Solving cubic polynomials

There are many sources for how to solve cubic polynomials, but a summary is helpful when implementing a new code. This page is an attempt to record that.

We will consider the polynomial

<math> x^3 + a x^2 + b x + c = 0 </math>

First, calculate the coefficients of the 'depressed cubic' <math>y^3 + p y - q = 0</math> obtained by the substitution <math>x = y - a/3</math>:

<math>p = b - \frac{a^2}{3}</math>

<math>q = \frac{2}{27} a^3 - \frac{1}{3} ab + c</math>

This can then be converted into a quadratic in <math>z^3</math> via the 'magic' Vieta substitution <math>y = z - \frac{p}{3 z}</math>,

<math>\left(z^3\right)^2 - q \left(z^3\right) - \left(\frac{p}{3}\right)^3 = 0</math>

for which the solution is

<math>z^3 = +\frac{q}{2} \pm \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}</math>

Firstly, it is possible that <math>p</math> and <math>q</math> equal zero, implying that <math>z^3 = 0</math> hence <math>y = p</math>

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