*Rational fraction*
R(x)
is called the fraction of the form:

If
n<m,
fraction is called proper.
*Common fractions*
is the rational fractions of the form:

, where

n, m
- natural numbers, coefficients
c, p, q,
A, B, C
are real numbers and the roots of the polinomial
x^{2}+p∙x+q
- is complex conjugate (i.e. ¼∙p^{2}−q<0).

If the denominator is decomposed
Q_{m}(x)
in the multiplication of the linear and/or quadratic multipliers:

Q_{m}(x)=(x−c_{1})^{n1}
∙...∙
(x−c_{r})^{nr}
∙...∙
(x^{2}+p_{1}∙x+q_{1}
)^{m1}
∙...∙
(x^{2}+p_{s}∙x+q_{s}
)^{ms},
where

c_{1}, c_{2}, ..., c_{r}
- real roots of the polinomial Q_{m}(x)
of order
n_{1}, n_{2}, ..., n_{r}
respectively, and
x^{2}+p_{k}∙x+q_{k}=(x−a_{k})∙(x−ã_{k}),
where
a_{k}
and
ã_{k}
complex conjugate roots of order
m_{k},
then initial fraction can be respesented such as:

Each linear multiplier of the form
,
contained in
Q_{m}(x)
corresponds the decomposition of the form:

,

Each quadratic multiplier of the form
,
contained in
Q_{m}(x)
corresponds to the decomposition of the form:

,

Our online calculator finds partial fraction decomposition of any (proper, improper) rational fraction. If initial fraction is the improper one, (i.e. order of polynomial in the numerator greaters of equals to the order of polynomial in the denominator) calculator divides numerator to the denominator and extracts the proper fraction.
*Partial fraction decomposition*
is usually used to
find the integrals
of the rational expressions.

The example of the step by step solution is here.

Fraction decomposition calculator

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