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Partial Fraction Decomposition

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numerator coefficients:
n
0
1
n
1
x
5
n
2
2
x
0
n
3
3
x
0
number of denominator roots:
d
4
denominator roots:
d
1
1
d
2
-2
d
3
0
d
4
0
The fraction
5x+1
(x-1)
2
x
(x+2)
can be decomposed into
-
1
2
2
x
-
11
4x
+
3
4(x+2)
+
2
x-1
Many rational functions can be expressed as a sum of simpler fractions. For example,
x+1
(x-3)(x-1)
can be expressed as the sum
A
x-3
+
B
x-1
. To find A and B, Heaviside's method can be used. First, multiply the original fraction by
(x-1)
, cancel, and substitute 1 for
x
:
x+1
(x-3)(x-1)
(x-1)=
x+1
x-3
⇒A=
1+1
1-3
=-1
. Similarly,
x+1
(x-3)(x-1)
(x-3)=
x+1
x-1
⇒B=
3+1
3-1
=2
. Thus,
x+1
(x-3)(x-1)
=
2
x-3
+
-1
x-1
. For more complicated fractions it is best to set up and solve a linear system of equations. There are many methods for partial fraction decomposition; all of them are performed by the Mathematica function Apart.
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