To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to check if the product of the expressions in each option is a polynomial.
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A. \((x^2/2)(1/x)\): This simplifies to \((x^2/2) \cdot (1/x) = (1/2) x^{2-1} = (1/2)x\), which is a polynomial. However, the expression \(1/x\) is not a polynomial since polynomials can only have non-negative integer exponents.
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B. \(x^2 + 2x + 1\): This is itself a polynomial. However, it is not an expression demonstrating multiplication, but rather just a polynomial.
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C. \((x^2 + 1)(x - 1/2)\): This product, when expanded, will be \(x^2(x) - \frac{1}{2}x^2 + x - \frac{1}{2}\). Both factors are polynomials, and the product of two polynomials is always a polynomial. Thus, this shows closure under multiplication.
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D. \((√x)(x + 1)\): This can be rewritten as \((x^{1/2})(x + 1)\). Since \(√x = x^{1/2}\) is not a polynomial (because it has a non-integer exponent), this does not demonstrate closure under multiplication of polynomials.
From this analysis, the correct response that demonstrates that polynomials form a closed system under multiplication is:
C. \((x^2 + 1)(x - 1/2)\).
1 answer