Thread: Relationship between roots of polynomial and derivative

1. Relationship between roots of polynomial and derivative

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.

2. Re: Relationship between roots of polynomial and derivative

From Wikipedia on Marden's Theorem:
"A more general version of the theorem, due to Linfield (1920), applies to polynomials p(z) = (z − a)i (z − b)j (z − c)k whose degree i + j + k may be higher than three, but that have only three roots a, b, and c. For such polynomials, the roots of the derivative may be found at the multiple roots of the given polynomial (the roots whose exponent is greater than one) and at the foci of an ellipse whose points of tangency to the triangle divide its sides in the ratios i : j, j : k, and k : i."

3. Re: Relationship between roots of polynomial and derivative

Thanks but this still only applies to polynomials with at most three distinct roots - I was looking for something for say, polynomials with up to ten complex roots.