To show that polynomials form a closed system under multiplication, we need to demonstrate that multiplying two polynomials always results in another polynomial.
Let's analyze each of the provided responses:
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\( 3x^2/4 - 1 \): This is a polynomial.
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\( (x)(12x) \): When you multiply these, you get \( 12x^2 \), which is also a polynomial.
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\( (1/4)x^3 \cdot 5 \): This simplifies to \( (5/4)x^3 \), which is a polynomial.
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\( (\sqrt{x})(6x \cdot -3) \): The multiplication of \( \sqrt{x} \) (which is \( x^{1/2} \)) and \( 6(-3)x \) does not yield a polynomial because \( x^{1/2} \), the square root of x, is not an integer exponent.
Based on this analysis, the first three responses demonstrate closure under multiplication as they result in polynomials, while the fourth does not.
So, the responses that show polynomials form a closed system under multiplication are:
- \( (x)(12x) \)
- \( (1/4)x^3 \cdot 5 \)
However, only one response can be chosen in the context of your question, and the response that most clearly indicates closure under multiplication is \( (x)(12x) \).
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