To determine which response demonstrates that polynomials form a closed system under multiplication, we need to check whether the product of the polynomials listed in each response is also a polynomial.
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Response: \((x^2 + 1)(x - \frac{1}{2})\)
- Both factors are polynomials. The product will also be a polynomial.
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Response: \((x - \sqrt{x})(x + 1)\)
- The factor \(\sqrt{x}\) is not a polynomial (it's a root), so this does not demonstrate closure under multiplication.
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Response: \(x^2 + 2x + 1\)
- This is a polynomial by itself, but does not involve multiplication of two polynomials.
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Response: \(\left( \frac{x^2}{2} \right)(1x)\)
- This is a product of a polynomial \(\frac{x^2}{2}\) and \(x\) (which is also a polynomial). The product is a polynomial.
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Response: \(\left( \frac{1}{x} \right)\)
- \(\frac{1}{x}\) is not a polynomial, so this does not demonstrate closure under multiplication.
The responses demonstrating that polynomials form a closed system under multiplication are:
- (1) \((x^2 + 1)(x - \frac{1}{2})\)
- (4) \(\left( \frac{x^2}{2} \right)(1x)\)
Thus, the correct responses are (1) and (4). If you need to select just one, you can choose (1).