Partial Fraction
MATH 101
Sahidul Islam
An algebraic fraction such as can often be broken down into simpler
parts called partial fractions. Specifically
An algebraic fraction is a fraction in which the numerator and denominator are
both polynomial expressions. A polynomial expression is one where every term is
a multiple of a power of x, such as
The degree of a polynomial is the power of the highest term in x. So in this case
the degree is 4.
The number in front of x in each term is called its coefficient. So, the coefficient of
is 5.
The coefficient of is 6.
Now consider the following algebraic fractions:
In both cases the numerator is a polynomial of lower degree than the
denominator. We call these proper fractions.
With other fractions the polynomial may be of higher degree in the numerator or
it may be of the same degree, for example
and these are called improper fractions.
Partial Fraction
MATH 101
Sahidul Islam
Key Point
If the degree of the numerator is less than the degree of the denominator the
fraction is said to be a proper fraction.
If the degree of the numerator is greater than or equal to the degree of the
denominator the fraction is said to be an improper fraction.
Procedure for Decomposing the proper rational function
Now consider a rational function where P and Q are polynomial.
To decompose the rational function , where the degree of is
less than
1. Factor into factors of the form and ,
where and are the multiplicities of and ,
respectively.
2. For each power of a linear factor , allow the decomposition to
include terms of the form
where are the real constants to be determined.
3. For each power of a irreducible quadratic factor , allow the
decomposition to include terms of the form
where and are the real constants to be determined.
Partial Fraction
MATH 101
Sahidul Islam
Case 1. The denominator Q(x) is a product of distinct linear factor
This means that we can write
In this case we have
where constants are given by
( )
Example 1. Decompose into partial fractions.
Solution:
Observe that the factors in the denominator are – and so we write
Using formula (2) we get
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Putting these results together we have
and we have expressed the given fraction in partial fractions.
Alternative Solution:
We multiply both sides by the common denominator (x − 1)(x + 2):
Partial Fraction
MATH 101
Sahidul Islam
Put in equation (i) and we get
If
Putting these results together we have
and we have expressed the given fraction in partial fractions.
Exercises 1
Express the following as a sum of partial fractions
Case 2. The denominator Q(x) is a product of linear factor, some being repeated
Suppose that contains a linear factor of multiplicity m.
In this case we have
where is the decomposition corresponding to the linear factor other than
(
where constants are given by
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Partial Fraction
MATH 101
Sahidul Islam
Example 2. Decompose into partial fraction.
Solution: Observe that the factors in the denominator are ,
and so we write
Using formula (4) we get
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Putting these results together we have
and we have expressed the given fraction in partial fractions.
Example 3. Decompose into partial fraction
Solution:
Observe that the factors in the denominator are
and so we write
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Partial Fraction
MATH 101
Sahidul Islam
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Exercises 2
Express the following as a sum of partial fractions
Case 3. The denominator Q(x) contains irreducible quadratic factor, none of which
is repeated
Example 3. Decompose into partial fraction
Here the two denominators of the partial fractions will be and
When the denominator contains a quadratic factor we have to consider the
possibility that the numerator can contain a term in x. This is because if it did, the
numerator would still be of lower degree than the denominator - this would still
be a proper fraction. So we write
Now
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Partial Fraction
MATH 101
Sahidul Islam
Hence and
And
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Putting these results together we have
and we have expressed the given fraction in partial fractions.
Exercises 3
Express the following as a sum of partial fractions
