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

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

(x22)(1x)

Answer 1

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

Let's analyze the provided responses:

1. **\(x^2 + 2x + 1\)** - This is a polynomial. However, this statement does not show multiplication and doesn't demonstrate closure.

2. **\((x^2 + 1)(x - \frac{1}{2})\)** - This expression represents the multiplication of two polynomials. When these are multiplied, the result will also be a polynomial. So, this response demonstrates that the set of polynomials is closed under multiplication.

3. **\((\sqrt{x})(x + 1)\)** - Here, \(\sqrt{x}\) can be rewritten as \(x^{1/2}\). Therefore, this expression represents the multiplication of a polynomial (\(x + 1\)) and a term that is not a polynomial (it has a fractional exponent). Thus, the product is not a polynomial, which does not demonstrate closure.

4. **\((x^2)(\frac{1}{x})\)** - In this case, multiplying \(x^2\) (a polynomial) by \(\frac{1}{x}\) results in \(x\), which is a polynomial. However, since \(\frac{1}{x}\) itself is not a polynomial, this example does not demonstrate closure of polynomials under multiplication.

Based on this analysis, the correct response that demonstrates that polynomials form a closed system under multiplication is:

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