Question

if alpha and beta are zeros of the quadratic polynomial f(x)=x^2-3x-2,find a polynomial whose zeroes are 1)2alpha+3beta and 3alpha+2 beta 2)alpha^2/beta and beta^2/alpha​

Answer 1

Answer:

(a)x²-15x+22 (b)x²-16x-2

Step-by-step explanation:

f(x)=x^2-3x-2,

αβ=-2,α+β=3

(1) The polynomial

=x²-(2α+3β+3α+2β)x+(2α+3β)(3α+2β)

=x²-5(α+β)x+6α²+4αβ+6αβ+6β²

=x²-5(α+β)x+6α²+10αβ+6β²

=x²-5*3x+6(α²+β²+2αβ)-2αβ

=x²-15x+6(α+β)²-2αβ

=x²-15x+6*3-2*-2

=x²-15x+18+4

=x²-15x+22

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(2) Zeros are α²/β and β²/α

g(x)=x²-(α²/β+ β²/α)x + α²/β* β²/α

x²-1/αβ(α³+ β³)x+αβ

=x²-1/αβ[ (α+β)(α²+β²-αβ)]x+αβ

=x²-1/αβ[(α+β){ (α+β)²-3αβ} ]+αβ

=x²-1/-2 [-2*(3²-3*(-2)]x+(-2)

=x²+1/2 [-2*(9+7)]x-2

=x²-16x-2