Cypress College Math Review: Partial Fraction Decomposition
 Partial fraction decomposition is a method of breaking up a complicated rational
expression into the sum of simpler ones.
 First factor the denominator of the rational expression.
 Based on the chart below determine what terms will be in your decomposition.
Factor in denominator:
ax  b
first power of a linear
 ax  b 
m
mth power of a linear
ax 2  bx  c
irreducible quadratic (has no
real zeros)
 ax
2
 bx  c

m
mth power of an irreducible
quadratic
Example:
3x
x2  x  2
CCMR Partial Fraction Decomposition
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Term(s) in decomposition:
A
ax  b
A3
A1
A2



2
ax  b  ax  b   ax  b 3

Am
 ax  b 
m
Ax  B
ax  bx  c
2
A3 x  B3
A1 x  B1
A2 x  B2



2
3
2
2
ax  bx  c ax  bx  c
ax 2  bx  c

 


Am x  Bm
 ax
2
 bx  c

m
 Before breaking up a rational expression using partial fraction decomposition you
must first make sure that the expression is proper.
 A proper rational expression is one in which the degree of the numerator is less
than the degree of the denominator.
 If the rational expression is improper, then divide first. Use partial fraction
decomposition to break apart the rational expression that you are left with.
x3  x 2  3
Example: 2
x  3x  4
CCMR Partial Fraction Decomposition
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Example:
Example:
x2  4 x  7
 x  1  x 2  2 x  3
x3
 x  2  x  12
CCMR Partial Fraction Decomposition
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Extra Practice – Try these on your own, then check with the answers below.
5
x2  x  6
x6
2.
2 x3  8 x
x3  x  3
3. 2
x  x2
2x  3
4.
 x  12
1.
Answers
1.
2.
1
1

x3 x2
3
3x  2

4 x 4 x2  4

3. x  1 
4.

or
1
1

x  2 x 1
1
2

x  1  x  12
CCMR Partial Fraction Decomposition
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3
1
3
x
4
2  4
2
x
x 4