Discriminants of Quadratics Generalized

Abstract

The definition of the discriminant Δ of quadratic polynomials is extended to higher degree polynomials using the leading three coefficients. The information Δ provides about the nature of the roots of a polynomial f of degree n > 2 is analyzed. It turns out that $\Delta <0$ still assures existence of complex roots. Additionally, if the roots are known to be real, then Δ = 0 if and only if $f(x)=a{(x-c)}^n$. Other Δ’s using all possible three consecutive coefficients of f are defined and the conditions for f(x) to be of the form $a{(x-c)}^n$ are established when the coefficients are not assumed to be real.

Keywords:

• MSC: Primary 11C08
• Secondary 26C10
• 30C15

Acknowledgments

The author wishes to thank the anonymous referee for suggesting a shorter proof for Theorems 1 and 2 and for pointing out that Theorem 3 is valid for any field. The author is also grateful to both referees and the editor for numerous corrections and helpful suggestions.

Additional information

Notes on contributors

Yusuf Z. Gürtaş

Yusuf Z. Gürtaş received his Ph.D. in mathematics from the University of California, Irvine, in 2003. He held visiting positions at various universities before joining the faculty at Queensborough Community College, CUNY. He is married and has two sons who don’t hate math.