General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings

Abstract

We propose some generalizations of the classical Division Algorithm for polynomials over coefficient rings (possibly non-commutative). These results provide a generalization of the Remainder Theorem that allows calculating the remainder without using the long division method, even if the divisor has degree greater than one. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. Finally, we will refer to some applications of these results for evaluating and dividing on skew polynomial rings. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities.