Which of the following responses shows that polynomials form a closed system under multiplication?

To determine if polynomials form a closed system under multiplication, we need to analyze if the product of two polynomials is also a polynomial.

1. **Response : \( (x)(\frac{1}{2}x) \)

- Here, we have \( x \) (which is a polynomial of degree 1) and \( \frac{1}{2}x \) (which is also a polynomial of degree 1).
- The product is:
\[
(x)(\frac{1}{2}x) = \frac{1}{2}x^2
\]
- \( \frac{1}{2}x^2 \) is a polynomial of degree 2.

Thus, Response A shows that the product of two polynomials is a polynomial.

2. **Response B**: \( (\sqrt{x})(6x \cdot -3) \)

- Here, \( \sqrt{x} \) is not a polynomial (it can be expressed as \( x^{1/2} \), which is not a non-negative integer power).
- The term \( 6x \cdot -3 \) simplifies to \( -18x \), which is a polynomial of degree 1.
- The product is:
\[
(\sqrt{x})(-18x) = -18x^{3/2}
\]
- The term \( -18x^{3/2} \) is also not a polynomial because \( 3/2 \) is not an integer.

Thus, Response B does not show that polynomials form a closed system under multiplication since the multiplication involves a term that is not a polynomial.

### Conclusion:
**Response is the correct choice as it demonstrates that polynomials form a closed system under multiplication.