One can call it a "complete factorization" of "P"("x") over Q (the rationals) if "S"("x") = 1. Otherwise, there is only a "partial factorization" of "P"("x") over Q, which may or may not be further factorable over the rationals but will certainly be further factorable over the reals or at worst the complex plane. (Note that a "complete factorization" of "P"("x") over Q means a factorization as a product of polynomials with rational coefficients, such that each factor is irreducible over Q, ans "irreducible over Q" means that the factor cannot be written as the product of two non-constant polynomials with rational coefficients and smaller degree.)