Consider the polynomial

where all the coefficients $a_{i}$ are integers.

If $p(x)$ has a rational zero $u/v$ where  $\gcd(u,\,v)=1$,  then  $u\mid a_{0}$  and  $v\mid a_{n}$.  Thus, for finding all rational zeros of $p(x)$, it suffices to perform a finite number of tests.

The theorem is related to the result about monic polynomials whose coefficients belong to a unique factorization domain. Such theorem then states that any root in the fraction field is also in the base domain.