Reducing Rational Expressions of the Form

Sometimes we need to simplify an expression such as the following:
In the expression, notice that the numerator and denominator are the same except for the signs of the terms.
To reduce the expression, we first factor -1 out of the numerator.
Then we cancel common factors.		= -1

Note:

Notice that

Formula — To Simplify a Rational Expression of the Form

Here, a and b are real numbers and a ≠ b.

Example 1

Reduce to lowest terms:

Solution

Step 1 Factor the numerator and denominator. For the numerator, find two numbers whose product is -27 and whose sum is 6. These numbers are -3 and 9. The denominator is the difference of two squares: 32 - x2.
Step 2 Cancel all pairs of factors common to the numerator and denominator.
Since has the form it reduces to -1.

Thus,

Note:

The answer may be written in several ways.