Question

Let P(x) = x² + 1/2x + b and Q(x) = x² + cx + d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) = 0.

Answer 1

P(x) Q (x) = Q (P(x))

P(x) Q (x) = (P(x))² + cP(x) + d

P(x) (Q (x) – P(x)) = cP(x) + d