Let's simplify the rational expression:

$\frac{x^2 - 2x - 15}{x^2 - 9}$

Step 1: Factor the numerator and denominator.

• Numerator: $x^2 - 2x - 15$

  • To factor, we look for two numbers that multiply to $-15$ and add to $-2$. These numbers are $-5$ and $3$.
  • So, the numerator factors as: $x^2 - 2x - 15 = (x - 5)(x + 3)$

• Denominator: $x^2 - 9$

  • This is a difference of squares, which factors as: $x^2 - 9 = (x - 3)(x + 3)$

Step 2: Simplify the expression.

Now we substitute the factored forms back into the rational expression: $\frac{(x - 5)(x + 3)}{(x - 3)(x + 3)}$

• We can cancel the common factor of $(x + 3)$ from both the numerator and the denominator: $\frac{x - 5}{x - 3}$

Final Answer:

The simplified expression in lowest terms is: $\frac{x - 5}{x - 3}$

1. What is an equivalent expression in lowest terms?

$\frac{x - 5}{x - 3}$

Would you like more details on any part of the solution or have other questions about rational expressions?

Here are some additional questions to explore:

1. How do you recognize a difference of squares in a rational expression?
2. What other factoring techniques can be used for more complex polynomials?
3. Can you cancel terms in rational expressions involving addition or subtraction?
4. How does factoring help simplify rational expressions in algebra?
5. What is the domain of the simplified rational expression?

Tip: Always check for common factors in both the numerator and denominator before simplifying any rational expression.