Use Newton's method starting from a close approximation.

First make sure your polynomial has no repeated roots. If you're working with polynomials f(x) with rational coefficients and gcd(f(x),f'(x)) is a nonzero constant then f(x) has no repeated roots. In general, if f(x) with rational coefficients has repeated roots then you can pass to a factor with the same roots all of multiplicity 1 by using the ratio f(x)/gcd(f(x),f'(x)).