To determine which of the provided responses demonstrates that polynomials form a closed system under multiplication, we need to check if the result of the multiplication of polynomials yields another polynomial.
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\(x^2 + 2x + 1\): This is a polynomial, but it does not demonstrate multiplication, so it's not applicable.
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\((x^2 + 1)(x - \frac{1}{2})\): This is a multiplication of two polynomials. The result will be \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\), which is also a polynomial. This does demonstrate that polynomials are closed under multiplication.
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\((\frac{x^2}{2})(\frac{1}{x})\): This involves multiplying a polynomial \(\frac{x^2}{2}\) by \(\frac{1}{x}\). The result is \(\frac{x^2}{2x} = \frac{x}{2}\), which is a polynomial, but also provides a fraction that resembles a rational function, and thus does not reinforce the concept of closure under polynomial multiplication in the traditional sense.
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\((x - \sqrt{x})(x + 1)\): This is also multiplication of two expressions. However, \((x - \sqrt{x})\) is not a polynomial (due to the \(\sqrt{x}\) term), and thus does not demonstrate closure under polynomial multiplication.
Based on this analysis, the response that demonstrates polynomials are closed under multiplication is:
(x^2 + 1)(x - \frac{1}{2})