A polynomial with integer coefficients

Setup. $P(p/q)=0$ gives
$$a_n(p/q)^n+a_{n-1}(p/q)^{n-1}+\cdots+a_0=0.$$
Multiply by $q^n$:
$$a_np^n+a_{n-1}p^{n-1}q+\cdots+a_0q^n=0.$$

$p\mid a_0$. Modulo $p$: $a_0q^n\equiv0\pmod p$. Since $\gcd(p,q)=1$, $p\mid a_0$.

$q\mid a_n$. Modulo $q$: $a_np^n\equiv0\pmod q$, so $q\mid a_n$.

Conclusion. The rational root test holds.