Which of the following equations is an example showing that polynomials form a closed system under multiplication?(1 point)

Question

Responses

(x22)(1x)=x2
open paren the fraction with numerator x squared and denominator 2 close paren times open paren 1 over x close paren is equal to x over 2

x2+2x+1=x2+x+x+1
x squared plus 2 x plus 1 is equal to x squared plus x plus x plus 1

(x−−√)(x+1)=xx−−√+x−−√
open paren square root of x close paren times open paren x plus 1 close paren is equal to x square root of x plus square root of x

(x2+1)(x−12)=x3−12x2+x−12

Answer 1

To determine which of the given equations shows that polynomials form a closed system under multiplication, we need to find an equation where the product of two polynomials yields another polynomial.

1. \((\frac{x^2}{2})(\frac{1}{x}) = \frac{x}{2}\)
   This is not a polynomial because \(\frac{1}{x}\) is not a polynomial.

2. \(x^2 + 2x + 1 = x^2 + x + x + 1\)
   This is an identity but does not represent a multiplication operation.

3. \((\sqrt{x})(x+1) = x\sqrt{x} + \sqrt{x}\)
   This equation involves the square root of \(x\), which is not a polynomial.

4. \((x^2 + 1)(x - 12) = x^3 - 12x^2 + x - 12\)
   This equation represents the multiplication of two polynomials \( (x^2 + 1) \) and \( (x - 12) \), and the product is \( (x^3 - 12x^2 + x - 12) \), which is also a polynomial.

Thus, the correct response is:

\((x^2 + 1)(x - 12) = x^3 - 12x^2 + x - 12\)