An algorithm for factoring non-monic quadratic polynomials Or: How I learned to stop using the quadratic formula and love undoing FOIL

TL;DR

This paper introduces a new algorithm for factoring quadratic polynomials over any unique factorization domain, providing proofs of correctness and demonstrating its application over integers and Gaussian integers.

Contribution

The paper presents a novel, general algorithm for factoring quadratic polynomials over arbitrary UFDs, moving beyond traditional methods like the quadratic formula.

Findings

Algorithm correctly factors quadratics over Z and Z[i]

Proof of correctness provided for the algorithm

Examples demonstrate practical application over different domains

Abstract

We give an algorithm for factoring quadratic polynomials over any UFD, Z in particular. We prove the correctness of this algorithm and give examples over Z and Z[i].