To determine which response shows that polynomials form a closed system under multiplication, we need to look for an expression where multiplying polynomials results in another polynomial.
Let's examine each response:
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\((\sqrt{x})(6x \cdot -3)\): This involves a square root, which is not a polynomial. Thus, this does not show closure.
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\(\frac{3x^2}{4} - 1\): This expression is itself a polynomial, but it doesn't represent multiplication of polynomials.
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\((x)(12x)\): This is the multiplication of two polynomials (since \(x\) is a polynomial and \(12x\) is also a polynomial). The product, \(12x^2\), is also a polynomial. This demonstrates that polynomials are closed under multiplication.
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\(14x^3 \cdot 5\): This is the multiplication of a polynomial \(14x^3\) and a constant \(5\), resulting in \(70x^3\), which is also a polynomial. This also demonstrates closure.
Both options 3 and 4 show closure under multiplication.
Final answer:
- \((x)(12x)\) (Option 3) demonstrates that polynomials form a closed system under multiplication.