Question

if alpha, beta and gamma are the zeroes of polynomial 6x^3 + 3x^2 - 5x + 1 then find the value of Alpha^-1 + beta^-1 + Gamma^-1

Answer 1

We have α , β And γ are the zeros of cubic polynomial 6x3 + 3x2 - 5x + 1
We know by the relationship between zeros and coefficient of a cubic polynomial , is
Sum of zeros = −Coefficient of x2coefficient of x3

α + β + γ = −36

α + β + γ = −12 ------------------- ( 1 )

Sum of the products of zeros taken two at a time = Coefficient of xcoefficient of x3
α β + β ​γ + γ α = −56 --------------------- ( 2 )

Product of zeros = −Constant termcoefficient of x3
α β γ = −16 --------------------- ( 3 )

Now for value of

α-1 + β-1 + γ-1

We can simplify it As :

⇒1α+ 1β+1γ
Taking L.C.M. and get

⇒αβ + βγ + γαα β γ

Now substitute values from equation 2 and 3 we get

⇒−56−16

⇒ α-1 + β-1 + γ-1 = 5


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