CK-12-Pre-Calculus Concepts

8.10. Partial Fractions http://www.ck12.org

5 =A+D

− 3 =− 2 A+B− 3 D+E

− 1 =A−B+C+ 3 D− 3 E

4 =−D+ 3 E

− 1 =E

This is a system of equations of five variables and 5 equations. Some of the equations can be solved using logic and
substitution likeE=− 1 ,D=− 7 ,A=12. You can use any method involving determinants or matrices. In this
case it is easiest to substitute known values into equations with one unknown value to get more known values and
repeat.

B= 1

C= 6

5 x^4 − 3 x^3 −x^2 + 4 x− 1

(x− 1 )^3 x^2 =

12

x− 1 +

1

(x− 1 )^2 +

6

(x− 1 )^3 +

− 7

x +

− 1

x^2

Example C
Use matrices to complete the partial fraction decomposition of the following rational expression.

2 x+ 4

(x− 1 )(x+ 3 )

Solution:

2 x+ 4

(x− 1 )(x+ 3 )=

A

x+ 1 +

B

x+ 3

2 x+ 4 =Ax+ 3 A+Bx+B

2 =A+B

4 = 3 A+B

[1 1

3 1

∣∣

∣∣^24

] →

→ − 3 ·I →

[ 1 1

0 − 2

∣∣

∣∣−^22

] →

→ ÷− 2 →

[1 1

0 1

∣∣

∣∣^21

]→ −II →

→

[1 0

0 1

∣∣

∣∣^11

]

A= 1 ,B= 1