Chapter 1: Cubic Equations
This WorkBook
is a gentle introduction to abstract algebra, and theory of equations, written
in homage to Gerolimo Cardano, who, with Tartaglia, established procedures for
solving cubic equations. We introduce here a novel approach to that technique
via the construction and description of what we call: Cubic numbers.
The aim is to
give a self-contained exposition of the basic facts about cubic equations
within the context of abstract algebra, and at the same time to provide many
opportunities to explore. The level is appropriate for an advanced
undergraduate. Our approach avoids matrix techniques, using instead a symbolic
approach to group representations which is amply illustrated in the interactive
exercises.
In this
Story, we will show you a way to learn (and remember) Cardano's Method for
solving cubic equations. It is amazingly simple to state. Given a cubic
equation to solve for complex z:
|
|
with a, b, and c complex numbers, all you have to do is attempt to solve the equation:
|
|
|
for cubic number w, instead. As soon as you write down what this means, you are led directly to Cardano's Method, and the solution. We actually develop here a general procedure that gives all solutions, and shows the relations among them, but you can see that the idea is very simple.
For those of you familiar with matrix representations, the
symbolic cubic numbers that we introduce are simply the elements of the algebra
of 3x3 circulant matrices. Our technique leads to an approach to solving cubic
and quartic equations that is straightforward and easy to remember.
Cubic equations are equations of the form:
|
|
to be solved for z where a, b, and c are given real or complex numbers.
There is a general procedure that
may be used to solve quadratic, cubic, and even quartic equations which places
them all in a unified context. This procedure relies on the on the use of
complex numbers, in particular, on the geometry and algebra of the roots of
unity. It discloses an interesting relationship between polynomial equations
and roots of unity, and it leads to a very easy way to learn (and remember) how
to solve cubic and quartic equations.
Almost everyone has learned the
quadratic formula, but not everyone remembers it. If
|
|
is a quadratic equation with a and b real numbers, then the solutions of this equation are:
|
|
Also, is the abscissa of the vertex (turning point) of the graph, and the number is called the discriminant of the equation. If it is non-negative then the equation has two real solutions equally placed about the abscissa of the vertex. If it is negative, the equation has no real solutions. And if it is zero, the equation has precisely one (real) solution.
This is illustrated below with
the equation:
|
|
The solutions: are equally placed about the abscissa of the vertex: x = 3.
An interesting and analogous
thing is true of cubic equations. For our purposes, a cubic function will be a
function of the form
|
|
with a, b, and c real for now (We consider the more general case later). Therefore a cubic equation will be an equation of the form: . Such an equation always has at least one real solution. It may have three real solutions, and it may not.
In any case, there is always an
interesting point associated with a cubic function called its inflection
point. This is the unique point on the graph where the concavity changes.
The abscissa of this point will be .
We illustrate with a picture below. where and the inflection point has abscissa x = 2.
We
may notice four important things about the inflection points of cubic functions
and their equations. We will prove them all later.
1. In the case that all roots are real, the abscissa of the inflection point is the average of the three real roots.
2. Every cubic function: may be transformed by a linear change of variable to the form
|
|
3. In the latter form, the abscissa of the inflection point is 0, so the average of the roots of the new equation is 0. Such an equation has all real solutions if and only if the following discriminant relation holds:
|
|
4. When the cubic equation has three real solutions, there is a circle centered at the abscissa of the inflection point on the real axis in the complex plane with the property that three equally spaced points on the circle "project" on the three real roots.
This last fact describes the
symmetric way in which roots are organized around the abscissa of the
inflection point. It is the generalization of the corresponding fact about
roots of quadratics. And it is not really surprising, since any three points in
the line can be so realized as the projection of 3 equally points on a circle
centered at their average. The radius of this circle is constrained, however,
to be twice the distance from the inflection to any critical point, and that is
interesting.
What is surprising is that this
symmetry does not extend to quartics. The real roots of quartics cannot in
general be obtained as the projections of four equally spaced points on a
circle centered at the analog of the inflection point. They require three
parameters. Let us begin by experimenting with these three facts about cubics,
beginning with graphing polynomials. Later, we
will show the surprising reason why they are true as we develop methods for
solving cubic and quartic equations.
