To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to evaluate them to see if the result is still a polynomial.
A. \( \frac{3x^2}{4} - 1 \)
- This expression is a polynomial, but it does not represent multiplication, so it doesn't directly demonstrate closure under multiplication.
B. \( \frac{1}{4}x^3 \cdot 5 \)
- Multiplying the constant \( \frac{1}{4} \) by \( 5 \) gives \( \frac{5}{4}x^3 \), which is a polynomial. This demonstrates a multiplication operation resulting in a polynomial, thus showing closure.
C. \( (x)(\frac{1}{2}x) \)
- Multiplying \( x \) and \( \frac{1}{2}x \) gives \( \frac{1}{2}x^2 \), which is also a polynomial. This also demonstrates closure under multiplication.
Both B and C demonstrate that polynomials form a closed system under multiplication. However, if we're to choose just one from the provided options, both B and C would be correct responses for showing closure under multiplication.
If only one response must be selected and based on a typical structure of such questions, B would be the clearer demonstration because it explicitly involves multiplying two polynomial expressions (the first being \( x^3 \) and the second being a constant). Therefore, I would identify B as the primary response that directly shows closure under multiplication.