Hello,

I was just wondering, is there a simple relationship between the (possibly complex) roots of a polynomial $\displaystyle f(x)$ and the roots of it's derivative $\displaystyle f'(x)$? As in is there a simple algorithm to extract the roots of $\displaystyle f'(x)$ by knowing only $\displaystyle f(x)$ and its roots (and without differentiating and solving $\displaystyle f'(x)$)? 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?

Thanks.