IMO 1976 — Shortlist Question 8

(SWE 3) Let $P$ be a polynomial with real coefficients such that $P(x)\gt 0$ if $x\gt 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x)=\frac{Q(x)}{R(x)}$ if $x\gt 0$.