Methods for ruling out rational roots of polynomials

Question

Given a polynomial $P(x)$ of degree $m>1$: $$P(x)=a_m x^m +...+ a_k x^k +...+ a_2 x^2 + a_1 x - \alpha$$ Where $\lvert a_m \rvert >...> \lvert a_k \rvert > ... > \lvert a_1 \rvert = 1$ and $\alpha$ are integer coefficients, such that $a_k <0$ $\forall k \neq m$. What methods are there to rule out a specific rational root $ \frac {1}{\beta} >0 $ ? (It is supposed that this case that $\beta$ divides $a_m$, so the rational root theorem applies, and thus this root $ \frac {1}{\beta} $ cannot be ruled out by contradiction with the theorem). According to Sturm's Theorem, one can assert that there is a root in the interval $[0,F]$, where F is a positive number. It is known that $P(x)$ has no irrational roots.