fastest way to find Zeros of a polynomial

Algorithm in simplified terms is,

Z(p(x))=Z(p(x)p'(x)) - Z(p'(x))

Z(p(x)) stands for simple Zeros of the polynomial p(x).

Note: One has to sieve out the multiple order zeros by using the conditions
 p'(x)=0 & p(x)=0 as a zero'th step.