Unit 7.4

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.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 4

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 )
.
Copyright © 2011 Pearson, Inc. Slide 7.4 - 7

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
.
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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
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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