Adding and Subtracting Rational Expressions
1. To add rational expressions you must have common denominators, then you add
the numerators. [The result must be checked to see if it will reduce to simpler form.]
Examples:
1.
2.
 Factor and simplify:
2. If the rational expressions do not have common denominators you must factor the
denominators and find the least common denominator (LCD).
Raise all fractions, so that they have common denominators, then
add the numerators.
NOTE: With different denominators find the LCD and add using one of two methods:
(1) raise each with LCD or (2) write LCD and find missing factor
for each numerator.
(1)
(2) Write the LCD then start with the first numerator – compare its denominator with
the LCD and multiply by its missing factor. Write the next sign and
next numerator
– compare its denominator and multiply by its missing factor.
(2)
Definitions and Notes
If necessary, factor the denominators of the examples in order to find the
LCD. Leave the
denominator of the results in factored form. [Always check and/or factor
the numerator of the results in
order to reduce the answer to simplest form.]
Method (1): Raise each fraction to have LCD = (x + 3)(x – 3)(x – 2)
[Do not cancel.] Remove parentheses and combine numerators. [Watch negative signs.]
To subtract
add the opposite or Negate second numerator: (-1)(2)(x+3) = –2x – 6.
Method (2): Compare each factored denominator with LCD = (x + 3)(x – 3)(x – 2)
 and find “missing factorsâ€.
[NOTE: “mf†means “missing factor†]
Remove parentheses and collect terms: [Watch negative signs.]
 [Leave in factored form.]
Check: Let x = 5 (unique prime) in the given example and in the answer:
Always check results by either method to see if the rational expression can be
reduced.
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