5 3 Partial Fractions

Section 5.3Partial Fractions

The Idea Behind Partial Fraction Decomposition

Each of the two fractions on the right is called a partial fraction. The sum of these fractions is called the partial fraction decomposition of the rational expression on the left-hand side.

The Partial Fraction Decomposition of a Rational Expression with Distinct Linear Factors in the

Denominator

Example

Find the partial fraction decomposition of

6 14

3 5

x

x x

The Partial Fraction Decomposition of a Rational Expression with Linear Factors in the Denominator, Some of

Which are Repeated

Example

2

Find the partial fraction decomposition

8

( 2)

x

x x

The Partial Fraction Decomposition of a Rational Expression with Prime,

Nonrepeated Quadratic Factors in the Denominator

Graphing Calculator

Example

2

2

Find the partial fraction decomposition of

3 4 2

2 3 5

x x

x x x

The Partial Fraction Decomposition of a Rational Expression with a

Prime, Repeated Quadratic Factor in the Denominator

Example

3

22

Find the partial fraction decomposition of

2 1

2

x x

x

(a)

(b)

(c)

(d)

Find the partial fraction decomposition of

14

2 5

x

x x

4 3

2 52 3

2 51 3

2 54 2

2 5

x x

x x

x x

x x

(a)

(b)

(c)

(d)

2

Find the partial fraction decomposition of

2 11

( 3)

x

x

2

2

2

2

5 2

( 3) ( 3)

3 2

( 3) ( 3)

5 4

( 3) ( 3)

2 5

( 3) ( 3)

x x

x x

x x

x x