To demonstrate that polynomials form a closed system under multiplication, we need to show that when two polynomials are multiplied together, the result is also a polynomial.
Letβs analyze each response:
1. **\((\sqrt{x})(x + 1)\)**: The square root of \(x\) is not a polynomial, so this product may not be a polynomial.
2. **\((x^2 + 1)(x - \frac{1}{2})\)**: Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. Their product will also be a polynomial. This demonstrates closure.
3. **\((\frac{x^2}{2})(\frac{1}{x})\)**: This product results in \( \frac{x}{2} \), which is a polynomial. However, it involves division by \(x\), which doesn't show closure under multiplication of polynomials specifically.
4. **\(x^2 + 2x + 1\)**: This is a polynomial, but it does not demonstrate a multiplication of two polynomials resulting in another polynomial.
From the analysis above, the response that demonstrates that polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - \frac{1}{2})\)**.