Real Polynomials with a Complex Twist

TL;DR

This paper introduces a method using modern 3D graphics to visualize complex polynomial functions, helping students understand both real and non-real solutions and the Fundamental Theorem of Algebra.

Contribution

The authors demonstrate a novel approach employing GeoGebra to create dynamic 3D visualizations of polynomials, integrating complex solutions into the graphical representation.

Findings

Enhanced student understanding of complex solutions

Effective use of GeoGebra for dynamic visualizations

Bridging the gap between symbolic and graphical representations

Abstract

Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the symbolic form. However, the real graph does not show the non-real solutions to polynomial equations. Instead of enhancing students idea of a function, the traditional graph implies a clear disconnect from the symbolic form. In order to fully appreciate the Fundamental Theorem of Algebra, and the non-real solutions of a polynomial equation, traditional graphs are inadequate. Since the early 20th century, mathematicians have tried to find a way to augment the traditional Cartesian graph of a polynomial to show its complex counterpart. Advancements in computer graphics allow us to easily illustrate a more complete graph of polynomial functions that is still accessible to students of many different levels. The authors will demonstrate a method using modern 3D graphical tools such as GeoGebra to create dynamic visualizations of these more complete polynomial functions.