Partial Fraction Decomposition

numerator coefficients:

n

0

1

n

1

x

5

n

2

x

0

n

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

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.