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 $f(\hat{I}¸)=a\cos \hat{I}¸+\cos 4\hat{I}¸+b$ on $[0,\ddot{I}€]$, obtained by matching the quartic to the Chebyshev identity $8{\cos }^4\hat{I}¸-8{\cos }^2\hat{I}¸+1=\cos 4\hat{I}¸$. 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.

Preliminary draft (working paper). Feedback welcome; may contain errors