If `alpha,beta,gamma,sigma` are the roots of the equation `x^4+4x^3-6x^2+7x-9=0,` then the value of `(1+alpha^2)(1+beta^2)(1+gamma^2)(1+sigma^2)` is a. `9` b. `11` c. `13` d. `5`

Options

A. 9
B. 11
C. 13
D. 5

Answer

3

Explanation

Since `alpha, beta, gamma, sigma` are the roots of the given equation, we have
`x^(4) + 4x^(3) - 6x^(2) + 7x - 9 = (x - alpha)(x - beta)(x - gamma)(x - sigma)`
Putting x = I and then x = -I, we get
`1 - 4i + 6 + 7i - 9 = (I - alpha) (I - beta)(i - gamma) (i - sigma)`
and `1 + 4i + 6 - 7i - 9 = (-i - alpha)(-i -beta)(-i -gamma)(-i - sigma)`
Multiplying these two equations, we get
`(-2 + 3i)(-2 - 3i) = (1 + alpha^(2))(1 + beta^(2))(1 + gamma^(2))(1 + sigma^(2))`
or `13 = (1 + alpha^(2))(1 + beta^(2))(1 + gamma^(2))(1 + sigma^(2))`