To demonstrate that polynomials form a closed system under multiplication, we need to show that multiplying two polynomials together results in another polynomial.
Let's examine each response:
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(x)(12x): This is a multiplication of two polynomials (x and 12x), resulting in \(12x^2\), which is a polynomial. This response shows closure.
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(x)(\(\frac{1}{2}x\)): This also results in \(\frac{1}{2}x^2\), which is a polynomial. This response shows closure.
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\(14x^3 \cdot 5 \cdot \frac{1}{4}x^3\): This simplifies to \( \frac{14 \cdot 5}{4} x^6\) or \(\frac{70}{4} x^6\), which is \(17.5 x^6\), a polynomial. This response shows closure.
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\(3x^2 \cdot 4 - 1\): This represents a polynomial \(12x^2 - 1\), which is a polynomial. This response shows closure as well.
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\((\sqrt{x})(6x \cdot -3)\): The term \(\sqrt{x}\) is not a polynomial; therefore, this combination does not show closure, as \(\sqrt{x}\) is not a polynomial.
The responses that show polynomials form a closed system under multiplication are:
- (x)(12x)
- (x)(\(\frac{1}{2}x\))
- \(14x^3 \cdot 5 \cdot \frac{1}{4}x^3\)
- \(3x^2 \cdot 4 - 1\)
The response containing \((\sqrt{x})(6x \cdot -3)\) does not demonstrate closure because \(\sqrt{x}\) is not a polynomial.