If alpha and beta are zeroes of a quadratic polynomial Fx = 2x square -5x 7. Find a polynomial who's zeroes are 2 alpha 3 beta and 3 alpha 2 beta?

Question

If alpha and beta are zeroes of a quadratic polynomial Fx = 2x square -5x 7. Find a polynomial who's zeroes are 2 alpha   3 beta and 3 alpha   2 beta?

Answer

Answer

Problem:

If alpha and beta are zeroes of a quadratic polynomial Fx = 2x square -5x 7. Find a polynomial who's zeroes are 2 alpha 3 beta and 3 alpha 2 beta? Explain in details.

Solution:

Given quadratic polynomial is Fx = 2x^2 - 5x + 7. Let alpha and beta be zeroes of Fx.

Sum and Product of roots

Sum of roots = alpha + beta = (-b/a) = -(-5)/2 = 5/2

Product of roots = alpha * beta = c/a = 7/2

Polynomial with roots 2alpha and 3beta

Let Gx be a polynomial whose zeroes are 2alpha and 3beta.

Then, Gx = (x - 2alpha) (x - 3beta)

Expanding, Gx = x^2 - (2alpha + 3beta) x + 6alpha beta

Substituting the values of alpha and beta, we get:

Gx = x^2 - 5x + 21/2

Polynomial with roots 3alpha and 2beta

Let Hx be a polynomial whose zeroes are 3alpha and 2beta.

Then, Hx = (x - 3alpha) (x - 2beta)

Expanding, Hx = x^2 - (3alpha + 2beta) x + 6alpha beta

Substituting the values of alpha and beta, we get:

Hx = x^2 - 13x/2 + 21/2

Final polynomial

Let Jx be the polynomial whose zeroes are 2alpha, 3beta, 3alpha and 2beta.

Then, Jx = Gx * Hx

Expanding, Jx = (x^2 - 5x + 21/2) (x^2 - 13x/2 + 21/2)

On simplifying, we get:

Jx = 2x^4 - 18x^3 + 61x^2 - 87x + 220/4

Jx = x^4 - 9x^3 + 61/2 x^2 - 87/2 x + 55/2

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