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+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

x2+2x+1

Answer 1

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

Let's evaluate the given responses:

1. Response 1: \((\sqrt{x})(x+1)\)

   • This expression includes \(\sqrt{x}\), which is not a polynomial since it is not expressed as a sum of non-negative integer powers of \(x\).

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

   • Both \(x^2 + 1\) and \(x - \frac{1}{2}\) are polynomials. The product of these two polynomials will also be a polynomial. Therefore, this response demonstrates closure under multiplication.

3. Response 3: \(\left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right)\)

   • This expression simplifies to \(\frac{x}{2}\), which is indeed a polynomial. However, involving \(\frac{1}{x}\) can cause confusion about closure, as one might argue it's not a polynomial in the standard form due to containing a term with a negative exponent.

4. Response 4: \(x^2 + 2x + 1\)

   • This is a polynomial, but it does not show multiplication between two polynomials.

The best choice to demonstrate that polynomials form a closed system under multiplication is Response 2: \((x^2 + 1)(x - \frac{1}{2})\), as the product of these polynomials remains a polynomial.