An elementary algebraic proof of the fundamental theorem of algebra

TL;DR

This paper presents an elementary algebraic proof of the fundamental theorem of algebra, emphasizing the use of long division, and extends to a more general version of the theorem.

Contribution

It introduces a new, elementary proof technique based on long division, simplifying the understanding of the fundamental theorem of algebra and generalizing recent results.

Findings

Elementary proof using long division

Quick derivation of a generalized theorem

Simplifies understanding of algebraic roots

Abstract

We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by Joseph Shipman.