As with integers, operations related to division are key to many computations with polynomials. Mathematica includes not only highly optimized univariate polynomial-division algorithms, but also state-of-the-art multivariate generalizations.
Cancel — cancel factors in a ratio of polynomials
PolynomialReduce — represent one polynomial as a sum of others
GroebnerBasis — find a reduced Groebner basis for a list of polynomials
PolynomialMod — reduce polynomial coefficients modulo m
Modulus — specify the modulus for a finite field