if alpha beta are the zeros of the polynomial x2

Question

If α, β are the zeros of the polynomial x² +8x +6 frame a Quadratic polynomial whose zeros are a)  1/α and  1/β b) 1+ β/α , 1+ α/β.

Ans.

P(x) = x² +8x +6

α + β = -8 and αβ = 6

a) Let two zeros are 1/α and 1/β

Sum = 1/α + 1/ β = α + β/α.β = -8/6

Product = 1/α × 1/β = α + β/ α.β = -8 /6

Required Q.P is

X² +4/3x +1/6

Sum = 1+ β/α +1+α/β

= 2 +α /β + β /α

= 2+ α² +β²/αβ

= 2+ (α +β)² -2αβ/αβ after solving this problem

We get = 32/3

Product = (1+β/α) (1+α/β)

= 1+ α/β +β/α +1

= 2+ α² + β²/αβ

Substitute this sum

We get = 32/3

Required Q.P is x2 -32/3 x +32/3