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The Cubic Equation
All the roots are given by the formula:
and the quartic
However there can be no other formulas – in part because
greater than 4 determines certain things called groups with properties related
to polynomials that can never be fixed. Here we will explore the
To begin with, any cubic polynomial can be written as
For convenience we will relabel our letters as
Notice the x2 term is gone ! This trick is incredibly
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