Question

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

(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis

(x2+1)(x−12)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis

x2+2x+1
x squared plus 2 x plus 1

(x−−√)(x+1)

Answers

GPT-4o mini
To determine which of the provided responses demonstrates that polynomials form a closed system under multiplication, we need to check if the product of the polynomials in each response is itself a polynomial.

1. **Response:** \((x^2)(1x)\)
- Product: \(x^2 \cdot x = x^{3}\), which is a polynomial.

2. **Response:** \(\left( \frac{x^2}{2} \right) \left( \frac{1}{x} \right)\)
- Product: \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x^2}{2x} = \frac{x}{2}\), which is a polynomial.

3. **Response:** \((x^2 + 1)(x - \frac{1}{2})\)
- Product: \((x^2 + 1)(x) - (x^2 + 1) \cdot \frac{1}{2} = x^3 + x^2 - \frac{1}{2}x - \frac{1}{2}\), which is a polynomial.

4. **Response:** \(x^2 + 2x + 1\)
- This expression is already a polynomial.

5. **Response:** \((\sqrt{x})(x + 1)\)
- Product: \(\sqrt{x} \cdot (x + 1) = x^{\frac{3}{2}} + \sqrt{x}\). This is not a polynomial because of the term \(x^{\frac{3}{2}}\) and \(\sqrt{x}\) (neither are whole number powers of \(x\)).

Considering the above responses, the first four responses (1 to 4) demonstrate that they include polynomials and their product results in another polynomial.

Thus, any response except the fifth one, particularly Responses 1, 2, 3, and 4, demonstrate that polynomials form a closed system under multiplication.

If you are looking for a single response that shows this, one clear example is:

**The response: \((x^2 + 1)(x - \frac{1}{2})\)** shows that polynomials multiply to yield another polynomial, demonstrating closure under multiplication.