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

1 answer

To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply any two polynomials, the result is also a polynomial.

Among the given options:

  1. (x1/2)(x+1) - This contains x1/2, which is not a polynomial (it's a root).

  2. (x2+1)(x12) - Both x2+1 and x12 are polynomials. Their product will also be a polynomial.

  3. (x22)(1x) - Here, while the terms involved are derived from polynomial expressions, the product simplifies to x2, which is a polynomial.

  4. x2+2x+1 - This is a polynomial by itself, but it doesn't demonstrate closure with another polynomial since it is a standalone expression.

The response that demonstrates closure under multiplication is:

(x2+1)(x12)

This indicates that the product of the two polynomials is also a polynomial, thereby establishing that polynomials form a closed system under multiplication.