To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to ensure that the product of two polynomial expressions results in another polynomial.
Let's analyze each of the responses:
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(√x)(x+1):
- Here, √x is not a polynomial, as it contains a fractional exponent (1/2). Therefore, their product is not guaranteed to be a polynomial.
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(x^2/2)(1x):
- This simplifies to (1/2)x^3, which is a polynomial.
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(x^2+1)(x−1/2):
- The first expression, x^2 + 1, is a polynomial, and x - 1/2 is also a polynomial. The product of two polynomials is also a polynomial.
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x^2 + 2x + 1:
- This is already a polynomial, but it is not a product of polynomials, so it doesn't demonstrate the closure property under multiplication.
Based on this reasoning, the responses that demonstrate that polynomials form a closed system under multiplication are:
- (x^2/2)(1x)
- (x^2+1)(x−1/2)
If you need a single option that best demonstrates the closure property, it would be (x^2+1)(x−1/2), as it explicitly shows the multiplication of two polynomials resulting in another polynomial.