Partial fraction decomposition involves writing rational functions as the sum of simpler fractional components. It is useful for integration in calculus. A rational function is broken into partial fractions by factoring the denominator and grouping terms with each linear or irreducible quadratic factor. Examples demonstrate decomposing fractions with linear factors and repeated or quadratic factors. Partial fraction decomposition allows rational functions to be integrated and graphs of rational functions to be sketched.

1. 7.4
Partial
Fractions
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Partial Fraction Decomposition
 Denominators with Linear Factors
 Denominators with Irreducible Quadratic Factors
 Applications
… and why
Partial fraction decompositions are used in calculus in
integration and can be used to guide the sketch of the
graph of a rational function.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 2

3. Partial Fraction Decomposition of
f(x)/d(x)
1. Degree of f  degree of d: Use the division algorithm
to divide f by d to obtain the quotient q and remainder
r and write
f (x)
d(x)
 q(x) 
r(x)
d(x)
.
2. Factor d(x) into a product of factors of the form (mx  n)u
or (ax2  bx  c)v , where ax2  bx  c is irreducible.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 3

4. Partial Fraction Decomposition of
f(x)/d(x)
3. For each factor (mx  n)u : The partial fraction
decomposition of r(x) / d(x) must include the sum
A1
mx  n

A2
mx  n2
 ... 
Au
mx  nu
,
where A1, A2 ,..., Au are real numbers.
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5. Partial Fraction Decomposition of
f(x)/d(x)
4. For each factor (ax2  bx  c)v : The partial fraction
decomposition of r(x) / d(x) must include the sum
Bx  C11

ax2  bx  c
B2x  C2
ax2  bx  c2
 ... 
Bvx  Cv
ax2   bx  cv
,
where B1,B2 ,...,Bv and C1,C2 ,...,Cv are real numbers.
The partial fraction decomposition of the original rational
function is the sum of q(x) and the fractions in parts 3 and 4.
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6. Example Decomposing a Fraction
with Distinct Linear Factors
Find the partial fraction decomposition of
3x  3
x 1x  2
.
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7. Example Decomposing a Fraction
with Distinct Linear Factors
Find the partial fraction decomposition of
3x  3
x 1x  2

A1
x 1

A2
x  2
3x  3
x 1x  2
3x  3  A1(x  2)  A2 (x 1) multiply both sides
by (x 1)(x  2)
3x  3  (A1  A2 )x  (2A1  A2 )
.
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8. Example Decomposing a Fraction
with Distinct Linear Factors
Find the partial fraction decomposition of
3x  3
x 1x  2
.
Compare coefficients on the left and right side of the
equation to find
A1  A2  3
2A1  A2  3
Solve the system of equations to find A1  2 and A2  1.
Thus
3x  3
x 1x  2

2
x 1

1
x  2
.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 8

9. Example Decomposing a Fraction
with Repeated Linear Factors
Use matrices to determine the values of A, B, and C so that
3x2  4x 11
A
B



x  32
x  2x  3
x  2
C
x  22
is true for all x, x  3, x  2.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 9

10. Example Decomposing a Fraction
with Repeated Linear Factors
Use matrices to determine the values of A, B, and C so that
3x2  4x 11
A
B



x  32
x  2x  3
x  2
C
x  22
is true for all x, x  3, x  2.
Multiply both sides by x  3x  22
3x2  4x 11  Ax  22
 Bx  3x  2 Cx  3
3x2  4x 11  A x2   4x  4 B x2   x  6 Cx  3
3x2  4x 11  A Bx2  4A B  Cx  4A 6B  3C
Copyright © 2011 Pearson, Inc. Slide 7.4 - 10

11. Example Decomposing a Fraction
with Repeated Linear Factors
3x2  4x 11  A Bx2  4A B  Cx  4A 6B  3C
A B  3
4A  B  C  4
4A 6B  3C  11
The augmented matrix for this system is:

