Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)

To determine which of the responses shows that polynomials form a closed system under multiplication, we need to see if the result of the multiplication in each case produces a polynomial.

1. **\((x)(\frac{1}{2}x)\)**:
- Multiplying: \(x \cdot \frac{1}{2}x = \frac{1}{2}x^2\) (This is a polynomial.)

2. **\(\frac{1}{4}x^3 \cdot 5\)**:
- Multiplying: \(\frac{1}{4} \cdot 5 \cdot x^3 = \frac{5}{4}x^3\) (This is a polynomial.)

3. **\((\sqrt{x})(6x \cdot -3)\)**:
- Multiplying: \(\sqrt{x} \cdot (6x \cdot -3) = \sqrt{x} \cdot -18x = -18x^{3/2}\) (This is not a polynomial because of the fractional exponent \(3/2\).)

4. **\(\frac{3x^2}{4} - 1\)**:
- This expression itself is a polynomial, but it is not a multiplication showing closure.

The correct responses that show polynomials form a closed system under multiplication are:

1. \((x)(\frac{1}{2}x)\)
2. \(\frac{1}{4}x^3 \cdot 5\)

So the responses that show closure are the first and second options.