Adding and Subtracting Rational Expressions Examples

Find the LCD of the rational expressions .

For each expression, write the equivalent expression where the denominator is the LCD.

The first thing we do is factor the expressions:

Neither of these expressions can be simplified, so now we need to find the LCD of .

The LCD must contain the factors (x + 1) and (x – 1). Since the denominator of the second rational expression contains two copies of the factor (x – 1), the LCD must also contain two copies of the factor (x – 1). That's a lot of stuff, so hopefully it's got plenty of storage space.

The LCD is (x + 1)(x – 1)(x – 1).

To rewrite the rational expressions so that each has the LCD as its denominator, we multiply each expression by a clever form of 1. Better make it extra-clever. We think it may be onto us.

What is ?

First we factor, which lets us rewrite the problem as:

Then cancel the indicated factors:

If we hadn't canceled those factors, we'd be hating life right about now. The LCD of these expressions is:

x(x – 3)(x + 1)

Rewriting the expressions to have a common denominator gives us: