Question

if alpha and beta are zeroes of polynomial 6x2-7x-3 then find a polynomial with zeroes 2alpha and 2 beta

Answer 1

polynomial is 3x^2 -7x-2

Answer 2

$\: \: \: \: 6 {x}^{2} - 7x - 3 \\=6 {x}^{2} + 2x - 9x - 3 \\ = 2x(3x + 1) - 3(x + 1) \\ = (3x + 1)(2x - 3) \\ \\ \\ \alpha = \frac{ - 1}{3}. \\ \beta = \frac{3}{2}.$



So,
Polynomial with Zeroes of 2alpha and 2beta is:-
$new \: \alpha = 2 \alpha = - \frac{2}{3}. \\ new \: \beta = 2 \beta = 3.$


We know that,
$\: \: \: \: \: \: \: {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0 \\ = >new \: polynomial \: is - > \\ \: \: \: \: \: \: \: {x}^{2} - ( \frac{8}{3})x + ( - 2 ) \\ \: \: = {x}^{2} - \frac{8}{3}x - 2.$






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