Let P(x) and Q(x) be two polynomials.Suppose that `f(x) = P(x^3) + x Q(x^3)` is divisible by `x^2 + x+1,` then

Options

A. P(x) is divisible by (x-1),but Q(x) is not divisible by x -1
B. Q(x) is divisible by (x-1), but P(x) is not divisible by x-1
C. Both P(x) and Q(x) are divisible by x-1
D. f(x) is divisible by x-1

Answer

C, D

Explanation

We have
`x^(2) + x+1 = (x-omega)(x- omega^(2))`
Simce `f(x)` is divisible by `x^(2) + x + 1,f(omega)=0, f(omega^(2)) = 0` so
`p(omega^(3)) + omegaQ(omega^(3)) = 0 rArr p(1) + omegaQ(1) = 0" "(2)`
`P (omega^(6)) + omega^(2)Q(omega^(6)) = 0 rArr P(1) + omega^(2) Q(1)= 0 " (2)`
Solving Eqs. (1) and (2), we obtain
`P(1) = 0 and Q(1) = 0`
Therefore, both P(x) and Q(x) are divisble by x-1. Hence,
`P(x^(3)) and Q(x^(3))` are divisible by `x^(3) -1` and soby x -1 Since
`f(x) = P(x^(3)) + xQ(X^(3))`, we get f(x) is divisible by x-1