Analytical formula for the roots of the general complex cubic polynomial

TL;DR

This paper introduces a new uniform analytical method for calculating roots of any complex cubic polynomial, avoiding case distinctions and standard root ambiguities present in traditional formulas.

Contribution

The paper proposes a novel approach based on variable transformations with an arbitrary parameter, providing a consistent formula for roots of complex cubics.

Findings

Provides a uniform formula for complex cubic roots

Avoids case distinctions in root calculations

Ensures correct standard convention usage

Abstract

We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this method is to calculate the roots of the cubic polynomial as uniform formula using the standard convention of thesquare and cubic roots. In contrast, the reference methods for this problem, as Cardan-Tartaglia and Lagrange, give theroots of the cubic polynomial as expressions with case distinctions which are incorrect using the standar convention.