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 on to us.
First we factor, which lets us rewrite the problem as
and cancel the indicated factors to find
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