variant of Cardano’s derivation

By a linear change of variable, a cubic polynomial over can be given the form x3+3bx+c. To find the zeros of this cubic in the form of surds in b and c, make the substitution x=y1/3+z1/3, thus replacing one unknown with two, and then write down identitiesPlanetmathPlanetmathPlanetmath which are suggested by the resulting equation in two unknowns. Specifically, we get

y+3(y1/3+z1/3)y1/3z1/3+z+3b(y1/3+z1/3)+c=0. (1)

This will be true if

y+z+c=0 (2)
3y1/3z1/3+3b=0, (3)

which in turn requires

yz=-b3. (4)

The pair of equations (2) and (4) is a quadratic system in y and z, readily solved. But notice that (3) puts a restrictionPlanetmathPlanetmathPlanetmathPlanetmath 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