The main purpose of this paper is to determine two new algorithms for the division of the polynomial matrix B(s) ε R[s]p×q by A(s) ε R[s]p×p (a) based on the Laurent matrix expansion at s = = of the inverse of A(s), i.e. A(s)−1, and (b) in a way similar to the one presented by Gantmacher (1959).