Partial fractions with a linear term to a power on the denominator

1. Linear term to a power on the denominator

2. Expressions with (ax + b)k in the denominator
decompose to terms of the form:
A B C
.....
2 3
ax b ( ax b) ( ax b)

3. To express
8x 42
3
(x 3)
as partial fractions.

4. Express as individual fractions with the factors on the denominator:
A B
2
(x 3) x 3

5. Express with a common denominator:
A B(x 3)
2
(x 3)

6. Substitution

7. 8x 42 A B(x 3)
Substitute x = 3
24 – 42 = A
A = -18
Now substitute x = 0
-42 = -18 - 3B
B=8

8. Hence:
8x 42 8 18
2 2
(x 3) (x 3) (x 3)