Partial Fraction Decomposition
Lesson · Intermediate
Algebra: Expressions & Identities
Partial Fraction Decomposition rewrites a rational expression as a sum of simpler fractions.
Two Ways to Calculate the Numerators
Approach 1: Comparing Coefficients
Approach 2: Heaviside Cover-Up Method
Multiply both sides by :
Expand:
Compare coefficients:
Therefore
Multiply both sides by :
Substitute :
Similarly, multiply both sides by :
Substitute :
Decompose
Using the cover-up method
Therefore
If the denominator contains
include every power:
Decompose
For
because is a repeated linear factor, decompose it as
Multiply by :
Set :
so
Set :
so
Now compare the coefficients:
so
Therefore
For a quadratic factor such as
the numerator must be linear:
Decompose
Multiply both sides by :
Expand:
So
Compare coefficients:
From
Then
so
and therefore
Hence