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'Partial fraction decomposition' is a theorem in algebra which states that a rational function can be decomposed into a polynomial plus a sum of proper fractions, each of which is either a constant over a power of a linear polynomial or a linear polynomial over a power of an irreducible quadratic polynomial.

Contents

Statement of theorem

Outline of proof

Lemma 1

Lemma 2

Generalization to Euclidean domains

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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 $a\_\{ij\}$ with such that

If , then $b=0$.

Outline of proof

Lemma 1

Let ''f'',''g'' and ''h'' be nonzero polynomials with ''f'' and ''g'' coprime. There are polynomials ''a'' and ''b'' such that

Lemma 2

Let ''f'' and ''g'' be nonzero polynomials and let ''n'' be a positive integer. There exist polynomials ''b'' and $a\_i$ with

Generalization to Euclidean domains

More generally, this is true in any Euclidean domain.

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