Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Simple Trigonometric Classification of Quartic Roots

Sawon Pratiher

Abstract

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a degree-six polynomial in the coefficients -- with an elementary analysis of the function $f(θ) = a\cosθ+ \cos 4θ+ b$ on $[0,π]$, obtained by matching the quartic to the Chebyshev identity $8\cos^4\!θ- 8\cos^2\!θ+ 1 = \cos 4θ$. The derivation is computationally light and conceptually natural, and has the potential to demystify the geometry of a quartic equation's roots from a trigonometric perspective.

A Simple Trigonometric Classification of Quartic Roots

Abstract

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a degree-six polynomial in the coefficients -- with an elementary analysis of the function on , obtained by matching the quartic to the Chebyshev identity . The derivation is computationally light and conceptually natural, and has the potential to demystify the geometry of a quartic equation's roots from a trigonometric perspective.

Paper Structure

This paper contains 16 sections, 7 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Derivation
  3. The depressed quartic
  4. The trigonometric substitution
  5. Why this works: the bijection
  6. What about roots outside $[-u,u]$?
  7. Classification of the roots
  8. Critical-point analysis
  9. The three cases
  10. Examples of use as a method
  11. The biquadratic case
  12. The case $m \geq 0$
  13. Discussion
  14. Comparison to the discriminant
  15. Extension to the quintic
  16. ...and 1 more sections

Key Result

Lemma 1

If $m < 0$, then $P$ is strictly convex on $(-\infty, -u)$ and on $(u, \infty)$.

Theorems & Definitions (14)

  • Lemma 1: Convexity outside $[-u,u]$
  • proof
  • Corollary 2
  • proof
  • Proposition 3
  • Lemma 4
  • proof
  • Theorem 5: All four roots complex
  • proof
  • Theorem 6: Two real, two complex roots
  • ...and 4 more