ABSTRACT
This chapter explores how computer algebra systems factor polynomials in one variable with integer coefficients. It shows that a procedure for factoring monic polynomials gives a technique for factoring polynomials with arbitrary leading coefficients. These are determined quickly by calculating the square-free decomposition. Factoring polynomials with rational coefficients is equivalent to factoring over the integers. Isaac Newton, in his Arithmetica Universalis, described a method for finding the linear and quadratic factors of a univariate polynomial. In 1793, the astronomer Friedrich von Schubert showed how to extend the technique to find factors of any degree. Convert each of the following to an “equivalent” monic polynomial for factoring. The chapter explains how the factorization of a polynomial with an arbitrary leading coefficient can be obtained from that of an “equivalent” monic polynomial—a polynomial whose leading coefficient is one. Use a computer algebra system to factor the monic polynomial, and then map an analogous result onto the factorization of the original polynomial.
