Question

If P(x) and Q(x) are two polynomials such that f(x) = P(x³) + x Q(x³ ) is divisible by x² + x + 1, then

(1) P(x) is divisible by (x-1) but Q(x) is notdivisible by (x-1) 

(2)Q(x) is divisible by (x-1) but P(x) is notdivisible by (x-1) 

(3) Both P(x) and Q(x) are divisible by (x-1) 

(4)f(x) is divisible by (x-1)

Answer 1

we have x2 + x + 1 = (x - ω)(x - ω²)

Since f(x) is divisible by x² + x + 1

P(ω³) + ωQ(ω³) = 0 ⇒   P(1) + ωQ(1) = 0.............(1)

P(ω⁶) + ω²Q(ω⁶) = 0 ⇒  P(1) + ω²Q(1) = 0................(2)

From (1) and (2), we have 

P(1) = 0 and Q(1) = 0

Both P(x) and Q(x) are divisible by x-1 

Since f(x) is divisible by (x-1)