To simplify the rational expression [tex]\(\frac{x^2 - 9}{x - 3}\)[/tex], follow these steps:
1. Factor the numerator: Notice that the numerator [tex]\(x^2 - 9\)[/tex] is a difference of squares. The difference of squares can be factored using the identity [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. In this case, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex], so:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
2. Rewrite the expression: Substitute the factored form of the numerator into the rational expression:
[tex]\[
\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}
\][/tex]
3. Simplify the fraction: Now we can cancel the common factor [tex]\((x - 3)\)[/tex] in the numerator and the denominator. Note that this cancellation is valid as long as [tex]\(x \neq 3\)[/tex] since division by zero is undefined:
[tex]\[
\frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad \text{for} \quad x \neq 3
\][/tex]
Thus, the simplified form of the rational expression [tex]\(\frac{x^2 - 9}{x - 3}\)[/tex] is:
[tex]\[
x + 3
\][/tex]
This is the final simplified expression.