1 1 0 3


4  1 1 4

4 6 3 11






Copyright © 2011 Pearson, Inc. Slide 7.4 - 11

12. Example Decomposing a Fraction
with Repeated Linear Factors
Find the reduced row echelon form for the augmented matrix.
1 1 0 3



4 1 1 4



4 6 3 11



























Copyright © 2011 Pearson, Inc. Slide 7.4 - 12







4R1  R2
4R1  Ruuuuuuuuuuur3
1 1 0 3
0 5 1 8
0 10 3 1




1
5

 

 
R2  R1
2R2  Ruuuuuuuuuuur3
1 0
1
5
7
5
0 5 1 8
0 0 5 15





1
5

 

 
R2

1
5

 

 
R2
uuuuuuuur
1 0
1
5
7
5
0 1 
1
5
8
5
0 0 1 3





13. Example Decomposing a Fraction
with Repeated Linear Factors
1 0
1
5
7
5
0 1 
1
5
8
5







0 0 1 3






Copyright © 2011 Pearson, Inc. Slide 7.4 - 13












1
5

 

 
R3  R1
1
5

 

 R3  R2
uuuuuuuuuuuur
1 0 0 2
0 1 0 1
0 0 1 3




The reduced row echelon form shows that
A  2, B  1, andC  3.
Interpret The original rational function can be
decomposed as
3x2  4x 11
x  3x  22

2
x  3

1
x  2

3
x  22

14. Example Decomposing a Fraction with
an Irreducible Quadratic Factor
Find the partial fraction decomposition of
x2  3x 1
x2  2x 1
.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 14

15. Example Decomposing a Fraction with
an Irreducible Quadratic Factor
Find the partial fraction decomposition of
x2  3x 1
x2  2x 1

A
x 1

Bx  C
x2  2
x2  3x 1
x2   2x 1
x2  3x 1  A x2   2 Bx  Cx 1
x2  3x 1  (A B)x2  (B  C)x  2A C
.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 15

16. Example Decomposing a Fraction with
an Irreducible Quadratic Factor
Compare coefficients to find the system of equations:
A B  1
B  C  3
2A C  1
Use any method to solve the system and find
A  1, B  0, and C  3.
Thus,
x2  3x 1
x2  2x 1

1
x 1

3
x2  2
.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 16

17. Quick Review
Perform the indicated operations and write your
answer as a single reduced fraction.
1.
1
x 1

2
x  2
2.
3
x 1

2
x  2
3. Divide f (x) by d(x) to obtain as quotient q(x) and
remainder r(x). Write a summary statement in fraction
form: q(x)  r(x) / d(x).
f (x)  x3  x2 1, d(x)  x  2
Copyright © 2011 Pearson, Inc. Slide 7.4 - 17

18. Quick Review
4. Write the polynomials as a product of linear and
irreducible quadratic factors with real coefficients.
x3  x2  2x  2
5. Assume that f (x)  g(x).
What can you conclude about A, B, C, and D?
f (x)  Ax2  Bx  C  2
g(x)  3x2  2x  3
Copyright © 2011 Pearson, Inc. Slide 7.4 - 18

19. Quick Review Solutions
Perform the indicated operations and write your
answer as a single reduced fraction.
1.
1
x 1

2
x  2
3x
x 1x  2
2.
3
x 1

2
x  2
x  8
x 1x  2
3. Divide f (x) by d(x) to obtain as quotient q(x) and
remainder r(x). Write a summary statement in fraction
form: q(x)  r(x) d(x).
f (x)  x3  x2 1, d(x)  x  2 x2  x  2  3 (x  2)
Copyright © 2011 Pearson, Inc. Slide 7.4 - 19

20. Quick Review Solutions
4. Write the polynomials as a product of linear and
irreducible quadratic factors with real coefficients.
x3  x2  2x  2
x2  2x 1
5. Assume that f (x)  g(x).
What can you conclude about A, B, C, and D?
f (x)  Ax2  Bx  C  2
g(x)  3x2  2x  3
A  3,B  2,C  5
Copyright © 2011 Pearson, Inc. Slide 7.4 - 20