Algebra II - NOTES 11.5 – Adding or Subtracting Rational Expressions
With LIKE Denominators
Step 1
Keep the Same Denominator, then add (or subtract) the numerators.
Step 2
Factor the numerator and denominator and reduce, if possible.
Example
x2
9


2x  6
2x  6


x2  9
2x  6
Step 1
 x  3 x  3
2  x  3
Step 2
x3
2
Step 2
With UNLIKE Denominators
Step 1
Factor all denominators to find the least common denominator. The least common denominator
is the product of all the factors that appear in either denominator-- (if a factor appears once in both
denominators, only include that factor once in the LCD, but if a factor occurs more than once in any
one denominator, it must be listed more than once in the LCD). Leave the least common denominator
in factored form.
Step 2
Write each rational expression as an equivalent rational expression with the LCD as the
denominator. Determine which factor(s) each denominator must be multiplied by in order to yield
the least common denominator--i.e. all the factors in the LCD that do not appear in each denominator.
Multiply numerator and denominator of each rational expression by the needed factor(s).
Step 3
Simplify the numerators, then add (or subtract).
Step 4
Factor the numerator and denominator and reduce, if possible.
Example
x2
x
 2

x  6x  8
x  3x  2
 x  4  x  2   x  1 x  2 
2
LCD   x  4  x  2  x  1

x2
x
 x 1 
 x4


 x  4  x  2   x  1   x  2  x  1  x  4 

x2  x  2
x2  4x

 x  4  x  2  x  1  x  4  x  2  x  1
2 x 2  3x  2

 x  4  x  2  x  1

 2 x  1 x  2   2 x  1  2 x  1
 x  4  x  2  x  1  x  4  x  1 x 2  5 x  4
Step 1
Step 2
Step 3
Step 4