If α, β are the zeros of the polynomial x^{2} +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) = x^{2} +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
X^{2 }+4/3x +1/6
Sum = 1+ β/α +1+α/β
= 2 +α /β + β /α
= 2+ α^{2} +β^{2}/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+ α^{2} + β^{2}/αβ
Substitute this sum
Required Q.P is x2 -32/3 x +32/3