To determine which response shows that polynomials form a closed system under multiplication, we need to identify which of the provided options results in a polynomial when multiplied.
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\( (x)(12x) \) = \( 12x^2 \) (a polynomial)
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\( \left(x\right) \left(\frac{1}{2} x\right) \) = \( \frac{1}{2} x^2 \) (a polynomial)
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\( \frac{3 x^2}{4} - 1 \) represents a polynomial, but it's not a multiplication operation.
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\( (x - \sqrt{})(6x \cdot -3) \) = ?
The product needs to be analyzed:
- \( x - \sqrt{} \) is not a polynomial due to the square root.
- Therefore, the multiplication of \( (x - \sqrt{}) \) with any polynomial will not yield a polynomial.
From the evaluation above, the responses that result in polynomials are:
- \( (x)(12x) \)
- \( \left(x\right) \left(\frac{1}{2} x\right) \)
Thus, both these options show that polynomials are closed under multiplication since the result of the products are polynomials.
To summarize: The responses that demonstrate the closure of polynomials under multiplication are:
- \( (x)(12x) \)
- \( \left(x\right) \left(\frac{1}{2} x\right) \)
However, if you need only one response, either of the first two will suffice as showing closure.
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