A Method for Locating the Real Roots of the Symbolic Quintic Equation Using Quadratic Equations

TL;DR

This paper introduces a novel method to locate roots of symbolic quintic polynomials using only quadratic equations, providing explicit intervals and analysis without numerical iteration or higher-degree equations.

Contribution

The method uniquely determines root locations of a symbolic quintic through resolvent quadratics, avoiding numerical methods and higher-degree equations, and analyzes root behavior based on coefficient variations.

Findings

Roots can be isolated within finite intervals using quadratic equations.

The method applies to various coefficient configurations of the quintic.

It reveals how coefficient changes influence root positions.

Abstract

A method is proposed with which the locations of the roots of the monic symbolic quintic polynomial $x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ can be determined using the roots of two resolvent quadratic polynomials: $q_1(x) = x^2 + a_4 x + a_3$ and $q_2(x) = a_2 x^2 + a_1 x + a_0$, whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of $q_1(x)$ and $q_2(x)$ and on some specific relationships between them. The method is illustrated with the full analysis of one of the possible cases. Some of the roots of the symbolic quintic equation for this case have their isolation intervals determined and, as this cannot be done for all roots with the help of quadratic equations only, finite intervals containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of the quintic are identified. Knowing the stationary points of the quintic polynomial, lifts the latter indeterminacy and allows one to find the isolation interval of each of the roots of the quintic. Separately, using the complete root classification of the quintic, one can also lift this indeterminacy. The method also allows to see how variation of the individual coefficients of the quintic affect its roots. No root finding iterations or any numerical approximations are used and no equations of degree higher than 2 are solved.