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:

g=\prod_{i=1}^k p_i^{n_i}.

There are (unique) polynomials b and aij with \deg\ a_{ij} < \deg\ p_i such that

\frac{f}{g}=b+\sum_{i=1}^k\sum_{j=1}^{n_i}\frac{a_{ij}}{p_i^j}.

If \deg\ f < \deg\ g, 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 \frac{h}{fg}=\frac{a}{f}+\frac{b}{g}.

[edit] Lemma 2

Let f and g be nonzero polynomials and let n be a positive integer. There exist polynomials b and ai with \deg\ a_i<\deg\ g such that \frac{f}{g^n}=b+\sum_{j=1}^n\frac{a_j}{g^j}.

[edit] Generalization to Euclidean domains

More generally, this is true in any Euclidean domain.


[edit] See also

Partial fraction