Polynomials are among the simplest of algebraic objects, yet they are also among the most useful. And they have many remarkable properties, as Dan Kalman of American University revealed in his recent lecture on "Provincial Polynomia: Uncommon Excursions for the Seasoned Visitor" at the MAA Carriage House Conference Center.
Kalman's itinerary began with a few quick, insightful jaunts involving polynomials. One foray led to a polynomial mind-reading trick. Ask someone in the audience to secretly make up a polynomial, p(x), of any degree but with positive integer coefficients less than 10. You then offer to reveal the polynomial, given only p(10).
The journey then continued with an extended look at Horner's form, a useful alternative method of representing polynomials. For example, the polynomial expression 5x4 – 11x3 + 6x2 + 7x – 3 can just as easily be represented in Horner's form as (((5x – 11)x +6)x + 7)x – 3.
This particular guise offers an efficient way to compute the value of a polynomial, working from the innermost parenthesis outward and accumulating the result as you go along. With a bit of practice, this is a feasible method for mentally computing values of polynomials, as Kalman demonstrated in competition with audience members wielding calculators.
Lill's method provides a novel way of visualizing the roots of a polynomial geometrically by tracing a coefficient-determined polygonal path on a square grid. Kalman showed that the proof that a Lill path gives a root is tied to Horner's form for a polynomial.
Kalman concluded his presentation with his favorite result in mathematics: Marden's theorem, which relates the roots of a polynomial p(x) to those of its derivative p'(x).
Rolle's theorem tells us that a root of a derivative must occur between any pair of roots of the original function. Marden's theorem, which is set in the complex plane, gives a 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, Kalman said. These roots are the vertices of a triangle, and a unique ellipse, he noted, 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, surprisingly, turn out to be the roots of p′(z).
Kalman had extolled the wonders of Marden's theorem in three simultaneous publications in April 2008: "The Most Marvelous Theorem in Mathematics" in the Journal of Online Mathematics and Its Applications; "The Most Marvelous Theorem in Mathematics" in Math Horizons; and "An Elementary Proof of Marden's Theorem" in the American Mathematical Monthly.
Kalman's journey covered a lot of territory, from Quebrûte to Horner Corner and Lill. If time had permitted, he could have gone on to other realms: palindromic polynomials, chaos theory, curve fitting, identities, and much more. As it was, he demonstrated that a deep understanding of polynomials offers rich rewards.
Kalman, who has longstanding ties to the MAA, received his undergraduate degree from Harvey Mudd College, and his M.A. and Ph.D. in mathematics from the University of Wisconsin–Madison. Before joining the faculty of American University in 1993, he worked as an applied mathematician at the Aerospace Corporation for eight years. Prior to that he had taught at University of Wisconsin–Green Bay and Augstana College in Sioux Falls, S. Dak. Kalman served for one year as an Associate Executive Director of the MAA.
Kalman based much of his presentation on material in his recent MAA-published book Uncommon Mathematical Excursions: Polynomia and Related Realms.—H. Waldman