Complexity of finding a rational root of a polynomial

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).

You do not need the full power of LLL algorithm for finding rational roots. In particular, Berlekamp-Zassenhaus algorithm exponential-time algorithm to compute the irreducible factorization can be used to get a polynomial-time algorithm for finding rational roots: The exponential part of this algorithm is the recombination of different modular factors to build true factors, but no such recombination is needed to build the linear factors.

A complete description and analysis of such an algorithm has been given by Loos in 1983:

Loos, R. Computing Rational Zeros of Integral Polynomials by P-Adic Expansion. SIAM Journal on Computing 12, no. 2 (May 1, 1983): 286–93. doi:10.1137/0212017.