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

Question

(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 results in another polynomial.

Let's analyze the given options:

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

   • This is the product of \(\sqrt{x}\) (which is not a polynomial) and \((x + 1)\) (which is a polynomial). The product is not a polynomial, so this does not demonstrate closure.

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, so this demonstrates closure.

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

   • \(\frac{x^{2}}{2}\) is a polynomial, but \(\frac{1}{x}\) is not (as it is a rational expression and not polynomial form). The product will not result in a polynomial, so this does not demonstrate closure.

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

   • This is simply a polynomial but does not show the multiplication of two polynomials, so it does not demonstrate closure in the context requested.

From this analysis, the only response that demonstrates that polynomials form a closed system under multiplication is:

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