To determine which response shows that polynomials form a closed system under multiplication, we need to recognize that multiplying two polynomials always produces another polynomial.
Let’s analyze the provided options:
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5x−1⋅2: This represents the multiplication of a polynomial \(5x - 1\) and a constant (which can also be considered a polynomial of degree 0). The result will be another polynomial. Hence, this shows closure.
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2x³ + x²: This is already a polynomial and doesn't represent multiplication.
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(1/3)(x⁴ - 2): Here, \(1/3\) is a constant, and multiplying it by the polynomial \(x^4 - 2\) results in another polynomial. Thus, this shows closure as well.
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5⋅3x: This also represents multiplication of constants, resulting in a constant polynomial. Therefore, it also shows closure.
In summary, options 5x−1⋅2, (1/3)(x⁴ - 2), and 5⋅3x all demonstrate that polynomials form a closed system under multiplication. However, only the first and third explicitly show a polynomial multiplied by another term, while the second is simply a polynomial, and the fourth is a product of constants.
Thus, the responses 5x−1⋅2 and (1/3)(x⁴ - 2) are proper representations of closure under multiplication for polynomials. If you want to select one, the response of 5x−1⋅2 directly illustrates this with two polynomial-like components.