variant of Cardano’s derivation

By a linear change of variable, a cubic polynomial over $ℂ$ can be given the form $x^3+3bx+c$. To find the zeros of this cubic in the form of surds in $b$ and $c$, make the substitution $x=y^{1/3}+z^{1/3},$ thus replacing one unknown with two, and then write down identities which are suggested by the resulting equation in two unknowns. Specifically, we get

$y+3(y^{1/3}+z^{1/3})y^{1/3}{z}^{1/3}+z+3b(y^{1/3}+z^{1/3})+c=0.$ (1)

This will be true if

$y+z+c=0$ (2)

$3y^{1/3}{z}^{1/3}+3b=0,$ (3)

which in turn requires

$yz=-b^3.$ (4)

The pair of equations (2) and (4) is a quadratic system in $y$ and $z$, readily solved. But notice that (3) puts a restriction on a certain choice of cube roots.

Title	variant of Cardano’s derivation
Canonical name	VariantOfCardanosDerivation
Date of creation	2013-03-22 13:38:43
Last modified on	2013-03-22 13:38:43
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Proof
Classification	msc 12D10