Methods for ruling out rational roots of polynomials
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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.
polynomials
