Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden’s Theorem, published in 1945, is one such — delightful — result.
Cubic with roots at x=1, x=2 and x=3.
For centuries, mathematicians have struggled to find roots of polynomials like
p(x) ≡ xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + … a1 x + a0 .
Roots are values of x for which p(x) = 0. Linear and quadratic equations, with n = 1 or n = 2, were solved in ancient times. Cubics and quartics, with n = 3 or n = 4, were cracked by Italian renaissance scholars in the sixteenth century. This led to the introduction of imaginary and complex numbers.
Gauss proved the Fundamental Theorem of Algebra: every polynomial equation p(z) = 0 has a root, that is, there is always a real or complex value z for which p(z) vanishes. It follows that every n-th degree polynomial has n roots (not all necessarily distinct).
A simple example
We consider the simple cubic equation
p(x) ≡ ( x – x1 ) ( x – x2 ) ( x – x3 ) = ( x – 1 ) ( x – 2 ) ( x – 3 ) .
Obviously this has three roots, x = 1, x = 2 and x = 3 (see picture above). It is easy to compute the derivative and get
p'(x) ≡ ( x – x’1 ) ( x – x’2 ) = ( x – ( 2 + 1/√3 ) ) ( x – ( 2 – 1/√3 ) ) .
Notice how the roots of p(x) and p'(x) are interwoven:
x1 < x’1 < x2 < x’2 < x3 .
In fact, a much more general pattern holds, described by the Gauss-Lucas Theorem. Suppose an n-th degree polynomial p(z) has n roots { z1 , z2 , … , zn }, and its derivative p'(z) has (n-1) roots, { z’1 , z’2 , … , z’n-1 }. Then all the roots of p'(z) fall within the convex hull of{ z1 , z2 , … , zn }, that is, the smallest convex set containing all these values. We can think of pins being placed at each point zk and an elastic band being stretched around all the pins.
Inellipse in a triangle
For a cubic, the three complex roots form a triangle in the complex z-plane. The roots of p'(z) both fall within this triangle. This is shown in the figure below. The special case in which all three roots are real, is a consequence of Rolle’s Theorem: between two zeros of a function there is a zero of its derivative.
Three roots of a cubic form a triangle. The roots of the derivative are at the foci of an inscribed ellipse. Image from Kalman, 2008.
In 1945, Morris Marden proved a much sharper result. He showed that the roots of p'(z) are at the foci of a unique conic section inscribed in the triangle. For any arbitrary triangle, there is a unique ellipse, the Steiner inellipse, that lies within it and that is tangent to the triangle at the mid-point of each of its three sides.
Marden showed that the two foci of the Steiner inellipse are the roots of the derivative: p'(z) = 0.
It is a simple matter to show that the average value of the roots of p(z) is equal to the average of the roots of p'(z), and also of p”(z) and so on. Thus, for a cubic, the average of the roots, which is at the centroid, and also at the centre of the inellipse, is the root of p”(z) = 0, which is z = – a2 / 3.
Earlier Proof
Marden’s Theorem is a delightful result – quite unexpected, and overlooked by the numerous scholars who have studied polynomials for many centuries. Marden provided a proof, but he gave credit for the result to Jörg Siebeck for discovering it about 80 years earlier (in 1864).
Readers may like to consider the various ways in which Marden’s theorem may be generalized. Surely, there are still rich seams of gold to be mined.
Sources
Kalman, Dan, 2008: The most marvelous theorem in mathematics. http://www.maa.org/mathhorizons. [PDF].
Marden, Morris (1945), A note on the zeros of the sections of a partial fraction”, Bulletin of the American Mathematical Society, 51 (12): 935–940, doi:10.1090/S0002-9904-1945-08470-5
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