The Ulam sequence U(1, n) is defined as the sequence starting with integers 1, n such that n > 1, and such that every subsequent term is the smallest integer that can be written as the sum of distinct previous terms in exactly one way. This family of sequences is notable for being the subject of several remarkable rigidity conjectures. We introduce an analogous notion of an Ulam sequence inside the integer polynomial ring, and use it both to give new, constructive proofs of old results as well as to produce a new conjecture that implies many of the other existing conjectures.