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.

Note that Algebra Solver can easily do all kinds of problems with rational expressions that you enter. Click here for a sample screenshot.

(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 factors”.

[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.