 3. The simplest choice is (x = 0), corresponding to the factor:
              [
              x
              ]<\/li>\n\n\n\n
            2. Final Polynomial<\/strong>: Now we combine all the factors:
              [
              f(x) = (x + 3)(x)(5x – 17)
              ]
              Multiply these together:
              [
              f(x) = x(x + 3)(5x – 17) = x(5x^2 + 15x – 17x – 51) = x(5x^2 – 2x – 51)
              ]
              Expanding further:
              [
              f(x) = 5x^3 – 2x^2 – 51x
              ]<\/li>\n<\/ol>\n\n\n\n

              Thus, the polynomial is:
              [
              f(x) = 5x^3 – 2x^2 – 51x
              ]<\/p>\n\n\n\n

              Solution to Problem 2: Zeros: -2, 0, 6; Degree: 3<\/h3>\n\n\n\n

              To form the polynomial with integer coefficients and a leading coefficient of 1, we follow these steps:<\/p>\n\n\n\n

                \n
              1. Zeros Given<\/strong>: -2, 0, 6<\/li>\n<\/ol>\n\n\n\n
                  \n
                • The polynomial must have these values as its zeros.<\/li>\n<\/ul>\n\n\n\n
                    \n
                  1. Degree<\/strong>: 3<\/li>\n<\/ol>\n\n\n\n
                      \n
                    • The degree of the polynomial is 3, so we expect three factors in the polynomial.<\/li>\n<\/ul>\n\n\n\n
                        \n
                      1. Factors<\/strong>:<\/li>\n<\/ol>\n\n\n\n
                          \n
                        • For the zero -2, the factor will be:
                          [
                          (x + 2)
                          ]<\/li>\n\n\n\n
                        • For the zero 0, the factor will be:
                          [
                          x
                          ]<\/li>\n\n\n\n
                        • For the zero 6, the factor will be:
                          [
                          (x – 6)
                          ]<\/li>\n<\/ul>\n\n\n\n
                            \n
                          1. Final Polynomial<\/strong>: Now we multiply all these factors:
                            [
                            f(x) = (x + 2)(x)(x – 6)
                            ]
                            First, multiply the first two factors:
                            [
                            (x + 2)(x) = x^2 + 2x
                            ]
                            Now, multiply this by the third factor:
                            [
                            f(x) = (x^2 + 2x)(x – 6) = x^3 – 6x^2 + 2x^2 – 12x = x^3 – 4x^2 – 12x
                            ]<\/li>\n<\/ol>\n\n\n\n

                            Thus, the polynomial is:
                            [
                            f(x) = x^3 – 4x^2 – 12x
                            ]<\/p>\n\n\n\n

                            Summary:<\/h3>\n\n\n\n
                              \n
                            1. For the zeros -3, 3.4, and degree 3, the polynomial is:
                              [
                              f(x) = 5x^3 – 2x^2 – 51x
                              ]<\/li>\n\n\n\n
                            2. For the zeros -2, 0, 6, and degree 3, the polynomial is:
                              [
                              f(x) = x^3 – 4x^2 – 12x
                              ]<\/li>\n<\/ol>\n\n\n\n

                              Now, let’s generate the image for these polynomials.<\/p>\n\n\n\n

                              Here are the graphs of the two polynomials:<\/p>\n\n\n\n

                                \n
                              1. The first polynomial, ( f(x) = 5x^3 – 2x^2 – 51x ), has zeros at -3, 3.4, and a degree of 3.<\/li>\n\n\n\n
                              2. The second polynomial, ( f(x) = x^3 – 4x^2 – 12x ), has zeros at -2, 0, and 6.<\/li>\n<\/ol>\n\n\n\n

                                These graphs illustrate the behavior of the functions based on their respective zeros. Let me know if you need further clarifications!<\/p>\n\n\n\n

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

                                The Correct Answer and Explanation is : Solution to Problem 1: Zeros: -3, 3.4; Degree: 3 To form the polynomial with integer coefficients and a leading coefficient of 1, we follow these steps: Thus, the polynomial is:[f(x) = 5x^3 – 2x^2 – 51x] Solution to Problem 2: Zeros: -2, 0, 6; Degree: 3 To form […]<\/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-191492","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191492","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=191492"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191492\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191492"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191492"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}