Properties of Integers and Polynomials Closure Operations

Published on May 26, 2026

Algebra

Determine if operations on integers and polynomials always, sometimes, or never result in the same type of number or expression. Covers closure properties for addition, subtraction, and multiplication.

Question

Answer

To solve this problem, we need to evaluate each mathematical statement based on the properties of integers and polynomials.

Part 1: Integers

The set of integers (

) is closed under addition, subtraction, and multiplication. This means that performing these operations on any two integers will always result in another integer.

An integer subtracted from an integer is an integer.
- If , then .
- Answer: always
An integer multiplied by an integer is an integer.
- If , then .
- Answer: always
An integer added to an integer is an integer.
- If , then .
- Answer: always

Part 2: Polynomials

The set of polynomials is closed under addition, subtraction, and multiplication. However, it is not closed under division.

A polynomial subtracted from a polynomial is a polynomial.
- If and are polynomials, then is also a polynomial.
- Answer: always
A polynomial divided by a polynomial is a polynomial.
- If we divide by , we get , which is a rational function, not a polynomial (because it has a negative exponent).
- However, if we divide by , we get , which is a polynomial.
- Since it is true in some cases but not all, it is “sometimes” true.
- Answer: sometimes
A polynomial multiplied by a polynomial is a polynomial.
- If and are polynomials, then is also a polynomial.
- Answer: always