Factorization of polynomials based on the multiplicities of their roots and applications

The process of factoring a polynomial in such a way that the multiplicities of its distinct roots can be determined, is considered. Central to this factorization process is the ability to compute the polynomial of the distinct roots of a given polynomial from the discriminants of the given polynomial. By subsequent extraction of, and division by these polynomials of distinct roots, a given polynomial is factored into a product of polynomials raised to powers that are equal to the multiplicities of the roots. The ability to compute polynomials of distinct roots and to factor polynomials based on multiplicity, is utilized to develop an algorithm for computing the greatest common divisor of a system of polynomials. Some numerical examples are provided to demonstrate these algorithms.