Download Partial Fraction Decomposition

Partial Fraction
Decomposition
Sec. 7.4a
First, remind me……………………..…what’s a rational function?
f  x
y  x 
g  x
with
g  x  0
In this section, we will write a rational function as a sum of
rational functions where each denominator is a power of a
linear factor or a power of an irreducible quadratic factor.
Example:
3x  4 2
1
 
2
x  2x x x  2
Each fraction in the sum is a partial fraction, and the sum is
a partial fraction decomposition of the original rational
function.
Steps to Partial Fraction Decomposition of f(x)/d(x)
1. If the degree of f > degree of d: Divide f by d to obtain the
quotient q and the 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
 ax
2

 bx  c , where ax  bx  c
v
2
 mx  n  or
 is irreducible
v
u
Steps to Partial Fraction Decomposition of f(x)/d(x)


3. For each factor mx  n : The partial fraction decomposition
of r(x)/d(x) must include the sum
u
A1
A2


2
mx  n  mx  n 
where
A1 , A2 ,

, Au

Au
 mx  n 
u
are real numbers

v
4. For each factor ax  bx  c : The partial fraction decomp.
of r(x)/d(x) must include the sum
2
B1 x  C1
B2 x  C2


2
2
ax  bx  c  ax 2  bx  c 
where
B1 , B2 ,
, BV
and
C1 , C2 ,

, Cv
Bv x  Cv
 ax
2
 bx  c 
v
are real numbers
Guided Practice
Write the terms for the partial fraction decomposition of the given
rational function. Do not solve for the corresponding constants.
5x 1
x 3  x  3  x 2  1
B1 x  C1
A1 A2 A3
A4
 2 3 +
+
2
x 1
x3
x x
x
Today, we’ll just focus on the
linear factors, like these…
Guided Practice
Find the partial fraction decomposition of the given function.
5x  1
2
x  2 x  15
Write the
partial fractions!
Factor the
denominator!
5x  1
 x  3 x  5
A1
A2
5x  1


 x  3 x  5 x  3 x  5
“Clear the fractions”
by multiplying everything
by the denominator!
5x 1  A1  x  5  A2  x  3
5 x  1  A1 x  5 A1  A2 x  3 A2
5x 1   A1  A2  x   5 A1  3 A2 
Guided Practice
Find the partial fraction decomposition of the given function.
5x  1
2
3


2
x  2 x  15
x 3 x 5
5x 1   A1  A2  x   5 A1  3 A2 
Equate the coefficients from
each side of the equation!
Solve the system!
(I don’t care how!!!)
A1  A2  5
5 A1  3 A2  1
A1  2 A2  3
Can we verify this answer algebraically? Graphically?
Guided Practice
Find the partial fraction decomposition of the given function.
5x  1
2
3


2
x  2 x  15
x 3 x 5
Another (easier?) way to
solve for the constants:
Plug in 5 for x, then
plug in –3 for x:
5x 1  A1  x  5  A2  x  3
A1  2 A2  3
Guided Practice
Find the partial fraction decomposition of the given function.
x  2x  4
A3
A2
 x  2 x  4 A1
 


3
2
2
2
x  4x  4x
x
x

2
x  x  2
 x  2
2
2
Clear fractions:
 x  2 x  4  A1  x  2   A2 x  x  2   A3 x
2
2
Expand and combine like terms:
 x2  2x  4   A1  A2  x2   4 A1  2 A2  A3  x  4 A1
Guided Practice
Find the partial fraction decomposition of the given function.
x  2x  4
1 2
2
 

3
2
x  4 x  4 x x x  2  x  2 2
2
 x2  2x  4   A1  A2  x2   4 A1  2 A2  A3  x  4 A1
Compare coefficients:
A1  A2  1
4 A1  2 A2  A3  2
4 A1  4
Solve the system:
A1  1 A2  2 A3  2
Guided Practice
Find the partial fraction decomposition of the given function.
x  2x  4
1 2
2
 

3
2
x  4 x  4 x x x  2  x  2 2
2
The other way to solve for the A’s:
 x  2 x  4  A1  x  2   A2 x  x  2   A3 x
2
Use x = 2, solve for A 3
Use x = 0, solve for A 1
Use any other x, solve for A 2
A1  1 A2  2 A3  2
2
More PFD:
Denominators with
Irreducible
Quadratic Factors
Now let’s apply a similar process
when working with irreducible
quadratic factors…
(see “Step 4” in your notes
from the previous slides!!)
Find the partial fraction decomposition of
A
Bx  C
x  4x 1

 2
3
2
x  x  x 1 x 1 x  1
2
Factor the denominator by grouping:
x  x  x  1  x  x 1   x  1   x  1  x 2  1
3
2
2
Clear fractions:
x  4 x  1  A  x  1   Bx  C  x  1
2
2
Expand and combine like terms:
x  4 x  1   A  B  x   B  C  x   A  C 
2
2
Find the partial fraction decomposition of
3
2 x  2
x  4x 1

 2
3
2
x  x  x 1 x 1 x  1
2
x  4 x  1   A  B  x   B  C  x   A  C 
2
2
Compare coefficients:
A B 1
B  C  4
AC 1
Solve the system:
A  3 B  2 C  2
Find the partial fraction decomposition of
2 x  x  5x
3
2
x
2
 1
2
B1 x  C1 B2 x  C2
 2

2
2
x  1  x  1
Clear fractions:
2 x  x  5 x   B1 x  C1   x  1  B2 x  C2
3
2
2
Expand and combine like terms:
2 x  x  5x  B1x  C1x   B1  B2  x  C1  C2 
3
2
3
2
Find the partial fraction decomposition of
2 x  x  5x
3
2
x
2
 1
2
2x 1
3x  1
 2

2
2
x  1  x  1
2 x  x  5x  B1x  C1x   B1  B2  x  C1  C2 
3
2
3
2
B1  2 C1  1
B1  B2  5 C1  C2  0
 B2  3 C2  1
Compare coefficients:
Find the partial fraction decomposition of
3x 2  4
x
2
 1
2
B1 x  C1 B2 x  C2
 2

2
2
x  1  x  1
Clear fractions:
3x  4   B1 x  C1   x  1  B2 x  C2
2
2
Expand and combine like terms:
3x2  4  B1 x3  C1 x2   B1  B2  x  C1  C2 
Find the partial fraction decomposition of
3x 2  4
x
2
 1
2
3
1
 2

2
2
x  1  x  1
3x  4  B1 x  C1 x   B1  B2  x  C1  C2 
2
3
2
B1  0 C1  3
B1  B2  0 C1  C2  4
 B2  0 C2  1
Compare coefficients: