This is inspired by this question. Let $f(x)=a_nx^n+…+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $a_0$ and all $q$ that divide $a_0$. This is not very complicated but involves factoring $a_0$ and $a_n$. The factoring problem is not known to be in P. If $n\le 4$, then the fact that the group $S_4$ is solvable and the well known formulas for roots of polynomials of degree $\le 4$ give easy polynomial time algorithm of finding rational roots.
Question Is the problem of finding a rational root of $f(x)$ in P for every $n$ (say, for $n=5$)?
Update 1 After I posted the question, I noticed an answer by Robert Israel to the previous question (of Joseph O'Rourke). That could give an answer to my question but I am still not sure how one can avoid factoring numbers $a_0,a_n$.
Update 2 Robert Israel's explanations (see his comment here ) convince me that his algorithm of checking whether a polynomial has a rational root (all roots rational) runs in polynomial time.
I removed Question 2 so that I can accept Michael Stoll's answer. I will post Question 2 as a separate question.
Best Answer
Didn't Lenstra, Lenstra and Lovász in their famous LLL paper prove that factorization of polynomials over $\mathbb Q$ can be done in polynomial time? You get a rational root if and only if there is a factor of degree 1, and the polynomial has only rational roots if and only if all factors have degree 1.
Lenstra, A.K.; Lenstra, H.W.jun.; Lovász, László: Factoring polynomials with rational coefficients. (English) Math. Ann. 261, 515-534 (1982).