{"id":10511,"date":"2022-04-23T19:57:59","date_gmt":"2022-04-23T14:27:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10511"},"modified":"2022-04-23T19:58:56","modified_gmt":"2022-04-23T14:28:56","slug":"find-a-quadratic-polynomial-whose-sum-of-zeroes-and-product-of-zeroes-are-respectively","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/find-a-quadratic-polynomial-whose-sum-of-zeroes-and-product-of-zeroes-are-respectively\/","title":{"rendered":"Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively."},"content":{"rendered":"<p><strong>Question<\/strong> : Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively.<\/p>\n<p><strong>(i)<\/strong>\u00a0 \\(1\\over 4\\), -1<\/p>\n<p><strong>(ii)<\/strong>\u00a0 \\(\\sqrt{2}\\), \\(1\\over 3\\)<\/p>\n<p><strong>(iii)<\/strong>\u00a0 0, \\(\\sqrt{5}\\)<\/p>\n<p><strong>(iv)<\/strong>\u00a0 1, 1<\/p>\n<p><strong>(v)\u00a0<\/strong> \\(-1\\over 4\\), \\(1\\over 4\\)<\/p>\n<p><strong>(vi)<\/strong>\u00a0 4, 1<\/p>\n<p><strong>Solution<\/strong> : Let the polynomial be \\(ax^2 + bx + c\\) and its zeroes be \\(\\alpha\\) and \\(\\beta\\).<\/p>\n<p><strong>(i)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = \\(1\\over 4\\)\u00a0 and \\(\\alpha\\).\\(\\beta\\) = -1<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (\\(1\\over 4\\))x &#8211; 1<\/p>\n<p>= \\(4x^2 &#8211; x &#8211; 4\\)<\/p>\n<p><strong>(ii)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = \\(\\sqrt{2}\\) \u00a0and \\(\\alpha\\).\\(\\beta\\) = \\(1\\over 3\\)<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (\\(\\sqrt{2}\\))x + \\(1\\over 3\\)<\/p>\n<p>= \\(3x^2 &#8211; 3\\sqrt{2}x + 1\\)\u00a0<\/p>\n<p><strong>(iii)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = 0\u00a0 and \\(\\alpha\\).\\(\\beta\\) = \\(\\sqrt{5}\\)\u00a0<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (0)x + \\(\\sqrt{5}\\)\u00a0<\/p>\n<p>= \\(x^2 + \\sqrt{5}\\)<\/p>\n<p><strong>(iv)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = 1\u00a0 and \\(\\alpha\\).\\(\\beta\\) = 1<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (1)x + 1<\/p>\n<p>= \\(x^2 &#8211; x + 1\\)<\/p>\n<p><strong>(v)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = \\(-1\\over 4\\)\u00a0 and \\(\\alpha\\).\\(\\beta\\) = \\(1\\over 4\\)\u00a0<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (\\(-1\\over 4\\))x + \\(1\\over 4\\)\u00a0<\/p>\n<p>= \\(4x^2 + x + 1\\)\u00a0<\/p>\n<p><strong>(vi)<\/strong>\u00a0 Here, \\(\\alpha\\) + \\(\\beta\\) = 4\u00a0 and \\(\\alpha\\).\\(\\beta\\) = 1<\/p>\n<p>Thus, the polynomial formed = \\(x^2\\) &#8211; (sum of zeroes) x + Product of zeroes<\/p>\n<p>= \\(x^2\\) &#8211; (4)x + 1<\/p>\n<p>= \\(x^2 &#8211; 4x + 1\\)<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Question : Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. (i)\u00a0 \\(1\\over 4\\), -1 (ii)\u00a0 \\(\\sqrt{2}\\), \\(1\\over 3\\) (iii)\u00a0 0, \\(\\sqrt{5}\\) (iv)\u00a0 1, 1 (v)\u00a0 \\(-1\\over 4\\), \\(1\\over 4\\) (vi)\u00a0 4, 1 Solution : Let the polynomial be \\(ax^2 + bx + c\\) and its zeroes be \\(\\alpha\\) and &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/find-a-quadratic-polynomial-whose-sum-of-zeroes-and-product-of-zeroes-are-respectively\/\"> <span class=\"screen-reader-text\">Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively.<\/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>Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/find-a-quadratic-polynomial-whose-sum-of-zeroes-and-product-of-zeroes-are-respectively\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. 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