Resumen de Remainder and quotient without polynomial long division

Francesco Laudano

• We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z[x] any integral polynomial with degree less than 6. The arguments are suitable for building classroom/homework activities in basic algebra courses.