A Remarkable Fact

Chapter 3: A Remarkable Fact

The Story…

We are finally in a position to unfold our strategy for solving cubic equations. We promised that it would be simple and easy to remember. But you must first remember what cubic numbers are. The presentation in this section is slightly complex, and will require the definition of the dt() function. But in the next section, Cardano’s Method, we will present a remarkably simple heuristic that only requires the definition of cubic number to remember and apply the strategy once and for all! This strategy also works for quartic equations, and it will be clear how that goes. What is the remarkable fact?

Fact: Suppose that is an arbitrary triple of complex numbers. Then there is a cubic number with the property that

where w is viewed as a quadratic polynomial, and is a primitive complex cube root of unity. Since the zeros of any cubic polynomial (with A, B, and C complex) are a triple of complex numbers, we will say that a cubic number w is associated to the polynomial if the triple consists of the zeros (with possible repetition) of the cubic polynomial. Obviously, such a w is not unique, since these zeros may be reordered.

Proof:

The proof of the assertion is simple, and it shows the direction for generalization to arbitrary degree. The following matrix is invertible. It is called a Vandermonde matrix.

Thus we may solve the system of 3 linear equations in unknowns: a, b, and c.

for a, b, and c. The solution is unique, remembering of course that the may be reordered. Now notice that

for cubic number . End of Proof

If we had chosen a 4th degree polynomial, then the corresponding Vandermonde would still be invertible, and we would seek a "quartic" number whose values on the 4th roots of unity: would be the zeros of that polynomial. The construction works for all degrees.

Now, given this fact, the trick is to find a cubic number that is associated with a given cubic polynomial. This is quite a trick, since the roots are not known. All that is known is the polynomial:

In order to find the cubic number w, and therefore to solve the polynomial equation,

we observe that is associated to a certain cubic number . Let x stand for a complex variable. Every complex number x may be thought of as the cubic number so consider the complex polynomial expression in x (with a, b, and c held fixed).

This is called the cubic characteristic polynomial of cubic number w. We will find a cubic number w such that that is, such that is the cubic characteristic polynomial of w. Then w will be shown to be associated to as described above, and will yield the roots of the cubic equation .

In the previous section, we defined for , to be

We now give a more perspicuous definition that will prepare us to use it in this context.

Observation: For any cubic number , is the complex number:

where is a primitive complex cube root of unity.

Proof of observation: Among other things this will clearly establish multiplicativity of dt, i.e. of the fact that

in light of the exercise at the end of the last section that indicated: evaluation at a cube root of unity defines a mapping from the cubic numbers to the complex numbers that preserves products. That is, the mapping for fixed complex z from the complex polynomial algebra taking

is an algebra homomorphism (in particular, is multiplicative) and, if z is a cube root of unity, is well-defined on the quotient algebra, the cubic numbers.

The observation follows from:

Now expand the entire trinomial using the facts that and to get

+ remainder

The remainder is seen to be 0 on collecting terms, and since , we see that this may be written in the original form:

End of Proof

Now given this observation, we see that the characteristic polynomial of w: also has a simple representation. The characteristic polynomial of cubic number w is the polynomial:

Therefore, the zeros of the characteristic polynomial of w are precisely counted with their multiplicity. Thus, we may solve the cubic equation

if we can find a cubic number

with the property that

And since

it is just a matter of solving

for a, b, and c.

We make two simplifications immediately. Recall that if we start with a cubic polynomial it may be transformed by a linear change of variable: to the form . This just means that we've translated the origin so that the sum (or average) of the roots is 0.

Since we are interested in finding a cubic number w so that and

is to be equal to , it follows that a must be zero for such a cubic. This is a great simplification. Thus, from now on we consider "traceless" cubic numbers of the form: to find solutions of cubics whose roots sum to 0.

The second simplification concerns the fact that we are only interested in solving cubic equations in which all roots are real. We will discuss this in the final section on discriminants.