Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point)

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:

1. **(√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.

2. **(x^2/2)(1x)**:
- This simplifies to (1/2)x^3, which is a polynomial.

3. **(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.

4. **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.