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

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

(x2/2)(1/x)

Answer 1

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

Let's examine the provided responses:

1. **(√x)(x + 1)**: This is not a polynomial because √x is not a polynomial; therefore, this response does not show closure.

2. **(x² + 1)(x - 1/2)**: Both x² + 1 and x - 1/2 are polynomials. Their product will also be a polynomial, hence this demonstrates closure.

3. **x² + 2x + 1**: This is a polynomial but doesn't demonstrate closure under multiplication by itself—it represents one specific polynomial.

4. **(x²/2)(1/x)**: Here, 1/x is not a polynomial. Therefore, this multiplication does not show closure.

The correct response that demonstrates that polynomials form a closed system under multiplication is **(x² + 1)(x - 1/2)**.