Prove that the Descartes's Criterion is correct. Descartes's criterion: If $a_nx^n + a_{n-1}x^{n-1}+...+a_0$ has a rational root $x = s/t$, where $s$ and $t$ are relatively prime, then t divides $a_n$ and $s$ divides $a_0$.
The hint says that I should factor $a_n$ and $a_0$ to get all possible values of $s/t$ and substitute to find which,if any, is a root. But I think that this hint only works for specific polynomials, not to prove the general statement.