{"id":10791,"date":"2022-05-30T01:41:32","date_gmt":"2022-05-29T20:11:32","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10791"},"modified":"2022-05-30T01:56:54","modified_gmt":"2022-05-29T20:26:54","slug":"obtain-all-the-zeroes-of-3x4-6x3-2x2-10x-5-if-two-of-its-zeroes-are-sqrt5over-3-and-sqrt5over-3","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/obtain-all-the-zeroes-of-3x4-6x3-2x2-10x-5-if-two-of-its-zeroes-are-sqrt5over-3-and-sqrt5over-3\/","title":{"rendered":"Obtain all the zeroes of \\(3x^4 + 6x^3 – 2x^2 – 10x – 5\\), if two of its zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\)."},"content":{"rendered":"

Solution :<\/h2>\n

Since two zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\),<\/p>\n

x = \\(\\sqrt{5\\over 3}\\) and x = -\\(\\sqrt{5\\over 3}\\)<\/p>\n

\\(\\implies\\) (x – \\(\\sqrt{5\\over 3}\\))(x + \\(\\sqrt{5\\over 3}\\)) = \\(3x^2 – 5\\) is a factor of the given polynomial. Now, we apply the division algorithm to the given polynomial and \\(3x^2 – 5\\).<\/p>\n

\"polynomial<\/p>\n

First term of quotient is \\(3x^4\\over 3x^2\\) = \\(x^2\\)<\/p>\n

Second term of quotient is \\(6x^3\\over 3x^2\\) = 2x<\/p>\n

Third term of the quotient is \\(3x^2\\over 3x^2\\) = 1<\/p>\n

So, \\(3x^4 + 6x^3 – 2x^2 – 10x – 5\\) = (\\(3x^2 – 5\\))(\\(x^2 + 2x + 1\\)) + 0 = (\\(3x^2 – 5\\))\\({(x + 1)}^2\\)<\/p>\n

Quotient = \\(x^2 + 2x + 1\\) = \\({(x + 1)}^2\\)<\/p>\n

Zeroes of \\({(x + 1)}^2\\) are -1 and -1.<\/p>\n

Hence, all its zeroes are \\(\\sqrt{5\\over 3}\\), -\\(\\sqrt{5\\over 3}\\), -1, -1.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : Since two zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\), x = \\(\\sqrt{5\\over 3}\\) and x = -\\(\\sqrt{5\\over 3}\\) \\(\\implies\\) (x – \\(\\sqrt{5\\over 3}\\))(x + \\(\\sqrt{5\\over 3}\\)) = \\(3x^2 – 5\\) is a factor of the given polynomial. Now, we apply the division algorithm to the given polynomial and \\(3x^2 – 5\\). First term …<\/p>\n

Obtain all the zeroes of \\(3x^4 + 6x^3 – 2x^2 – 10x – 5\\), if two of its zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\).<\/span> Read More »<\/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":"\nObtain all the zeroes of \\(3x^4 + 6x^3 - 2x^2 - 10x - 5\\), if two of its zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\). - 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\/obtain-all-the-zeroes-of-3x4-6x3-2x2-10x-5-if-two-of-its-zeroes-are-sqrt5over-3-and-sqrt5over-3\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Obtain all the zeroes of \\(3x^4 + 6x^3 - 2x^2 - 10x - 5\\), if two of its zeroes are \\(\\sqrt{5\\over 3}\\) and -\\(\\sqrt{5\\over 3}\\). - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Since two zeroes are (sqrt{5over 3}) and -(sqrt{5over 3}), x = (sqrt{5over 3}) and x = -(sqrt{5over 3}) (implies) (x – (sqrt{5over 3}))(x + (sqrt{5over 3})) = (3x^2 – 5) is a factor of the given polynomial. Now, we apply the division algorithm to the given polynomial and (3x^2 – 5). First term … Obtain all the zeroes of (3x^4 + 6x^3 – 2x^2 – 10x – 5), if two of its zeroes are (sqrt{5over 3}) and -(sqrt{5over 3}). 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