Linear term on the denominator

Expressions with ax + b in the denominator decompose to terms of the form:

A

ax b

2

8 42

3 18

x

x x

To express

as partial fractions.

8 42

( 6)( 3)

x

x x

First factorise the denominator:

6 3

A B

x x

Express as individual fractions with the factors on the denominator:

( 3) ( 6)

( 6)( 3)

A x B x

x x

Express with a common denominator:

There are two methods to proceed from here.

Comparing coefficients

2

8 42 ( 3) ( 6)

3 18 ( 6)( 3)

x A x B x

x x x x

( ) 3 6

( 6)( 3)

A B x A B

x x

8 42 ( ) 3 6x A B x A BComparing the coefficients:

A + B = 8-3A + 6B = -42

Solving gives A = 10 and B = -2

Looking at the numerators:

2

8 42 10 2

3 18 ( 6) ( 3)

x

x x x x

Hence:

Substitution

8 42 ( 3) ( 6)x A x B x

Any value of x can be substituted, but in this case x = 3 and then x = -6, for obvious reasons, are best.

x = 3

24 – 42 = 9BB = -2

x = -6

-48 – 42 = -9AA = 10

2

8 42 10 2

3 18 ( 6) ( 3)

x

x x x x

Hence: