Final answer:
Expressions are considered closed under an operation if performing that operation on any two expressions results in an expression of the same form. For the given expressions x^3 and 1, they are closed under addition and subtraction, but not under multiplication as dividing 1 by x^3 doesn't maintain the same form.
Step-by-step explanation:
To determine if expressions are open or closed for addition, subtraction, division, and multiplication, we need to understand if performing these operations on the expressions will result in an expression of the same form. For the expressions x^3, 1, and x^3, we observe the following:
- Addition or subtraction: Combining x^3 with 1 or another x^3 through addition or subtraction will still yield polynomial expressions, therefore it is closed for these operations.
- Multiplication: Multiplying any combination of the expressions x^3, 1, and x^3 results in a polynomial, so the set of expressions is closed for multiplication as well.
- Division: Dividing x^3 by 1 is closed, not changing the form of the expression. However, dividing 1 by x^3 results in a rational expression, which is different from the given expressions. So division is not closed.
Based on these observations, the correct answer would be B. Open for addition and subtraction only.