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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.