Chapter 3 Partial Fractions
Chapter 3
Partial Fractions
3.1 Proper and Improper Fractions
An algebraic fraction is a fraction in which the numerator and denominator are both
polynomial expressions. An algebraic fraction, for example
in which the numerator is a polynomial of lower degree than the denominator. We call this a
proper fraction.
With other fractions the polynomial may be of higher degree in the numerator or it may be of
the same degree as denominator, for example
are called improper fractions.
Definition (Proper and Improper fraction)
The algebraic fraction (that represents the rational function )
( )
( ) ( )
( )
is called a proper fraction if the following two conditions are satisfied:
1) and has no common factors (no common zeros)
2) degree of degree of
Otherwise it is called improper.
Combining fractions over a common denominator is a familiar operation from algebra, for
instance
( )
30
, Chapter 3 Partial Fractions
From the standpoint of integration, the left side of equation (*) would be much easier to work
with than the right side. So, when integrating rational functions it would be helpful if we
could undo the simplification going from left to right in equation (*). Reversing this process
is referred to as finding the partial fraction decomposition of a rational function.
Important note
The method for computing partial fraction decompositions only applies to all rational
functions in a proper fraction form
3.2 Decomposition of a Fraction into Partial Fractions [4 steps]
Step 1 : Divide if improper
If ( ) ( ) is an improper fraction (degree of degree of ), perform long division to
obtain
( ) ( )
( )
( ) ( )
where ( ) is the quotient, and it is a polynomial of degree equals the difference between
degree of and degree of , and the remainder from the division is ( ) . Now apply the
following steps on the proper fraction
( )
( )
Step 2 : Factor the denominator :
Completely factor the polynomial ( ) into factors of linear and quadratic forms
( ) ( )
where
the quadratic term is irreducible ( )
Step 3 : Partial Fractions :
For each factor of the form ( ) , the partial fraction decomposition must
include the sum of the fractions
( ) ( ) ( )
31
Chapter 3
Partial Fractions
3.1 Proper and Improper Fractions
An algebraic fraction is a fraction in which the numerator and denominator are both
polynomial expressions. An algebraic fraction, for example
in which the numerator is a polynomial of lower degree than the denominator. We call this a
proper fraction.
With other fractions the polynomial may be of higher degree in the numerator or it may be of
the same degree as denominator, for example
are called improper fractions.
Definition (Proper and Improper fraction)
The algebraic fraction (that represents the rational function )
( )
( ) ( )
( )
is called a proper fraction if the following two conditions are satisfied:
1) and has no common factors (no common zeros)
2) degree of degree of
Otherwise it is called improper.
Combining fractions over a common denominator is a familiar operation from algebra, for
instance
( )
30
, Chapter 3 Partial Fractions
From the standpoint of integration, the left side of equation (*) would be much easier to work
with than the right side. So, when integrating rational functions it would be helpful if we
could undo the simplification going from left to right in equation (*). Reversing this process
is referred to as finding the partial fraction decomposition of a rational function.
Important note
The method for computing partial fraction decompositions only applies to all rational
functions in a proper fraction form
3.2 Decomposition of a Fraction into Partial Fractions [4 steps]
Step 1 : Divide if improper
If ( ) ( ) is an improper fraction (degree of degree of ), perform long division to
obtain
( ) ( )
( )
( ) ( )
where ( ) is the quotient, and it is a polynomial of degree equals the difference between
degree of and degree of , and the remainder from the division is ( ) . Now apply the
following steps on the proper fraction
( )
( )
Step 2 : Factor the denominator :
Completely factor the polynomial ( ) into factors of linear and quadratic forms
( ) ( )
where
the quadratic term is irreducible ( )
Step 3 : Partial Fractions :
For each factor of the form ( ) , the partial fraction decomposition must
include the sum of the fractions
( ) ( ) ( )
31
