A qualitative study of polynomials in high school

Abstract

A method is suggested for getting students acquainted with polynomials of degree higher than 2. The goal is to develop students’ intuition for polynomials as a basis for further algebraic and analytical study of polynomials. In particular, existence, number and type of real and complex roots of polynomials of a given degree n >2 are under consideration. Polynomials are constructed and treated inductively as products of linear and quadratic functions. Computer‐generated graphs are used for gathering observation‐based qualitative data. Finally, difficulties in solving polynomial equations of degree higher than 4 are illustrated graphically and the applications of the proposed methods to the demonstration of the fundamental theorem of algebra are discussed.