A Simple Trigonometric Classification of Quintic Roots
This work addresses a specific mathematical problem for researchers and practitioners dealing with quintic equations, offering an incremental extension of existing trigonometric methods from quartics to quintics.
The paper tackles the problem of classifying the number of real and complex roots of quintic equations without solving them, by introducing a simple trigonometric method that transforms a depressed quintic into a trigonometric equation using Chebyshev identities, providing a computationally light alternative given the Abel-Ruffini theorem's restrictions.
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 $f(θ) = α\cos^2\!θ+ β\cosθ+ \cos 5θ+ γ= 0$ via the Chebyshev identity $16\cos^5\!θ- 20\cos^3\!θ+ 5\cosθ= \cos 5θ$. 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.