To simplify the rational expression [tex]\(\frac{x^2 - 121}{x + 11}\)[/tex], we can follow these steps:
1. Factor the numerator: The numerator [tex]\(x^2 - 121\)[/tex] can be factored using the difference of squares formula, which states that [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex]. In this case, we identify [tex]\(a = x\)[/tex] and [tex]\(b = 11\)[/tex], because [tex]\(121 = 11^2\)[/tex].
[tex]\[
x^2 - 121 = (x + 11)(x - 11)
\][/tex]
2. Rewrite the rational expression: Substitute the factored form of the numerator back into the expression.
[tex]\[
\frac{(x + 11)(x - 11)}{x + 11}
\][/tex]
3. Cancel the common factor: Since [tex]\(x\)[/tex] cannot be equal to [tex]\(-11\)[/tex] (otherwise the denominator would be zero, which is undefined), we can cancel out the common factor [tex]\((x + 11)\)[/tex] in the numerator and denominator.
[tex]\[
\frac{(x + 11)(x - 11)}{x + 11} = x - 11
\][/tex]
Therefore, the simplified form of the rational expression is [tex]\(x - 11\)[/tex].
Thus, the correct answer is:
C. [tex]\(x - 11\)[/tex]
