Q: If f(x) and g(x) are two polynomials such that the polynomial $h(x) = x f(x^3) + x^2 g(x^6)$ is divisible by x^{2} +x +1 , then

(A) f(1) = g(1)

(B) f(1) = – g( 1)

(C) f(1) = g(1) ≠ 0

(D) f(1) = -g(1) ≠ 0

Sol. Roots of x^{2} + x +1 = 0 are complex cube roots of unity,

so h(w) = h(w^{2}) =0

⇒ w f(1) + w^{2} g(1) = 0 and w^{2} f(1) + wg(1) = 0

⇒ f(1) = g(1) = 0.

Hence (A) and (B) are the correct answers.