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Adding and Subtracting Rational Expressions with Different Denominators
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Adding and Subtracting Rational Expressions with Different Denominators

In the next example we must first factor polynomials to find the LCD.


Example 1

Different denominators

Perform the indicated operations.


a) Because x2 - 1 = (x + 1)(x + 1) and x2 + x = x(x + 1), the LCD is x(x - 1)(x + 1). The first denominator is missing the factor x, and the second denominator is missing the factor x - 1.

The LCD is x(x - 1)(x + 1).
  Build up the denominators to the LCD.
  Add the numerators.

For this type of answer we usually leave the denominator in factored form. That way, if we need to work with the expression further, we do not have to factor the denominator again.

b) Because -1(2 - a) = a - 2, we can convert the denominator 2 - a to a - 2.

  The LCD is a - 2.
  Subtract the numerators.

Note that if we had changed the denominator of the first expression to 2 - a, we would have gotten the answer

but this rational expression is equivalent to the first answer.


Helpful hint

It is not actually necessary to identify the LCD. Once the denominators are factored, simply look at each denominator and ask,“What factor does the other denominator have that is missing from this one?” Then use the missing factor to build up the denominator and you will obtain the LCD.

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