KöMaL 2017 A687

Let $f(x)$ and $g(x)$ be nonzero polynomials such that the degree of $f(x)$ is higher than that of $g(x)$. Suppose that for infinitely many prime numbers $p$, the polynomial $pf(x)+g(x)$ also has a rational root. Show that $f(x)$ has a rational root.