Question

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

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

(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

(x−−√)(x+1)

Answer 1

To determine which of the provided responses demonstrates that polynomials form a closed system under multiplication, we need to check if the result of the multiplication of polynomials yields another polynomial.

1. **\(x^2 + 2x + 1\)**: This is a polynomial, but it does not demonstrate multiplication, so it's not applicable.

2. **\((x^2 + 1)(x - \frac{1}{2})\)**: This is a multiplication of two polynomials. The result will be \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\), which is also a polynomial. This does demonstrate that polynomials are closed under multiplication.

3. **\((\frac{x^2}{2})(\frac{1}{x})\)**: This involves multiplying a polynomial \(\frac{x^2}{2}\) by \(\frac{1}{x}\). The result is \(\frac{x^2}{2x} = \frac{x}{2}\), which is a polynomial, but also provides a fraction that resembles a rational function, and thus does not reinforce the concept of closure under polynomial multiplication in the traditional sense.

4. **\((x - \sqrt{x})(x + 1)\)**: This is also multiplication of two expressions. However, \((x - \sqrt{x})\) is not a polynomial (due to the \(\sqrt{x}\) term), and thus does not demonstrate closure under polynomial multiplication.

Based on this analysis, the response that demonstrates polynomials are closed under multiplication is:

**(x^2 + 1)(x - \frac{1}{2})**