A Mathematical Approach to Constructing Special Four-Way Factorable Quadratics

Overview

When I first learned how x2±5x±6 can factor with interchangeable signs, I was fascinated with its simplicity and orderliness. I was curious to know if there would be more quadratics that could factor in four ways like this and how we could find them. Inspired by that, this paper aims to find a mathematical approach to construct four- way factorable quadratics and explore the relationships between factors and polynomials. It was hypothesized that there are unlimited numbers of such four-way factorable quadratics which share certain patterns. To test this hypothesis and study the relationships, a solvable mathematical model was established with six variables. Geometric visualization and computer programming were also utilized for identifying patterns and inspiring future exploration. By proving the rationale of the mathematical model, simplifying the six variables back into two, and analyzing the relationship of the remaining two variables, the existence of unlimited numbers of four- way factorable quadratics was confirmed. During the investigation, the patterns of such four-way factorable quadratics were also identified, which can be applied to look for all the admissible quadratics given a certain range. This research serves as a catalyst for further investigations of various natures and other patterns of polynomial factoring. In addition to using a complex formula to solve the roots of a quadratic, factoring can be connected to the shape of a polynomial, geometry, and computer programming. The thinking process and conclusion of this study hopefully will inspire middle school and high school students to view factoring in a different light.

Competition history

• JSHS 2022 Category not listed

Resources

• National JSHS abstract book ↗

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