Extended Quotient-Remainder Theorem on a Polynomial Ring

Abstract

There have been many researches on the polynomial division. With the fast development of computer calculation, the classical problem of a polynomial division with a remainder has been changed to find fast algorithms to computing the coefficients of the quotients and of remainder on the division of by , . In [3], D. Bini and V. Pan introduced various known polynomial division algorithms(see [2], [4], [5], [8]), and compared them with their new algorithm. From these algorithms, we obtain motivation to consider extended division algorithm of polynomials with related to the remainder theorem. In this thesis, we shall be interested in the dividing by of degree higher than 1. Moreover, we use the determinant of coefficient matrix to obtain extended quotient and remainder theorems in a polynomial ring. The main theorem is the following. Extended Quotient-Remainder Theorem : If are all distinct, and we divide by , then we obtain quotient and remainder where .