PolynomialRemainder[p, q, x] gives the remainder from dividing p by q, treated as polynomials in x.
The degree of the result in x is guaranteed to be smaller than the degree of q.
Unlike PolynomialMod, PolynomialRemainder performs divisions in generating its results.
See The Mathematica Book: Section 3.3.4.
See also: PolynomialQuotient, Apart, Cancel, PolynomialMod, Mod, PolynomialReduce.