If `alpha, beta, gamma` the roots of the equation `x^(3)+ 4x 1=0,` then find the value of `(alpha + beta)^(-1)+(beta + gamma)^(-1) + (gamma + alpha)^(-1).`

Options

A. 2
B. 3
C. 4
D. 5

Answer

C

Explanation

We have, `alpha + beta + gamma = 0, alpha beta + beta gamma + gamma alpha = 4 and alpha beta gamma = -1`
`therefore" "(alpha + beta)^(-1) + (beta+gamma)^(-1) +(gamma + alpha)^(-1)`
`=(1)/(alpha+beta)+(1)/(beta+gamma)+(1)/(gamma+alpha)`
`=-(1)/(gamma)-(1)/(alpha)-(1)/(beta)=-((alpha beta + beta gamma + gamma alpha)/(alpha beta gamma))=(-4)/(-1)=4`