I was just wondering, is there a simple relationship between the (possibly complex) roots of a polynomial and the roots of it's derivative ? As in is there a simple algorithm to extract the roots of by knowing only and its roots (and without differentiating and solving )? Because I'm in the process of writing a little fractal renderer and the existence of such a relationship could potentially increase the speed of my program by several orders of magnitude (of course I could always probabilistically find the roots of the derivative using Newton-Rapshon or whatever but it wouldn't be as scalable).
Edit: I see there is one result (Marden's theorem) where the roots of the derivative of a cubic are located at the foci of the ellipse formed by the roots of the cubic, but I'm looking for a more general result. Does such a result exist?