Partial Fractions
MATH 109 - Precalculus
S. Rook
Overview
• Section 7.4 in the textbook:
– Decomposition with linear factors
– Decomposition with quadratic factors
– Decomposition with improper rational expressions
2
Partial Decomposition with
Linear Factors
Decomposition in General
• Partial fraction decomposition: the process
of breaking up a large fraction into smaller,
more manageable (hopefully) fractions
– We know how to combine multiple fractions into a
single fraction:
• e.g.
2
3
5x  1
5x  1


 2
x  1 x  2 x  1x  2 x  x  2
– Decomposition reverses the process
• e.g.
5x  1
5x  1
2
3



x 2  x  2 x  1x  2 x  1 x  2
• Depends on the denominator
– Attempt to factor the denominator as completely
4
Unique Linear Factors
• A linear factor has the form px + q where p
and q are constants
• Each unique linear factor in the denominator
A
contributes a partial fraction of px  q to the
decomposition where A is a constant
– Writing the partial fraction contributed by each
factor in the denominator is called writing the
general form of the decomposition
– e.g. What would be the general form of the
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decomposition of
x2 x  3
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Writing the Basic Equation
• Once we have written the general form of the
decomposition, we need to solve for the constants in
the numerator
• The original fraction is equivalent to the sum of the
partial fractions: A  B  6
x 2 x  3 x2 x  3
• Eliminate the fractions by multiplying both sides by
the LCM: A2x  3  Bx  6
– This is called the basic equation and it holds true for
any value of x
6
Solving the Basic Equation for
Linear Factors
• For linear factors, pick convenient values for x in order
find the value of each constant:
– e.g. What value of x makes it easiest to find A?
– e.g. What value of x makes it easiest to find B?
– This process is straightforward when all factors of the
denominator in the original fraction are linear
• We will discuss how to handle solving for the constants
of the partial fractions for quadratic factors later
• After solving for the value of each constant, write the
final form of the decomposition
• Can check whether decomposition matches the
original fraction
7
Decomposition with Unique Linear
Factors (Example)
Ex 1: Write the partial fraction decomposition
of the rational expression:
1
a) 2
x 1
3
b) 2
x  x2
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Repeated Linear Factors
• Each factor of the form (px + q)n in the
factored denominator of the original fraction
contributes the sum of partial fractions
A
A
A
A



to
the
decomposition
 px  q   px  q 
 px  q   px  q 
1
n 1
2
2
n 1
n
n
n > 1 where n is an integer; if n = 1, the factor is
unique
– e.g. What would be the general form of the
decomposition of 1
x  43
• We can solve for the constants using the
technique developed for unique linear factors
9
Decomposition with Repeated
Linear Factors (Example)
Ex 2: Write the partial fraction decomposition
of the rational expression:
a)
3x
x2  6x  9
b)
4x  2x 1
x3  x 2
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Decomposition with Quadratic
Factors
Unique Quadratic Factors
• A quadratic factor has the form ax2 + bx + c where
the factor is irreducible and a, b, and c are constants
– i.e. Cannot be factored further over the reals
– e.g. (x2 – 1) is NOT a quadratic factor since it can be
reduced to (x + 1)(x – 1)
– e.g. (x2 + 1) IS a quadratic factor since it is irreducible
• Each unique quadratic factor in the denominator
Bx  C
contributes a partial fraction of ax  bx  c to the
decomposition where B and C are constants
2
– e.g. What would be the general form of the
x4
decomposition of
2

x x 2

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Solving the Basic Equation for
Quadratic Factors
• When working with partial fractions contributed by
quadratic factors, it is often difficult to find
convenient values of x like we were able to with the
linear factors
• Set up the basic equation: Ax2  2 Bx  C x  x  4
• Expand and group by powers of x:
Ax2  2 A  Bx2  Cx  x  4   A  Bx2  Cx  2 A  x  4
• Construct a system of equations by equating the
coefficients of each power of x on the left side with
0
its partner on the right side:  A  B

2A

C
1
4
13
Repeated Quadratic Factors
• Each factor of the form (ax2 + bx + c)n in the
factored denominator of the original fraction
contributes the sum of partial fractions
Bn1 x  Cn1
Bn  Cn
B1 x  C1
B2 x  C2





2
n 1
n
ax2  bx  c
ax2  bx  c
ax2  bx  c
ax2  bx  c

 





to the decomposition n > 1 where n is an integer;
if n = 1, the factor is unique
– e.g. What would be the general form of the
decomposition of 1
x  4
2
4
• We can solve for the constants using the
technique developed for unique quadratic factors
14
Decomposition with Quadratic
Factors (Example)
Ex 3: Write the partial fraction decomposition of
the rational expression:
a)
b)
x
16 x 4  1
2x2  x 1
x
2

1
2
15
Decomposition with Improper
Rational Expressions
Improper Rational Expressions
• An improper rational expression occurs when the
degree of its numerator is greater than OR equal to the
degree of its denominator
4
2
– e.g.
x2
x2  2x  9
or
x  5x  1
x3  x 2  4
• Thus far, we have worked exclusively with proper
rational expressions
– i.e. the degree of the numerator is less than the
degree of the denominator
17
Decomposition with Improper
Rational Expressions
• Before decomposing a rational expression, we MUST
check to see whether it is proper
– If the rational expression is improper, we MUST
perform a long division
• Perform a decomposition on the remainder (now proper)
• Do not forget the quotient of the initial long division
– If the rational expression is proper, proceed as normal
• i.e. factor the denominator and look for linear and
quadratic factors
• The first step in a decomposition problem should
ALWAYS be to check whether the rational expression
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is proper
Decomposition with Improper
Rational Expressions (Example)
Ex 4: Write the partial fraction decomposition
of the rational expression:
2 x3  x 2  x  5
a)
x 2  3x  2
x4
b) 3
x 1
19
Summary
• After studying these slides, you should be able to:
– Decompose rational expressions
• Additional Practice
– See the list of suggested problems for 7.4
• Next lesson
– Sequences & Series (Section 9.1)
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