To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to find a response that involves the multiplication of two polynomials and shows that the result is also a polynomial.
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\((x^2 + 1)(x - \frac{1}{2})\): This expression is the product of two polynomials. The result will also be a polynomial. Thus, it demonstrates closure under multiplication.
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\(\sqrt{x}(x + 1)\): The term \(\sqrt{x}\) is not a polynomial because polynomials must have whole number exponents. Hence, this does not demonstrate closure under multiplication.
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\(x^2 + 2x + 1\): This is a polynomial, but it does not result from multiplying two separate polynomials in the expression given. Therefore, it doesn't specifically demonstrate closure.
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\((x^2)(\frac{1}{x})\): This expression results in \(x\), which is a polynomial, but \(\frac{1}{x}\) is not a polynomial. Therefore, it does not demonstrate closure.
The first option, \((x^2 + 1)(x - \frac{1}{2})\), is the correct response that demonstrates that polynomials form a closed system under multiplication.