To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomials is also a polynomial.
Let's analyze the provided responses:
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2x³ + x²: This is a polynomial, so it is valid, but it doesn't explicitly show multiplication.
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5x - 1 ⋅ 2: This represents multiplying the polynomial \(5x - 1\) by the constant \(2\). The product will still be a polynomial \((10x - 2)\).
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(1/3)(x⁴ - 2): This expression shows multiplying a polynomial \(x⁴ - 2\) by the constant \(1/3\). The result is \((1/3)x⁴ - (2/3)\), which is still a polynomial.
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5 ⋅ 3x: This shows multiplying the constant \(5\) by the polynomial \(3x\). The result is \(15x\), which is also a polynomial.
From the responses, the second and third responses explicitly show a multiplication involving polynomials and confirm that the product is a polynomial.
However, none of the options alone clearly illustrate the idea that the product remains within the set of polynomials. To clearly demonstrate closure, you would need explicit multiplication between two nonlinear polynomials creating another polynomial.
Thus, any of the responses reflecting multiplication of a polynomial will suffice to show that polynomials are closed under multiplication, but option 2 or 3 more directly asserts this concept through multiplication.