Rings: Polynomials and factorization

Abstract

We begin with the properties of polynomials found in school algebra and then move in Section 5.2 to a relatively formal construction of the polynomial ring. Particularly interesting to students ought to be the altered properties of polynomials when the ring of coefficients is not a field. Section 5.3 deals with polynomial functions arising from polynomials emphasizing the distinction between these two concepts. Although this is a small point to a professional mathematician, an understanding of such distinctions helps mature the student of mathematics. The matters at stake in the factor theorem are then clearer. We are then led to the important results counting the number of roots of a polynomial and the concept of multiplicity of roots.