A Simple Trigonometric Classification of Quintic Roots

TL;DR

This paper introduces a straightforward trigonometric method to determine the number of real and complex roots of a depressed quintic equation without solving it, extending the approach used for quartics.

Contribution

It presents a simple, computationally light trigonometric criterion for analyzing quintic roots, based on transforming the equation using Chebyshev identities.

Findings

Provides a natural extension of quartic root analysis to quintics.

Offers a light computational method for root classification.

Avoids solving the quintic explicitly.

Abstract

This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5 + mt^3 + nt^2 + pt + q = 0$ with $m < 0$ into the trigonometric equation $f(\theta) = \alpha\cos^2\!\theta + \beta\cos\theta + \cos 5\theta + \gamma = 0$ via the Chebyshev identity $16\cos^5\!\theta - 20\cos^3\!\theta + 5\cos\theta = \cos 5\theta$. The derivation is computationally light and conceptually natural, extending the quartic case to fifth-degree equations. As the Abel--Ruffini theorem forbids a general algebraic solution for the quintic, having a simple trigonometric criterion for the nature of its roots is especially appealing.