# The Cubic Equation

[Pinter, Charles C. A Book of Abstract Algebra ,
McGraw-Hill, Inc. New

York, 1990.]

Recall the quadratic formula solved for the roots of a quadratic polynomials;

that is polynomials of the form

All the roots are given by the formula:

There is a general formula for cubic polynomials , those of the form

and the quartic

However there can be no other formulas – in part because
any number

greater than 4 determines certain things called groups with properties related

to polynomials that can never be fixed. Here we will explore the

cubic equation.

To begin with, any cubic polynomial can be written as

For convenience we will relabel our letters as

1. Begin by substituting
in for x. You may want to try this with

some actual numbers first . Once you have checked you will realize

that

N**otice the **x^{2}** term is gone **! This trick is incredibly
useful because

now instead of finding roots of f(x) we will only care about roots of

Keep in mind the roots of g will be different form those
of f, but we

know that
so we can go back at the end.

2. Finally g(x) is a nicer polynomial, nice enough to have
a cubic formula.

One of the roots will be given as

It is particularly more difficult to derive this formula
but you may check

with some polynomials to see that you agree.

3. You have one root so you can divide you polynomial g(x) by

to get a quadratic polynomial . Here you simply use the quadratic

equation to solve for the remaining roots.

4. Now you have to translate all the roots into roots for f(x) by subtracting

from each.

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