Question

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

(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+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

x2+2x+1

Answer 1

To determine which response demonstrates that polynomials form a closed system under multiplication, we need to check whether multiplying the two expressions results in another polynomial.

1. **(x²)(1/x)**: This results in \(x^{2 - 1} = x\), which is a polynomial. So this product is valid.

2. **(x² + 1)(x - 1/2)**: When you expand this, you'll get a polynomial. The result will be \(x^{3/2} - \frac{1}{2}(x^2 + 1)\), which is not a polynomial because it has a fractional exponent.

3. **(√x)(x + 1)**: When you multiply this, you get \(x^{1/2}(x + 1) = x^{3/2} + x^{1/2}\), which is also not a polynomial because of the fractional exponents.

4. **x² + 2x + 1**: This is already a polynomial.

Given these evaluations, the one that clearly demonstrates multiplication within polynomials yielding another polynomial is:

- **(x²)(1/x)**

Thus, the answer that best demonstrates polynomials forming a closed system under multiplication is **(x²)(1/x)** (or simply x).