Partial fraction decomposition

partial fraction decomposition

1. Partial fraction decomposition

Partial fraction decomposition is the process of starting with a simplified
solution and reversing it by decomposing the final expression into its initial
polynomial fractions.

Partial fraction decomposition examples

1) Determine the partial fraction decomposition of



Step 1: Factorise the bottom expression




Step 2: Write a partial fraction for each of those factors with A1 and A2 since
the constants are unknown



Step 3: Cross-multiply the expressions with the letters A1 and A2



Step 4: Now, find the constants by substituting the roots of the expression

2. Then, we have our answer:









2) Determine the partial fraction decomposition of
??????
2
+4
3??????
3
+4??????
2
−4??????


Step 1: Factorise the denominator and split the fraction into partial fraction
decomposition

??????
2
+4
??????(??????+2)(3??????−2)
=
�
??????
+
�
??????+2
+
�
3??????−2


Step 2: Set the numerators equal by cross-multiplying with the bottom
expressions

??????
2
+4=�(??????+2)(3??????−2)+�??????(3??????−2)+�??????(??????+2)

Step 3: Find the constants by substituting the roots of the expression

??????=0 4=�(2)(−2) ⇒ �=−1

??????=−2 8 =�(−2)(−8) ⇒ �=
1
2


??????=
2
3

40
9
=�(
2
3
)(
8
3
) ⇒ �=
40
16
=
5
2



Then, we have our answer:
−1
??????
+
1
2(??????+2)
+
5
2(3??????−2)

3. Practice questions

Determine the partial fraction decomposition of each of the following:

1)
8??????−42
??????
2
+3??????−18



2)
9−9??????
2??????
2
+7??????−4



3)
9??????+25
(??????+3)
2









Solutions

1)
10
??????+6
−
2
??????−3



2)
1
2??????−1
−
5
??????+4



3)
9
??????+3
−
2
(??????+3)
2