Partial fraction decomposition
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Partial fraction decomposition is a theorem in algebra which states that a rational function over a field can be decomposed into a polynomial plus a sum of proper fractions, each of which has the form p(x) / q(x)n, where the degree of p(x) is smaller than the degree of q(x) and q(x) is irreducible.
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[edit] Statement of theorem
Let f and g be nonzero polynomials. Write g as a product of powers of distinct irreducible polynomials:
There are (unique) polynomials b and aij with such that
If , then b = 0.
[edit] Outline of proof
[edit] Lemma 1
Let f,g and h be nonzero polynomials with f and g coprime. There are polynomials a and b such that
[edit] Lemma 2
Let f and g be nonzero polynomials and let n be a positive integer. There exist polynomials b and ai with such that
[edit] Generalization to Euclidean domains
More generally, this is true in any Euclidean domain.