A vectorial approach to generalize the remainder theorem

Abstract

We propose a new computational proof for the division algorithm that, usingvector algebra, generalizes the remainder theorem to divisions for polynomials of any degreeover a generic integral domain. Then, we extend this result to calculate the pseudo-divisions.Later, starting from the previous theorems, we obtain some algorithms that calculate thepseudo-remainder and the pseudo-quotient while avoiding long division. Finally, we provideexamples and comparisons indicating that these algorithms are efficient in divisions by sparsepolynomials and their divisors, as cyclotomic polynomials.2010 Mathematics Subject Classification. Primary 13B25; Secondary 13F20.Key words and phrases. polynomial pseudo-division, pseudo-remainder, algorithm, matlabcode.