Partial fractions with a linear term on the denominator

1. Linear term on the denominator

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

3. To express
8x 42
2
x 3x 18
as partial fractions.

4. First factorise the denominator:
8x 42
(x 6)( x 3)

5. Express as individual fractions with the factors on the denominator:
A B
x 6 x 3

6. Express with a common denominator:
A( x 3) B(x 6)
(x 6)( x 3)

7. There are two methods to proceed from here.

8. Comparing coefficients

9. 8x 42 A( x 3) B(x 6)
2
x 3x 18 (x 6)( x 3)
(A B)x 3A 6B
(x 6)( x 3)

10. Looking at the numerators:
8x 42 (A B)x 3A 6B
Comparing the coefficients:
A+B =8
-3A + 6B = -42
Solving gives A = 10 and B = -2

11. Hence:
8x 42 10 2
2
x 3x 18 (x 6) (x 3)

12. Substitution

13. 8x 42 A( x 3) B(x 6)
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 = 9B
B = -2
x = -6
-48 – 42 = -9A
A = 10

14. Hence:
8x 42 10 2
2
x 3x 18 (x 6) (x 3)