The expansion by the Partial Fractions can be used to convert any algebraic rational function
of the form p (x) / q (x) where p and q are polynomials into a function which will be of the form
of summation of pj (x) / qj (x) with respect to j and where qj (x) are the polynomials which are
the factors of q (x) and generally they have a lower degree.
Therefore the decomposition by the partial fractions can be understood as the reverse method
of the elementary addition operation of the algebraic rational fractions which gives just one
rational function generally having a numerator and the denominator which have high degree.
The complete decomposition forces the reduction to go as far as possible or we can say that
in a different way that the factorization of q is utilized as more as it can be done.
Hence, the result of the complete partial fractions shows that function as an addition of
fractions in which the denominator of every single term is the power of such a polynomial
which cannot be factorized and the numerator will be a polynomial of a degree which is
smaller than that of the polynomial which is not reducible.
The algorithm of Euclidean can be utilized to directly reduce the degree of the numerator but
this algorithm is not useful if the numerator p is already having a lower degree than the
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The main motive of decomposing an algebraic rational function into an addition of the simpler
functions with the help of the partial fractions is that it makes it easier to do the linear
operations on it.
Hence the problems which are faced in the computation of the integrals, derivatives and
antiderivatives, in the expansions of the Power series and the Fourier series and in the linear
and the functional conversion of the rational functions can be decreased to a very great level
with the help of the decomposition by the partial fractions.
The computation can be made on every single element which is used in the process of the
decomposition by the partial fractions. We can take an example of the extreme use of the
partial fractions in the problems of the integration.
The partial fractions can be used in the problems of the integration for calculating the
antiderivatives. The field of the scalars we adopt tells us that which polynomials are not
Therefore we can conclude that the degree of the polynomials which are not reducible will
either be 1 or 2 if we take only the real numbers. However, if we allow the complex numbers,
then only the polynomials which are of degree equal to one can be irreducible.
But, some of the polynomials of the higher degree are also irreducible if we allow only the
rational numbers or a field which is finite.
partial fraction decomposition or partial fraction expansion is a procedure used to reduce the
degree of either the numerator or the denominator of a rational function (also known as a
rational algebraic fraction).
In symbols, one can use partial fraction expansion to change a rational function in the form
where ƒ and g are polynomials, into a function of the form
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where gj (x) are polynomials that are factors of g(x), and are in general of lower degree. Thus
the partial fraction decomposition may be seen as the inverse procedure of the more
elementary operation of addition of algebraic fractions, that produces a single rational function
with a numerator and denominator usually of high degree.
The full decomposition pushes the reduction as far as it will go: in other words, the
factorization of g is used as much as possible. Thus, the outcome of a full partial fraction
expansion expresses that function as a sum of fractions, where:
the denominator of each term is a power of an irreducible (not factorable) polynomial and
the numerator is a polynomial of smaller degree than that irreducible polynomial. To decrease
the degree of the numerator directly, the Euclidean algorithm can be used, but in fact if ƒ
already has lower degree than g this isn't helpful.
The main motivation to decompose a rational function into a sum of simpler fractions is that it
makes it simpler to perform linear operations on it. Therefore the problem of computing
derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and
linear functional transformations of rational functions can be reduced, via partial fraction
decomposition, to making the computation on each single element used in the decomposition.
Partial fractions in integration for an account of the use of the partial fractions in finding
antiderivatives. Just which polynomials are irreducible depends on which field of scalars one
adopts. Thus if one allows only real numbers, then irreducible polynomials are of degree
either 1 or 2.
If complex numbers are allowed, only 1st-degree polynomials can be irreducible. If one allows
only rational numbers, or a finite field, then some higher-degree polynomials are irreducible.
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