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A Simple Trigonometric Classification of Quintic Roots

Sawon Pratiher

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 $f(θ) = α\cos^2\!θ+ β\cosθ+ \cos 5θ+ γ= 0$ 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.

A Simple Trigonometric Classification of Quintic Roots

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 with into the trigonometric equation via the Chebyshev identity . 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.

Paper Structure

This paper contains 21 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Derivation
  3. The depressed quintic
  4. The Chebyshev identity
  5. The trigonometric substitution
  6. Why this works: the bijection
  7. What about roots outside $[-u,u]$?
  8. Critical-point analysis
  9. Classification of the roots
  10. A practical decision procedure
  11. Examples of use as a method
  12. A perfect example: $P(t) = t^5 - 5t^3 + 5t$
  13. Example with three real roots
  14. Example with one real root
  15. The reduced quintic ($n = 0$)
  16. ...and 6 more sections

Key Result

Proposition 2

For all $\theta \in \mathbb{R}$,

Theorems & Definitions (19)

  • Remark 1
  • Proposition 2
  • proof
  • Corollary 3
  • Lemma 4: Third-derivative positivity
  • proof
  • Corollary 5
  • Proposition 6: Exterior root detection
  • Proposition 7
  • proof
  • ...and 9 more