We study the number of irreducible factors (over Q) of the n-th iterate of a polynomial of the form f_r(x) = x² + r for r ∈ Q. When the number of such factors is bounded independent of n, we call f_r(x) eventually stable (over Q). Previous work of Hamblen, Jones, and Madhu [8] shows that f_r is eventually stable unless r has the form 1/c for some c ∈ Z\{0,-1}, in which case existing methods break down. We study this family, and prove that several conditions on c of various flavors imply that all iterates of f_1/c are irreducible. We give an algorithm that checks the latter property for all c up to a large bound B in time polynomial in log B. We find all c-values for which the third iterate of f_1/c has at least four irreducible factors, and all c-values such that f_1/c is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of f_1/c, for all c with absolute value at most 10⁹.

We thank Jennifer Balakrishnan for conversations related to the proof of Theorem 1.6, and the anonymous referee for many helpful suggestions.