To determine which of the given options is equal to the rational expression [tex]\(\frac{x^2 - 9}{x + 3}\)[/tex] when [tex]\(x \neq -3\)[/tex], let's simplify the expression step-by-step.
1. Start with the given rational expression:
[tex]\[
\frac{x^2 - 9}{x + 3}
\][/tex]
2. Notice that the numerator, [tex]\(x^2 - 9\)[/tex], is a difference of squares. We can factor it as:
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
3. Substitute the factored form of the numerator back into the rational expression:
[tex]\[
\frac{(x + 3)(x - 3)}{x + 3}
\][/tex]
4. We can now cancel the common factor [tex]\((x + 3)\)[/tex] in the numerator and the denominator, provided [tex]\(x \neq -3\)[/tex] to avoid division by zero:
[tex]\[
\frac{(x + 3)(x - 3)}{x + 3} = x - 3
\][/tex]
Therefore, the simplified form of the rational expression [tex]\(\frac{x^2 - 9}{x + 3}\)[/tex] when [tex]\(x \neq -3\)[/tex] is:
[tex]\[
x - 3
\][/tex]
Looking at the given options:
A. [tex]\(x - 3\)[/tex]
B. [tex]\(\frac{1}{x - 3}\)[/tex]
C. [tex]\(x + 3\)[/tex]
D. [tex]\(\frac{x - 3}{x + 3}\)[/tex]
The correct answer is:
A. [tex]\(x - 3\)[/tex]
