Question

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

(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 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

(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+2x+1

Answer 1

To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial.

Let's evaluate the given responses:

1. **Response 1**: \((\sqrt{x})(x+1)\)
- This expression includes \(\sqrt{x}\), which is not a polynomial since it is not expressed as a sum of non-negative integer powers of \(x\).

2. **Response 2**: \((x^2 + 1)(x - \frac{1}{2})\)
- Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. The product of these two polynomials will also be a polynomial. Therefore, this response demonstrates closure under multiplication.

3. **Response 3**: \(\left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right)\)
- This expression simplifies to \(\frac{x}{2}\), which is indeed a polynomial. However, involving \(\frac{1}{x}\) can cause confusion about closure, as one might argue it's not a polynomial in the standard form due to containing a term with a negative exponent.

4. **Response 4**: \(x^2 + 2x + 1\)
- This is a polynomial, but it does not show multiplication between two polynomials.

The best choice to demonstrate that polynomials form a closed system under multiplication is **Response 2: \((x^2 + 1)(x - \frac{1}{2})\)**, as the product of these polynomials remains a polynomial.