Polynomial Greatest Common Divisor

The polynomial greatest common divisor of two polynomials is the polynomial of highest possible degree that divides both of them, the same notion as the integer greatest common divisor transposed into polynomial algebra. For univariate polynomials over a field it is computed with a version of the Euclidean algorithm, repeatedly replacing a pair of polynomials with the second polynomial and the remainder of dividing the first by the second until a zero remainder is reached, with the last nonzero remainder being the GCD, defined only up to multiplication by an invertible constant and so conventionally taken as monic. The computation has several standard uses in symbolic computation: the GCD of a polynomial and its derivative reveals repeated roots without solving the polynomial, computer algebra systems use it to simplify rational expressions, and it enables square-free factorization, which separates a polynomial's factors by root multiplicity; the concept extends to multivariate polynomials and general unique factorization domains.