When a given rational function f(x)/g(x), where
f(x) and g(x) are polynomials in x, is expressed as the sum of two or more simpler fractions according to certain
rules, it is said to be expressed in partial fractions.

A rational function f(x)/g(x) is proper
if deg f(x) < deg g(x). Otherwise
it is called an improper fraction.

If f(x)/g(x) is improper,
then by long division

f(x)

= g(x)Q(x)
+ R(x)

\

f(x)
¾¾
g(x)

= Q(x) +

R(x)
¾¾
g(x)

where R(x)/g(x) is
a proper fraction.

ules For Partial Fractions

Before expressing a rational function into partial fractions, the following steps should be performed:

f(x), g(x) are factorised to eliminate common factors.

If f(x)/g(x) is improper, it has to be broken down as Q(x) + R(x)/g(x).

g(x) must be completely factorised into its linear or irreducible quadratic factors.

If a rational function f(x)/g(x) is a proper fraction, we can express it into partial fractions according to the following rules.

Every non-repeated linear factor (ax + b) in g(x) corresponds to a fraction of the form:

A ¾¾¾
ax + b

.

Every linear factor (ax + b) in g(x) that is repeated n times corresponds to a sum of n partial fractions:

A_{1} ¾¾¾
ax + b

+

A_{2} ¾¾¾¾
(ax + b)^{2}

+ ¼ +

A_{n} ¾¾¾¾
(ax + b)^{n}

.

Every non-repeated irreducible quadratic factor (ax^{2} + bx + c) in g(x) corresponds to a fraction of the form: