SUMMARY OF PARTIAL FRACTIONS
Proper and Improper Rational Functions. A rational function
is a quotient f (x)/g(x) of polynomials (with real coefficients), where
f (x) is called the numerator and g(x) is called the denominator. The
degree d of a polynomial is the highest power that x is raised to in that
polynomial. The coefficient of xd is called the leading coefficient. Thus
the degree of a nonzero constant polynomial (i.e. one with no x at all)
is zero. If the degree of f (x) is strictly less than the degree of g(x)
then f (x)/g(x) is called a proper rational function. A rational function
which is not proper is improper. If f (x)/g(x) is improper then there
are polynomials q(x) and r(x), such that f (x)/g(x) = q(x)+r(x)/g(x),
where r(x)/g(x) is a proper rational function. q(x) and r(x) can be
computed using polynomial division of f (x) by g(x); q(x) will be the
quotient and r(x) will be the remainder. We only do partial fraction
expansions of proper rational functions.
Factoring the Denominator. Suppose f (x)/g(x) is a proper rational function. Suppose r1 , . . . , rn are the distinct real roots of g(x),
and a1 ± ib1 , . . . , am ± ibm are the distinct complex roots of g(x).
Thus b1 , . . . , bm are all positive real numbers and the points in R2
(a1 , b1 ), . . . , (am , bm ) are distinct. There are positive integers k1 , . . . , kn
and l1 , . . . , lm , called the multiplicities of these roots such that g(x) has
the factorization
g(x) = c(x − r1 )k1 · · · (x − rn )kn [(x − a1 )2 + b21 ]l1 · · · [(x − am )2 + b2m ]lm ,
where c is the leading coefficient of g(x). By absorbing c into f (x) we
may assume the leading coefficient of g(x) is 1.
Partial Fraction Expansions. For each factor (x − r)k in the factorization of g(x) include the terms
A2
Ak
A1
+
+
·
·
·
+
x − r (x − r)2
(x − r)k
in the partial fraction expansion of f (x)/g(x), where A1 , . . . , Ak are
undetermined constants. The constant A1 for r1 is different from A1
for r2 , and so they should be given different names. For each factor
1
2
[(x − a)2 + b2 ]l in the factorization of g(x) include the terms
B2 x + C2
Bl x + Cl
B1 x + C 1
+
+ ··· +
2
2
2
2
2
(x − a) + b
[(x − a) + b ]
[(x − a)2 + b2 ]l
in the partial fraction expansion of f (x)/g(x), where B1 , C1 , . . . , Bl , Cl
are undetermined constants. The constants B1 , C1 for a1 ± ib1 are
different from B1 , C1 for a2 ± ib2 , and so they should be given different
names. The terms for each factor are then added together to get the
entire partial fraction expansion.
To find all the undetermined constants get a common denominator
in the partial fraction expansion and then equate f (x) with the numerator. Some of the constants may be found by setting x = r1 or x = r2 ,
etc. Some may be found by looking at the leading coefficients of both
sides. Any constants which have been found can be substituted into
the equation and the result simplified. Any remaining undetermined
constants can be found by solving a system of equations obtained by
equating the coefficients of xh from both sides of the equation, where
each value of h leads to one equation in the system. Another approach
derives equations to be solved by setting x equal to various real constants; usually x = 0 leads to a simple equation. Usually such systems
are manageable to solve in smaller examples.