Partial Fraction Decomposition

Sec. 7.4a HW: p. 612 13-21 odd, 25, 27

First, remind me……………………..…what’s a rational function?

f x

y xg x

0g x with

In this section, we will write a rational function as a sum ofrational functions where each denominator is a power of alinear factor or a power of an irreducible quadratic factor.

Example:2

3 4

2

x

x x

2 1

2x x

Each fraction in the sum is a partial fraction, and the sum isa partial fraction decomposition of the original rationalfunction.

Steps to Partial Fraction Decomposition of f(x)/d(x)

f x r x

q xd x d x

umx n

1. If the degree of f > degree of d: Divide f by d to obtain the quotient q and the remainder r and write

2. Factor d(x) into a product of factors of the form or

, where is irreducible 2 vax bx c 2 v

ax bx c

Steps to Partial Fraction Decomposition of f(x)/d(x)

1 2

2u

u

AA A

mx n mx n mx n

umx n3. For each factor : The partial fraction decomposition of r(x)/d(x) must include the sum

where are real numbers1 2, , , uA A A

1 1 2 2

22 2 2

v vv

B x CB x C B x C

ax bx c ax bx c ax bx c

2 vax bx c 4. For each factor : The partial fraction decomp.

of r(x)/d(x) must include the sum

where and are real numbers1 2, , , VB B B1 2, , , vC C C

Guided PracticeGuided Practice

3 2

5 1

3 1

x

x x x

Write the terms for the partial fraction decomposition of the givenrational function. Do not solve for the corresponding constants.

31 22 3

AA A

x x x 4

3

A

x 1 12 1

B x C

x

+ +

Today, we’ll just focus on thelinear factors, like these…

Guided PracticeGuided Practice

2

5 1

2 15

x

x x

Find the partial fraction decomposition of the given function.

5 1

3 5

x

x x

Factor thedenominator!

1 25 1

3 5 3 5

A Ax

x x x x

Write the

partial fractions!

1 25 1 5 3x A x A x “Clear the fractions”by killing the denominator!

1 1 2 25 1 5 3x A x A A x A

1 2 1 25 1 5 3x A A x A A

Guided PracticeGuided Practice

2

5 1

2 15

x

x x

Find the partial fraction decomposition of the given function.

1 2 5A A Equate the coefficients fromeach side of the equation!

1 25 3 1A A

1 2 1 25 1 5 3x A A x A A

Solve the system!(I don’t care how!!!) 1 2A 2 3A

2 3

3 5x x

Can we verify this answer algebraically? Graphically?Can we verify this answer algebraically? Graphically?

Guided PracticeGuided Practice

2

5 1

2 15

x

x x

Find the partial fraction decomposition of the given function.

Another (easier?) way tosolve for the constants:

1 2A 2 3A

2 3

3 5x x

1 25 1 5 3x A x A x

Substitute in 5 for x, thensubstitute in –3 for x:

(these values come from thedenominator, x=-3, and 5)

Guided PracticeGuided Practice

2

3 2

2 4

4 4

x x

x x x

Find the partial fraction decomposition of the given function.

Clear fractions:

221 2 32 4 2 2x x A x A x x A x

2

2

2 4

2

x x

x x

31 2

22 2

AA A

x x x

Expand and combine like terms:

2 21 2 1 2 3 12 4 4 2 4x x A A x A A A x A

Guided PracticeGuided Practice

2

3 2

2 4

4 4

x x

x x x

Find the partial fraction decomposition of the given function.

Compare coefficients:

21 2 2

2 2x x x

1 2 1A A

1 2 34 2 2A A A

14 4A

2 21 2 1 2 3 12 4 4 2 4x x A A x A A A x A

Solve the system: 3 2A 1 1A 2 2A

Guided PracticeGuided Practice

221 2 32 4 2 2x x A x A x x A x

2

3 2

2 4

4 4

x x

x x x

Find the partial fraction decomposition of the given function.

The other way to solve for the A’s:

21 2 2

2 2x x x

3 2A 1 1A 2 2A Use x = 2, solve for A 3

Use x = 0, solve for A 1

Use any other x, solve for A2

More PFD:

Denominators with Irreducible Quadratic

Factors

Now let’s apply a similar processwhen working with irreducible

quadratic factors…(see “Step 4” in your notesfrom the previous slides!!)

Find the partial fraction decomposition of

2

3 2

4 1

1

x x

x x x

Factor the denominator by grouping:3 2 1x x x 2 1 1x x x

21 1

A Bx C

x x

21 1x x Clear fractions:

2 24 1 1 1x x A x Bx C x Expand and combine like terms:

2 24 1x x A B x B C x A C

Find the partial fraction decomposition of

2

3 2

4 1

1

x x

x x x

2

3 2 2

1 1

x

x x

Compare coefficients: 1A B 2 24 1x x A B x B C x A C

4B C 1A C

Solve the system: 3A 2B 2C

Find the partial fraction decomposition of

3 2

22

2 5

1

x x x

x

1 1 2 2

22 21 1

B x C B x C

x x

Clear fractions:

3 2 21 1 2 22 5 1x x x B x C x B x C

Expand and combine like terms:

3 2 3 21 1 1 2 1 22 5x x x B x C x B B x C C

Find the partial fraction decomposition of

3 2

22

2 5

1

x x x

x

22 2

2 1 3 1

1 1

x x

x x

Compare coefficients:

3 2 3 21 1 1 2 1 22 5x x x B x C x B B x C C

1 2B 1 1C

1 2 5B B 1 2 0C C

2 3B 2 1C

Find the partial fraction decomposition of

2

22

3 4

1

x

x

1 1 2 2

22 21 1

B x C B x C

x x

Clear fractions:

2 21 1 2 23 4 1x B x C x B x C

Expand and combine like terms:

2 3 21 1 1 2 1 23 4x B x C x B B x C C

Find the partial fraction decomposition of

2

22

3 4

1

x

x

22 2

3 1

1 1x x

Compare coefficients:

2 3 21 1 1 2 1 23 4x B x C x B B x C C

1 0B 1 3C

1 2 0B B 1 2 4C C

2 0B 2 1C