New Five Roots to Solve Quantic Equation in General Forms by Using Radical Expressions Along With New Theorems

TL;DR

This paper introduces a novel method to solve fifth-degree polynomial equations by deriving five roots simultaneously using new theorems and radical expressions, building on solutions for quartic equations.

Contribution

It proposes five new formulas for solving quintic equations directly, reducing quintics to quartics and enabling nearly simultaneous root calculation.

Findings

Five solutions for fifth degree polynomials with real coefficients

Reduction of quintic equations to quartic form

Approximate simultaneous calculation of roots

Abstract

This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate the five roots of any quantic equation nearly simultaneously. The proposed roots for fifth degree polynomials in this paper are structured basing on new proposed solutions for fourth degree polynomial equations, which we developed in order to reduce the expression of any quantic polynomial to an expression of quartic polynomial.