{"id":222414,"date":"2025-05-31T10:17:14","date_gmt":"2025-05-31T10:17:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222414"},"modified":"2025-05-31T10:17:16","modified_gmt":"2025-05-31T10:17:16","slug":"divide-a-polynomial-ax-x4-7-5x3-14-5x2-3x-20-by-the-monomial-factor-x-2-is-x-2-a-root","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/31\/divide-a-polynomial-ax-x4-7-5x3-14-5x2-3x-20-by-the-monomial-factor-x-2-is-x-2-a-root\/","title":{"rendered":"Divide a polynomial A?(x) = x4 \u2013 7.5×3 + 14.5×2 + 3x \u2013 20 by the monomial factor x \u2013 2. Is x = 2 a root"},"content":{"rendered":"\n

Divide a polynomial A?(x) = x4 \u2013 7.5×3 + 14.5×2 + 3x \u2013 20 by the monomial factor x \u2013 2. Is x = 2 a root?<\/p>\n\n\n\n

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

To divide the polynomial A(x)=x4\u20137.5×3+14.5×2+3x\u201320A(x) = x^4 \u2013 7.5x^3 + 14.5x^2 + 3x \u2013 20<\/p>\n\n\n\n

by the monomial factor x\u22122x – 2, we can use polynomial (or synthetic) division<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Use Synthetic Division<\/strong><\/h3>\n\n\n\n

We divide A(x)A(x) by x\u22122x – 2, so we use 2<\/strong> in synthetic division.<\/p>\n\n\n\n

Write down the coefficients of the polynomial: 1\u22127.514.53\u2212201 \\quad -7.5 \\quad 14.5 \\quad 3 \\quad -20<\/p>\n\n\n\n

Now perform synthetic division:<\/p>\n\n\n\n

   2 |   1   -7.5   14.5    3    -20\n     |        2   -11     7   20\n     ----------------------------\n       1   -5.5    3.5   10     0\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
Result:<\/strong><\/h3>\n\n\n\n
    \n
  • Quotient<\/strong>:<\/li>\n<\/ul>\n\n\n\n
    x3\u22125.5×2+3.5x+10x^3 – 5.5x^2 + 3.5x + 10<\/p>\n\n\n\n
      \n
    • Remainder<\/strong>:<\/li>\n<\/ul>\n\n\n\n
      00<\/p>\n\n\n\n
      Since the remainder is zero<\/strong>, this means that x\u22122x – 2 is a factor<\/strong>, and therefore:<\/p>\n\n\n\n
      \n
      \u2705 x = 2 is a root of the polynomial A(x)<\/strong>.<\/p>\n<\/blockquote>\n\n\n\n
      \n\n\n\n
       Explanation:<\/strong><\/h3>\n\n\n\n
      To determine whether x=2x = 2 is a root of the polynomial A(x)=x4\u22127.5×3+14.5×2+3x\u221220A(x) = x^4 – 7.5x^3 + 14.5x^2 + 3x – 20, we divide the polynomial by x\u22122x – 2. A root of a polynomial is a value of xx for which the polynomial evaluates to zero, meaning A(x)=0A(x) = 0.<\/p>\n\n\n\n
      A practical method for this is synthetic division<\/strong>, which is faster than long division when dividing by a linear binomial like x\u2212cx – c. In this case, we use c=2c = 2 and write the coefficients of the polynomial: 1, -7.5, 14.5, 3, and -20.<\/p>\n\n\n\n
      Applying synthetic division, we compute a new row of values by multiplying the divisor (2) successively and adding to the next coefficient. The process ends with a final value, called the remainder. If the remainder is zero, then x\u22122x – 2 divides evenly into the polynomial, and x=2x = 2 is a root.<\/p>\n\n\n\n
      In this problem, the synthetic division yields a remainder of 0<\/strong>, which confirms that x\u22122x – 2 is a factor of A(x)A(x). Therefore, x=2x = 2 is<\/strong> a root of the polynomial.<\/p>\n\n\n\n
      The quotient from the division, x3\u22125.5×2+3.5x+10x^3 – 5.5x^2 + 3.5x + 10, is the remaining polynomial when x\u22122x – 2 is factored out. This result is useful for further factorization or solving higher-degree polynomials.<\/p>\n\n\n\n
      Thus, we conclude that x=2x = 2 is a valid solution to the equation A(x)=0A(x) = 0.<\/p>\n\n\n\n
      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
      Divide a polynomial A?(x) = x4 \u2013 7.5×3 + 14.5×2 + 3x \u2013 20 by the monomial factor x \u2013 2. Is x = 2 a root? The Correct Answer and Explanation is: To divide the polynomial A(x)=x4\u20137.5×3+14.5×2+3x\u201320A(x) = x^4 \u2013 7.5x^3 + 14.5x^2 + 3x \u2013 20 by the monomial factor x\u22122x – 2, […]<\/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-222414","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222414","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=222414"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222414\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222414"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222414"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}