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Title: Cubic Equations
Book
Description: This
WorkBook summarizes an original work by James White and Dan Kalman and
aims to present it in a form that will be accessible to an advanced undergraduate.
It develops an approach to the study of Cardano’s method for solving cubic
equations that discloses certain new symmetries and points the way to
generalization to higher degree equations. Those generalizations (to the
quartic case) and remarks on difficulties with higher-degree cases are
developed in detail in a forthcoming paper by White and Kalman.
The approach taken in this WorkBook is to develop the algebra of “cubic
numbers” which may be thought of either as 3x3 complex circulant matrices,
or as the complex group algebra on the cyclic group of order 3. Pedagogically,
of course, we do not present them in this rarified light here, but rather
construct and explore them in our laboratories. Another new feature of
this WorkBook is that it gives the reader an opportunity to experiment
with abstract algebra using certain rings of numbers, and to verify empirically
the key facts that we visit along the way to Cardano’s algorithm.
In fact, the principal aim of the WorkBook is to bring the reader to such
a clear understanding of the technique, its hidden structure and symmetry,
that she will be tempted to explore higher-degree cases on her own thereafter.
This WorkBook covers some elements of the theory of equations (but not
many), and is meant to guide the reader to her own questions by presenting
an old and venerable idea in a slightly new light.
There are many exercises in the laboratories, some more challenging than
others, but all aimed at clarifying some point of the exposition. The
author will be quite interested in learning how the material fares in
the classroom, as well as how it is received for private study and recreation.
Please send comments to the author.
Author: James E. White
Suggested Use: The WorkBook is fully documented in the 40 page documentaion file that accompanies it. That printable file contains hardcopy of all text and lectures.
Topics: Cubic Equations, equations, Inflection Points, graphing, factorization of polynomials, maxima and minima, cubic polynomials, complex numbers
Number of Pages: 12
Animation: Yes
Grade
Level: 