Question

if alpha and beta are the zeros of the polynomial 3x square + 5 x minus 2 then form a quadratic polynomial whose zeros are 2 alpha and 2 Beta​

Answer 1

This is the answer of the question

Answer 2

The required polynomial is

$q(x) = x^2 + \dfrac{10}{3}x-\dfrac{8}{3}$

Step-by-step explanation:

We are given the following in the question:

$p(x) = 3x^2+5x-2$

To find the zeroes of the polynomial:

$p(x) = 3x^2+5x-2 = 0\\3x^2 + 6x-x-2 = 0\\3x(x+2)-1(x+2) = 0\\(3x-1)(x+2) = 0\\x = -2, x = \dfrac{1}{3}\\\alpha = -2\\\beta = \dfrac{1}{3}$

New roots of polynomial are:

$\alpha' = 2\alpha = -4\\\beta' = 2\beta = \dfrac{2}{3}$

Sum of roots:

$\alpha' + \beta' = -4 + \dfrac{2}{3} = -\dfrac{10}{3}$

Product of roots:

$\alpha'\beta' = -4\times \dfrac{2}{3} = -\dfrac{8}{3}$

New Polynomial:

$q(x) = x^2 - (\alpha' + \beta')x + \apha'\beta'\\\\q(x) = x^2 + \dfrac{10}{3}x-\dfrac{8}{3}$

is the required polynomial.

#LearnMore

If alpha,beta are the zeroes of polynamial 6x^2-5x+7, then find the polynomial whose zeroes are 2alpha+3beta, 3alpha+2beta

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