Which of the following responses shows that polynomials form a closed system under multiplication? (1 point) () (1/2 * x); (3x ^ 3)/(4 - 1); 1/(4x ^ 3) * 5; (sqrt(x))(6x * (- 3))

To determine whether the given responses show that polynomials form a closed system under multiplication, we need to check if the results of the multiplications are polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, where the variables have non-negative integer exponents.

Let's evaluate each of the responses:

1. **(1/2 * x)**: This expression simplifies to \(0.5x\), which is a polynomial of degree 1.

2. **(3x^3)/(4 - 1)**: This simplifies to \((3x^3)/3 = x^3\), which is also a polynomial of degree 3.

3. **1/(4x^3) * 5**: This simplifies to \(5/(4x^3)\), which is not a polynomial because it involves a negative exponent (i.e., \(x^{-3}\)), meaning it’s a rational function rather than a polynomial.

4. **(sqrt(x))(6x * (-3))**: The term \(\sqrt{x}\) is equivalent to \(x^{1/2}\), and multiplying it by \(6x(-3)\) gives \(-18x^{3/2}\). This is also not a polynomial because \(x^{1/2}\) has a fractional exponent.

In summary, the only responses that yield polynomials are the first two. The last two do not yield polynomials. Therefore, polynomials do not form a closed system under multiplication in every case presented in the responses.

If I had to select an example that indicates closure under multiplication, I would pick the responses that show valid polynomial results:

- **(1/2 * x)**
- **(3x^3)/(4 - 1)**

are valid examples, and they demonstrate that polynomials are closed under multiplication for these specific cases. However, since not all provided options yield polynomials, one cannot definitively conclude that polynomials form a closed system under multiplication based on all the given responses.