Integer roots of quadratic and cubic polynomials with integer coefficients

TL;DR

This paper characterizes when quadratic and cubic polynomials with integer coefficients have all roots as integers, providing explicit coefficient conditions and describing families of such polynomials, mainly for educational purposes.

Contribution

It offers new theorems with explicit coefficient conditions for quadratic and cubic polynomials to have integer roots, including families with specific root multiplicities.

Findings

Theorem3 provides conditions for quadratic trinomials to have two integer roots.

Theorem4 describes cubic polynomials with a double or single integer root.

Theorem5 characterizes quadratic trinomials with two integer roots parameterized by odd positive integers.

Abstract

The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of mathematics,grades K12 to K20.The results of this work are expressed in Theorems3,4,and5. Of these theorems, Theorem3, is the one that most likely, the general reader of this article will have some familiarity with.In Theorem3, precise coefficient conditions are given;in order that a quadratic trinomial(with integer) have two integer roots or zeros.On the other hand, Theorems4 and5 are largely unfamiliar territory. In Theorem4, precise coefficient conditions are stated; for a monic cubic polynomial to have a double(i.e.of multiplicity 2) integer root, and a single integer root(i.e.of multiplicity 1).The entire family of such cubics can be described in terms of four groups or subfamilies; each such group being a two-integer parameter subfamily. In Theorem5, a one-integer parameter family of quadratic trinomials(with integer coefficients) with two integer roots; is described.The parameter can take any odd positive integer values.