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

Question

Responses

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

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)

Answer 1

To demonstrate that polynomials form a closed system under multiplication, we need to show that multiplying two polynomials results in another polynomial.

Let's analyze each response:

1. \((x−−√)(x+1)\):

   • This involves a square root, which is not a polynomial. Thus, this does not demonstrate closure under multiplication of polynomials.

2. \(x^2 + 2x + 1\):

   • This is a polynomial, but it does not result from multiplying two other polynomials in this context, so it does not demonstrate closure.

3. \((x^2 + 1)(x−\frac{1}{2})\):

   • This is the multiplication of two polynomials: \(x^2 + 1\) and \(x - \frac{1}{2}\). The product will be another polynomial, demonstrating closure.

4. \((\frac{x^2}{2})(\frac{1}{x})\):

   • This expression involves rational functions (terms like \(\frac{1}{x}\)), which are not polynomials. Thus, this does not demonstrate closure under polynomial multiplication.

Conclusion: The response that demonstrates that polynomials form a closed system under multiplication is:

\((x^2 + 1)(x−\frac{1}{2})\).