CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 8. Systems and Matrices

2 x+ 4

(x− 1 )(x+ 3 )=

1

x+ 1 +

1

x+ 3

Concept Problem Revisited
To decompose the rational expression into the sum of two simpler fractions you need to use partial fraction decom-
position.

4 x− 9

x^2 − 3 x=

A

x+

B

x− 3

4 x− 9 =A(x− 3 )+Bx

4 x− 9 =Ax− 3 A+Bx

Notice that the constant term -9 must be equal to the constant term− 3 Aand that the terms withxmust be equal as
well.

− 9 =− 3 A

4 =A+B

Solving this system yields:
A= 3 , B= 1
Therefore,

x^42 x−− 39 x=^3 x+x−^13

Vocabulary

Partial fraction decompositionis a procedure that undoes the operation of adding fractions with unlike denomina-
tors. It separates a rational expression into the sum of rational expressions with unlike denominators.

Guided Practice

1. Use matrices to help you decompose the following rational expression.

( 2 x−^51 x)(− 32 x+ 4 )

2. Confirm Example C by adding the partial fractions.

(x−^21 x)(+x^4 + 3 )=x+^11 +x+^13

3. Confirm Guided Practice #1 by adding the partial fractions.

( 2 x−^51 x)(− 32 x+ 4 )= 2 x^111 − 1 + 3 −x+^11264
Answers: