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Algebra Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

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.

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