{"id":10796,"date":"2022-05-30T01:59:45","date_gmt":"2022-05-29T20:29:45","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10796"},"modified":"2022-05-30T01:59:49","modified_gmt":"2022-05-29T20:29:49","slug":"on-dividing-x3-3x2-x-2-by-a-polynomial-gx-the-quotient-and-the-remainder-were-x-2-and-2x-4-respectively-find-gx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/on-dividing-x3-3x2-x-2-by-a-polynomial-gx-the-quotient-and-the-remainder-were-x-2-and-2x-4-respectively-find-gx\/","title":{"rendered":"On dividing \\(x^3 &#8211; 3x^2 + x + 2\\) by a polynomial g(x), the quotient and the remainder were x &#8211; 2 and -2x + 4, respectively. Find g(x)."},"content":{"rendered":"<p><strong>Question<\/strong> : On dividing \\(x^3 &#8211; 3x^2 + x + 2\\) by a polynomial g(x), the quotient and the remainder were x &#8211; 2 and -2x + 4, respectively. Find g(x).<\/p>\n<p>p(x) = \\(x^3 &#8211; 3x^2 + x + 2\\)<\/p>\n<p>q(x) = x &#8211; 2 and r(x) = -2x + 4<\/p>\n<p><strong>Solution<\/strong> : By division algorithm, we know that<\/p>\n<p>p(x) = q(x) \\(\\times\\) g(x) + r(x)<\/p>\n<p>Therefore, \\(x^3 &#8211; 3x^2 + x + 2\\) = (x &#8211; 2) \\(\\times\\) g(x) + (-2x + 4)<\/p>\n<p>\\(\\implies\\) \\(x^3 &#8211; 3x^2 + x + 2 + 2x &#8211; 4\\) = (x &#8211; 2) \\(\\times\\) g(x)<\/p>\n<p>\\(\\implies\\)\u00a0 g(x) = \\(x^3 &#8211; 3x^2 + 3x &#8211; 2\\over x &#8211; 2\\)<\/p>\n<p>On dividing\u00a0 \\(x^3 &#8211; 3x^2 + x + 2\\) by x &#8211; 2, we get g(x)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-10800 aligncenter\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/05\/polynomial2-3-4.jpeg?resize=332%2C302&#038;ssl=1\" alt=\"polynomial image\" width=\"332\" height=\"302\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/05\/polynomial2-3-4.jpeg?resize=300%2C273&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/05\/polynomial2-3-4.jpeg?w=523&amp;ssl=1 523w\" sizes=\"(max-width: 332px) 100vw, 332px\" data-recalc-dims=\"1\" \/><\/p>\n<p>Hence, <strong>g(x) = \\(x^2 &#8211; x + 1\\)<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Question : On dividing \\(x^3 &#8211; 3x^2 + x + 2\\) by a polynomial g(x), the quotient and the remainder were x &#8211; 2 and -2x + 4, respectively. Find g(x). p(x) = \\(x^3 &#8211; 3x^2 + x + 2\\) q(x) = x &#8211; 2 and r(x) = -2x + 4 Solution : By division &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/on-dividing-x3-3x2-x-2-by-a-polynomial-gx-the-quotient-and-the-remainder-were-x-2-and-2x-4-respectively-find-gx\/\"> <span class=\"screen-reader-text\">On dividing \\(x^3 &#8211; 3x^2 + x + 2\\) by a polynomial g(x), the quotient and the remainder were x &#8211; 2 and -2x + 4, respectively. Find g(x).<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,910],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>On dividing \\(x^3 - 3x^2 + x + 2\\) by a polynomial g(x), the quotient and the remainder were x - 2 and -2x + 4, respectively. 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