To determine which response demonstrates that polynomials form a closed system under multiplication, we need to check which response results in a polynomial after performing the multiplication. A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers with coefficients.
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(13)(x⁴−2): This expression involves multiplying a constant (13) by a polynomial (x⁴−2). The result, \( 13x^4 - 26 \), is a polynomial.
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(1/3)(x/4 - 2): Here, we are multiplying a constant (1/3) by another polynomial (x/4 - 2). The result will be \( \frac{1}{12}x - \frac{2}{3} \), which is also a polynomial.
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5⋅3x: This results in \( 15x \), which is a polynomial as it has a variable raised to the first power.
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5x^{-1}⋅2: This results in \( \frac{10}{x} \) or \( 10x^{-1} \), which includes the variable raised to a negative power, making it not a polynomial.
From the analysis, the first three responses keep the product as a polynomial, while the fourth does not. Therefore, the responses that show polynomials form a closed system under multiplication are:
- (13)(x⁴−2)
- (1/3)(x/4 - 2)
- 5⋅3x
The response showing closure under multiplication specifically is (13)(x⁴−2) because it clearly highlights multiplying a polynomial by a constant.