Question

Whole numbers (positive integers and zero) are said to be a closed set under addition: if you add two whole numbers, you always get a whole number. Whole numbers are not a closed set under subtraction: if you subtract two whole numbers, you do not always get a whole number. For example, 2 - 5 = -3 and -3 is not a whole number. a. Investigate with your team whether the set of integers is a closed set under addition and under subtraction. Then investigate whether the integers are a closed set under multiplication and under division. Give examples. If you think the set is closed, explain why. If, not, give counterexamples. b. Are single-variable polynomials closed under addition, subtraction, and multiplication? In other words, if you add, subtract, or multiply two polynomials that have the same variable, will you always get a polynomial as your answer?

Answer 1

The set of integers is closed under addition, subtraction, and multiplication, but not division. Single-variable polynomials are closed under addition, subtraction, and multiplication, resulting in another polynomial after these operations.

Step-by-step explanation:

Investigating the Closure of Integers under Various Operations

The set of integers is indeed a closed set under addition and subtraction. No matter what two integers you add or subtract, the result will always be an integer. For example, 5 + (-3) = 2 and 7 - 10 = -3. This is due to the fundamental rules of addition and subtraction, which stay consistent regardless of the sign of the numbers involved.

Furthermore, integers are closed under multiplication. When two integers are multiplied, their product is always an integer, as demonstrated by examples such as (-2) × 4 = -8 and 6 × (-3) = -18.

However, integers are not closed under division, because dividing two integers does not always result in an integer. For instance, 1 / 2 is 0.5, which is not an integer. Thus, for division, integers are not a closed set.

Closure of Single-variable Polynomials under Operations

Single-variable polynomials are indeed closed under addition, subtraction, and multiplication. Adding, subtracting, or multiplying any two polynomials will result in another polynomial. Examples include:

• Addition: (x^2 + 2x + 1) + (3x^2 - x + 4) = 4x^2 + x + 5
• Subtraction: (2x^3 - x^2 + x - 7) - (x^3 + 4x^2 - 3x + 2) = x^3 - 5x^2 + 4x - 9
• Multiplication: (x - 1)(x + 2) = x^2 + x - 2

This closure is due to algebraic properties that dictate polynomial expressions remain polynomials after such operations.