Skill: Simplify rational expressions by factoring

To simplify a rational expression by factoring:

1. Find restrictions by setting the original denominator equal to zero.
2. Factor the numerator and denominator completely.
3. Cancel common factors—not individual terms separated by addition or subtraction.
4. Write the simplified expression and keep the original restrictions.

Example:

$$\frac{x^2-9}{x^2+x-6}$$

Step 1: Find the restrictions

The denominator cannot equal zero:

$$x^2+x-6\neq 0$$

Factor the denominator:

$$x^2+x-6=(x+3)(x-2)$$

Therefore,

$$x\neq -3 \quad \text{and} \quad x\neq 2$$

These restrictions come from the original denominator.

Step 2: Factor the numerator and denominator

The numerator is a difference of squares:

$$x^2-9=(x-3)(x+3)$$

So the expression becomes

$$\frac{(x-3)(x+3)}{(x+3)(x-2)}$$

Step 3: Cancel common factors

The factor $(x+3)$ appears in both the numerator and denominator, so it can be canceled:

$$\frac{\cancel{(x-3)}(x+3)}{\cancel{(x+3)}(x-2)} = \frac{x-3}{x-2}$$

The simplified expression is

$$\boxed{\frac{x-3}{x-2}}, \qquad x\neq -3,\ 2$$

Remember that you can cancel factors, such as $(x+3)$, but not terms from a sum or difference. For example, you cannot cancel the $x$ in $\frac{x+3}{x+2}$.