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
x squared plus 2 x plus 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 analyze the given responses one by one:

1. (x−−√)(x+1):

   • Here, \( \sqrt{x} \) is not a polynomial; hence, the product is not a polynomial.

2. (x²+1)(x−12):

   • Both \( x^2 + 1 \) and \( x - 12 \) are polynomials. Their product will be \( x^3 - 12x^2 + x - 12 \), which is also a polynomial. This demonstrates closure.

3. (x²/2)(1/x):

   • The term \( 1/x \) is not a polynomial; thus, the product is not a polynomial.

4. x² + 2x + 1:

   • This expression is a polynomial but does not show multiplication between two separate polynomials.

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

(x²+1)(x−12).