On Nonnegativity of the Generalized Discriminants of Quadratics

Abstract

Discriminants of quadratics have been recently generalized as Δ k = k ( n − k ) c n − k 2 − ( k + 1 ) ( n − k + 1 ) c n − k − 1 c n − k + 1 for a polynomial f ( x ) = c n x n + ⋯ + c 1 x + c 0 of degree n ≥ 2 for 1 ≤ k ≤ n − 1 and it has been shown that Δ 1 ≥ 0 if f has real roots only, [1]. In this article we extend this result to Δ k . Namely, we show that Δ k ≥ 0 if f has real roots only for 1 < k ≤ n − 1 . As an application, we also demonstrate how to graph a plane quartic without using any calculus tools other than continuity.