trigonometric cubic formula

Given a cubic polynomial of the form f(X)=X3+aX2+bX+c=0, one may reduce f(x) via the substitution X(x-a/3) to obtain f~(x)=f(x-a/3) where the reduced polynomialMathworldPlanetmathPlanetmathPlanetmath may be represented

f~(x)=x3+qx+r (1)

The roots to (1) are given by Viéte in the following cases:

Case I The roots of f~(x) are real:
Define t-4q/3 and α=arccos(-4r/t3). Then the roots of f~(x) are


Case II The roots of f~(x) are complex:
Keeping the definition of t from Case I, if -4q/30, then the real root of f~(x) is


If -4q/3<0, then the real root of f~(x) is


One may then inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath transform the roots of f~(x) to obtain the roots of the desired cubic f(x)

We note there are no other cases for the possibilities of the roots of a cubic (i.e. there is no instance where one finds one complex and two real roots). This result is intuitively obvious after graphing cubic polynomials and taking into account that imaginaryPlanetmathPlanetmath roots may only occur in conjugate pairs.

Title trigonometric cubic formula
Canonical name TrigonometricCubicFormula
Date of creation 2013-03-22 15:02:07
Last modified on 2013-03-22 15:02:07
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Theorem
Classification msc 12D10
Synonym Alternate cubic formula
Related topic CardanosFormulae