#### Date of Award

12-2021

#### Document Type

Thesis

#### Department

Mathematics

#### First Advisor

Dr. Jing Zhang

#### Second Advisor

Dr. Andrius Tamulis

#### Third Advisor

Lauren Ryba

#### Abstract

Recently, many mathematics educators have tried to explore the potential for aspects of abstract algebra to improve the teaching of high school algebra. As a high school math teacher, I am specifically interested in how I can tie the ring theory in abstract algebra into a high school level algebra course. More precisely, this thesis concerns the possible application of the irreducibility of polynomials (with integer coefficients over rational field in high school algebra. We will explore the differences between a reducible polynomial and an irreducible polynomial. In detail, concepts such as the following will be explored and addressed: the comparison of the ring of integers and the rings of polynomials over (or), including the Division Theorem, Fundamental Theorem of Arithmetic, Factorization Theorem for Polynomials, and long division. Secondly, we will discuss how to verify a polynomial with integer coefficients is irreducible over, such as Eisenstein’s Criterion. In the final section, we will include several strategies and lesson proposals to introduce these topics at a secondary level tied in with appropriate common core state standards.

#### Recommended Citation

Nowobilski, Gabriella, "Integers and Polynomials with Integer Coefficients for High School Students" (2021). *Mathematics Capstone Projects*. 1.

https://opus.govst.edu/capstones_math/1