Depressed cubic polynomial

Function to obtain the "depressed" cubic polynomial, from the general cubic form.

Syntax:
p2 = depressedcubic(p1)

If the general cubic polynomial p1(y) has the form p1=[a b c d], meaning:

p1(y) = a*y^3 + b*y^2 + c*y +d

The Matlab function depressedcubic.m transform it into the "depressed" cubic form p2(x)

p2(x) = a*x^3 + k1*x + k2

by applying the change of variable

y = x - b/(3a)

where k1 = c - b^2/(3a)
and k2 = d - bc/(3a) + (2*b^3)/(27a^2)

This function checks if the input is a valid cubic polynomial (expressed as a 4-vector as in Matlab format) with the coefficient of the cubic term non zero.

The method is the work of Nicolo' Fontana (a.k.a. Tartaglia; 1500-1557) and Scipione Del Ferro (1465-1526). Later published by Cardano in his Ars Magna.