Which of the following responses demonstrates that polynomials form a closed system under multiplication ? (1 point ) (x ^ 2 + 1)(x - 1/2); (sqrt(x))(x + 1); x ^ 2 + 2x + 1; ((x ^ 2)/2)(1/x)

To determine which of the responses demonstrates that polynomials form a closed system under multiplication, we should remember that the product of two polynomials must also be a polynomial in order for the system to be considered closed under that operation.

Let's analyze each option:

1. **(x^2 + 1)(x - 1/2)**: This is the product of two polynomials. The result will be a polynomial: \( (x^2 + 1)(x - 1/2) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \).

2. **(sqrt(x))(x + 1)**: The term \( \sqrt{x} \) is not a polynomial; it is a radical expression. Therefore, this does not demonstrate closure under multiplication of polynomials.

3. **x^2 + 2x + 1**: This is a polynomial by itself but does not demonstrate closure since it does not involve a multiplication of two polynomials.

4. **((x^2)/2)(1/x)**: The term \( 1/x \) is not a polynomial; it is a rational function. Thus, this option does not demonstrate closure under multiplication of polynomials.

Only option 1, **(x^2 + 1)(x - 1/2)**, demonstrates that polynomials form a closed system under multiplication since it results in another polynomial.

Therefore, the correct response is:
**(x^2 + 1)(x - 1/2)**.