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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
The result is -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.