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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: