{"id":257146,"date":"2025-07-16T18:53:43","date_gmt":"2025-07-16T18:53:43","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=257146"},"modified":"2025-07-16T18:53:45","modified_gmt":"2025-07-16T18:53:45","slug":"use-synthetic-division-to-determine-the-multiplicity-of-the-root","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/16\/use-synthetic-division-to-determine-the-multiplicity-of-the-root\/","title":{"rendered":"Use synthetic division to determine the multiplicity of the root."},"content":{"rendered":"\n

The value x=-2 is a zero of the function g(x)=x^(4)+3x^(3)-4x. Use synthetic division to determine the multiplicity of the root. The value x=-2 is a zero of the function g(x)-x^(4)+3x^(3)-4x+1 the multiplicity of the root.<\/p>\n\n\n\n

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

To determine the multiplicity of the root x=\u22122x = -2x=\u22122 for the function g(x)=x4+3×3\u22124x+1g(x) = x^4 + 3x^3 – 4x + 1g(x)=x4+3×3\u22124x+1, we will use synthetic division.<\/p>\n\n\n\n

Step 1: Set up the synthetic division.<\/h3>\n\n\n\n

First, we write the coefficients of the polynomial g(x)=x4+3×3+0x2\u22124x+1g(x) = x^4 + 3x^3 + 0x^2 – 4x + 1g(x)=x4+3×3+0x2\u22124x+1. The coefficient of x2x^2×2 is zero because there is no x2x^2×2 term in the polynomial.<\/p>\n\n\n\n

The coefficients are:
1, 3, 0, -4, 1.<\/p>\n\n\n\n

We are dividing by x+2x + 2x+2 because x=\u22122x = -2x=\u22122 is a root, so the divisor is \u22122-2\u22122.<\/p>\n\n\n\n

Step 2: Perform synthetic division.<\/h3>\n\n\n\n

We use the synthetic division method to divide the polynomial by x+2x + 2x+2:\u22122130\u221241\u22122\u221224011\u2212201\\begin{array}{r|rrrrr} -2 & 1 & 3 & 0 & -4 & 1 \\\\ & & -2 & -2 & 4 & 0 \\\\ \\hline & 1 & 1 & -2 & 0 & 1 \\\\ \\end{array}\u22122\u200b11\u200b3\u221221\u200b0\u22122\u22122\u200b\u2212440\u200b101\u200b\u200b<\/p>\n\n\n\n

    \n
  • Bring down the first coefficient (1).<\/li>\n\n\n\n
  • Multiply 111 by \u22122-2\u22122 and place it under the next coefficient: 1\u00d7(\u22122)=\u221221 \\times (-2) = -21\u00d7(\u22122)=\u22122.<\/li>\n\n\n\n
  • Add 3+(\u22122)=13 + (-2) = 13+(\u22122)=1.<\/li>\n\n\n\n
  • Multiply 111 by \u22122-2\u22122 and place it under the next coefficient: 1\u00d7(\u22122)=\u221221 \\times (-2) = -21\u00d7(\u22122)=\u22122.<\/li>\n\n\n\n
  • Add 0+(\u22122)=\u221220 + (-2) = -20+(\u22122)=\u22122.<\/li>\n\n\n\n
  • Multiply \u22122-2\u22122 by \u22122-2\u22122 and place it under the next coefficient: \u22122\u00d7(\u22122)=4-2 \\times (-2) = 4\u22122\u00d7(\u22122)=4.<\/li>\n\n\n\n
  • Add \u22124+4=0-4 + 4 = 0\u22124+4=0.<\/li>\n\n\n\n
  • Multiply 000 by \u22122-2\u22122 and place it under the next coefficient: 0\u00d7(\u22122)=00 \\times (-2) = 00\u00d7(\u22122)=0.<\/li>\n\n\n\n
  • Add 1+0=11 + 0 = 11+0=1.<\/li>\n<\/ul>\n\n\n\n

    The result of the synthetic division is:x3+x2\u22122x+0+1x+2x^3 + x^2 – 2x + 0 + \\frac{1}{x + 2}x3+x2\u22122x+0+x+21\u200b<\/p>\n\n\n\n

    Step 3: Interpretation of the result.<\/h3>\n\n\n\n

    Since the remainder is 1, x=\u22122x = -2x=\u22122 is not a factor<\/strong> of g(x)g(x)g(x), meaning it is not a root of the function. Thus, the root x=\u22122x = -2x=\u22122 does not have multiplicity in this function.<\/p>\n\n\n\n

    Conclusion:<\/h3>\n\n\n\n

    The value x=\u22122x = -2x=\u22122 is not a zero<\/strong> of g(x)=x4+3×3\u22124x+1g(x) = x^4 + 3x^3 – 4x + 1g(x)=x4+3×3\u22124x+1, and there is no multiplicity associated with it.++-=<\/p>\n\n\n\n

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

    The value x=-2 is a zero of the function g(x)=x^(4)+3x^(3)-4x. Use synthetic division to determine the multiplicity of the root. The value x=-2 is a zero of the function g(x)-x^(4)+3x^(3)-4x+1 the multiplicity of the root. The Correct Answer and Explanation is: To determine the multiplicity of the root x=\u22122x = -2x=\u22122 for the function g(x)=x4+3×3\u22124x+1g(x) […]<\/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-257146","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257146","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=257146"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257146\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257146"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257146"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}