Partial fraction decomposition

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Partial fraction decompostion is a theorem in algebra which says 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.

1 Statement of theorem
2 Outline of proof
- 2.1 Lemma 1
- 2.2 Lemma 2
3 Generalization to Euclidean domains

[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 $a i j$ 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 $b = 0$ .

Alternative Statement:

Hopefully this will provide a simpler, and more accessible version of the proof.

$\frac{1}{r+1}=\frac{A}{r}+(\frac{B}{r}+1)$

$1 = A(r + 1) + Br \$

$1 = Ar + A + Br \$

$1 = (A + B)r + A \$

$0r + 1 = (A + B)r + A \$

$A = 1 \$

$A + B = 0 \$

$B = -1 \$

$\frac{1}{r(r+r)}=\frac{1}{r}-\frac{1}{r+1}$

[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 $a i$ with $\deg\ a_i<\deg\ g$ such that $\frac{f}{g^n}=b+\sum_{j=1}^n\frac{a_j}{g^j}.$