In symbols, one can use partial fraction expansion to change a rational fraction in the form
where ƒ and g are polynomials, into an expression of the form
where gj x are polynomials that are factors of g'x, and are in general of lower 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 fraction 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 division can be used, but in fact if ƒ already has lower degree than g this isn't helpful.
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