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2x2-5x + 7, find a polynomial whoseIfα and β are the zeros of the quadratic polynomialzeros are 2α + 3β and 3α + 2β21.(x) |
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Answer» α² +β² can be written as (α +β)² - 2αβ p(x) = 2x² - 5x + 7a = 2 , b =- 5 , c = 7 α andβ are the zeros of p(x) we know that ,sum of zeros =α +β = -b/a = 5/2 product of zeros = c/a = 7/2 2α + 3β and 3α + 2β are zeros of a polynomial. sum of zeros = 2α + 3β+ 3α + 2β = 5α + 5β = 5 [α +β] = 5× 5/2 = 25/2 product of zeros = (2α + 3β)(3α + 2β) = 2α [3α + 2β] + 3β [3α + 2β] = 6α² + 4αβ + 9αβ + 6β² = 6α² + 13αβ +6β² = 6 [α² +β² ] + 13αβ = 6 [ (α +β)² - 2αβ ] + 13αβ = 6 [ ( 5/2)² - 2× 7/2 ] + 13× 7/2 = 6 [ 25/4 - 7 ] + 91/2 = 6 [ 25/4 - 28/4 ] + 91/2 = 6 [ -3/4 ] + 91/2 = -18/4 + 91/2 = -9/2 + 91/2 = 82/2 = 41 -18/4 = -9/2 [ simplest form ] a quadratic polynomial is given by:- k { x² - (sum of zeros)x + (product of zeros) } k {x² - 5/2x + 41} k = 2 2 {x² - 5/2x + 41 ] 2x² - 5x + 82 -----> is the required polynomial Like my answer if you find it useful! |
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