Chebyshev cubic roots

1. Jul 10, 2009

junglebeast

Say I have a monic polynomial,

x^3 + ax^2 + bx + c

with a=-2.372282, b=1.862273, c=-0.483023

The discriminant is given by

a^2 b^2 - 4 b^3 - 4 a^3 c - 27 c^2 + 18 ab c

which is < 0, indicating 1 real root and 2 complex conjugates.

A method for solving a general cubic using the Chebyshev root is explained here,
http://www.statemaster.com/encyclopedia/Cubic-equation

but (a^3 - 3b) is negative, which means that "t" will be imaginary. But t is then used in the Chebyshev cubic root, which is only defined for real numbers [-2, inf].

2. Jul 10, 2009

HallsofIvy

Staff Emeritus
The site you give does NOT say that the Chebyshev cubic root is "only defined for real numbers [-2, inf]". It gives simple formulas for the root in the case that the argument is in that interval, and then tells how to expand it to complex arguments.

3. Jul 10, 2009

junglebeast

EDIT - deleted previous response.

Let me clarify my remaining confusion from here:
http://www.exampleproblems.com/wiki/index.php/Cubic_equation

They define s = t^2, then say

Well, as they already pointed out t is imaginary, but then they say $$S_{1\over3}(t)$$ is a function of a real variable! But it's still defined in terms of t only! So how can I calculate $$S_{1\over3}(t)$$ given that I can't compute the imaginary t?

Last edited: Jul 11, 2009