The Journal of Online Mathematics and Its Applications

Volume 8. March 2008. Article ID 1663

The Most Marvelous Theorem in Mathematics

Dan Kalman


My favorite theorem in all of mathematics is what I call Marden's theorem.  It relates the roots of a polynomial p(x) and those of its derivative p′(x).  This is a familiar idea in calculus, where Rolle's theorem tells us that a root of the derivative must occur between any pair of roots of the original function.  That is one kind of relationship between the roots of  p and the roots of its derivative.

Figure 1: Rolle's Theorem: Between roots of p(x) there is a root of p′(x).

Rolle's Theorem

However, Marden's theorem is set in the complex plane.  In that context, the polynomial p(z) has the same algebraic form that we meet in calculus, for example, p(z) might be given by z3 + a2 z2 + a1 z + a0. But now, the coefficients aj are permitted to be fixed complex numbers, and the variable z likewise can take on any complex value.   We can still discuss roots of p -- values of z where p(z) = 0 -- and we compute p′(z) in the familiar way.  For the example, we have p′(z) = 3z2 + 2a2 z + a1.  If you know where the roots of p are, what can you say about the roots of p?  Does Rolle's theorem still hold?

Where the real numbers are envisioned as lying along a line, complex numbers occupy a plane.  The roots of  p and  p are points in that plane.  One might ask whether roots of p must be found between roots of p, as in Rolle's theorem. But note that there is some ambiguity about the meaning of between when dealing with points in a plane.  One obvious idea would be to ask whether a root of p must lie on the line segment between roots of p.  But this cannot be true. Consider, if p is a cubic (that is, of third degree), and if the roots are not all in a line, then these roots determine a triangle. It is not possible for there to be a root of the derivative on the line between each pair of roots of p, as p has only two roots. If some version of Rolle's Theorem is to hold, the two roots of p′ must somehow be shared among the three sides of the triangle.

To see how this might occur, let us examine an example. Suppose p(z) = (z2 + 1)(z − 1) =  z3z2 + z − 1.  The roots are at 1, i, and i.  Meanwhile, p′(z) = 3z22z + 1, which has roots at   1/3 +/− i (root 2) / 3. As expected, these roots of the derivative do not lie on the line segments joining the roots of p, as illustrated in Figure 2 below, where the roots of p appear as black dots and the roots of p as blue dots.

Figure 2: Roots of p (blue dots) do not lie on segments joining the roots of p (black dots).

roots of p and p′

On the other hand, notice how the roots of  p' can be found near to each side of the triangle, and as a result, they are completely surrounded by the roots of  p.  It turns out that this is always the case, as asserted by  Lucas' Theoremall of the roots of the derivative must lie in the convex hull of the roots of the original polynomial. In particular, if p(z) is a third degree polynomial with roots forming a triangle (as in the example) then the roots of the derivative p′(z) must lie inside or on this triangle.   That is what Lucas' theorem says. But we can say more, and that is where Marden's theorem comes in.

Marden's theorem gives a wonderfully geometric recipe for finding the exact positions of the roots of p when p is a cubic polynomial with distinct noncollinear roots in the complex plane. These roots are the vertices of a triangle, and a unique ellipse can be inscribed in the triangle tangent to the sides at their midpoints.  Like any ellipse, it has two special points called foci.  The foci are the roots of p′(z).  This situation is illustrated below in Figure 3.  The roots of p are at the vertices of the triangle, the midpoints of the sides of the triangle are indicated in red, and the foci of the ellipse are shown in blue.

Figure 3: A Marvelous Theorem: p(z) = 0 at the vertices of the triangle; p′(z) = 0 at the foci of the inscribed ellipse.

Marden's Theorem

When I first read of this result in my third year of college, I thought it was the most marvelous theorem in mathematics, a feeling that persists to this day. I call it Marden's theorem because I read it in a book by Marden, although Marden himself credits Siebeck with the original discovery. My purpose in this exposition is to celebrate Marden's theorem and to share it with a broad audience. I will present an interactive dramatization of Marden's theorem, as well as some of the history, and a proof. The proof is completely elementary and yet draws on background information from several diverse parts of elementary mathematics. To fully understand the proof, the reader must be familiar with the geometric properties of ellipses, fundamentals of complex analysis, linear transformations, and algebraic properties of polynomials. I have endeavored to provide the necessary background for all of these topics, so that anyone with a command of first year calculus can work through whatever details are unfamiliar or interesting. Most of this background is included as hyper−text, meaning that the reader can reach it through links, at his or her option. You may already have noticed this in connection with Rolle's Theorem above, as well as some of the terminology. The presentation of the background topics also includes some interactive dramatizations which illustrate fundamental ideas.

The overall organization comprises four independent sections, plus a list of references.  You can choose which of these are of interest, and may read them in any order.  One possibility is to systematically explore the mathematical background. Alternatively, as you read the proof, you can follow links to each background topic at the point where it is relevant.

Note that one of the motivations for presenting this material in a hyper−text format is to include much more detail than would be possible in a print presentation.  You are encouraged to explore the details that you find most interesting, without feeling compelled to read everything.  For a more traditional printed presentation, or an overview in general terms, see Kalman [2] and [3].


To help you decide what pages to view, here is an annotated version of the Table of Contents.

  1. Animated Dramatization -- dynamic geometry shows how the ellipses and foci relate to roots of derivative
  2. Historical Background -- brief account of history of Marden's theorem, and my interest in the topic
  3. Mathematical Background -- several sections covering ellipses, complex numbers, linear transformations, and other topics necessary for a complete understanding of the proof. Under the heading Ellipses, the Optical Property and Extended Optical Property are particularly recommended.
  4. Existence and uniqueness of the inscribed ellipse  including a nice animated dramatization of the existence of the inscribed ellipse
  5. Proof of Marden's Theorem-- Combining aspects of arguments from two earlier proofs
  6. References