MathNet 2023-part-06 — 191 problems (EN–VI) Question 136

Let $n$ be a positive integer and $P, Q$ be polynomials with real coefficients such that $P(x) = x^n Q(1/x)$ and $P(x) \ge Q(x)$ for all real numbers $x$. Prove that $P(x) = Q(x)$ for all real numbers $x$.