This lets us reduce the problem from one with repeated roots to one with simple roots.

We say that is ** square-free** if there does * not*
exist another polynomial of degree bigger than 0 such that
(recall that ** c | d ** means that **c** divides
evenly into **d**).

The ** square-free factorization** of is

where each is square-free, and the GCD if .

For example, has
but otherwise splits **a** into factors containing the simple roots, the
roots of degree 4, and the roots of degree 5.

** Remark** It is easy to prove that has repeated roots
if and only if the GCD . You should prove
that for yourself as an exercise, as it is fundamental to the
following algorithm for computing the square-free factorization.

Thu Nov 16 13:46:20 PST 1995