To determine which response shows that polynomials form a closed system under multiplication, we need to examine each option to see if multiplying the expressions results in another polynomial.
-
\((\sqrt{x})(6x \cdot -3)\):
- \(\sqrt{x}\) is not a polynomial (it is a radical), so the product \((\sqrt{x})(-18x)\) is not a polynomial.
- Not closed under multiplication.
-
\( \frac{1}{4}x^3 \cdot 5\):
- This simplifies to \(\frac{5}{4}x^3\), which is a polynomial.
- Closed under multiplication.
-
\((x)(12x)\):
- This simplifies to \(12x^2\), which is a polynomial.
- Closed under multiplication.
-
\(\frac{3x^2}{4} - 1\):
- This expression is a polynomial (specifically, a polynomial with a constant term). However, it describes an expression rather than a product.
- Not a multiplication result.
From the analysis, the responses that demonstrate closure under multiplication are \(\frac{1}{4}x^3 \cdot 5\) and \((x)(12x)\).
The clear acknowledgment of a closed system under multiplication is shown in:
- \( \frac{1}{4}x^3 \cdot 5 \): resulting in a polynomial.
- \((x)(12x)\): resulting in a polynomial.
Thus, the answers indicating closure under multiplication are the second and third responses. The first and fourth do not demonstrate that polynomials are closed under multiplication.