Let
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
be two nonzero polynomials with integer coefficients and
![\deg f>\deg g](/media/m/6/a/b/6ab90c3fc2a15d4a3036796a9e567a7d.png)
. Suppose that for infinitely many primes
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
the polynomial
![pf+g](/media/m/3/2/8/3282303c2759bb599ddf22b5587193c3.png)
has a rational root. Prove that
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
has a rational root.
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Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.