Editing Partial fraction decomposition (section)

===Statement===
{{math_theorem|name=Theorem|Let {{math|''f''}} and {{math|''g''}} be nonzero polynomials over a field {{math|''K''}}. Write {{math|''g''}} as a product of powers of distinct irreducible polynomials :
<math display="block">g=\prod_{i=1}^k p_i^{n_i}.</math>

There are (unique) polynomials {{math|''b''}} and {{math|''a''<sub>''ij''</sub>}} with {{math|deg ''a''<sub>''ij''</sub> < deg ''p''<sub>''i''</sub>}} such that
<math display="block">\frac{f}{g}=b+\sum_{i=1}^k\sum_{j=1}^{n_i}\frac{a_{ij}}{p_i^j}.</math>

If {{math|deg ''f'' < deg ''g''}}, then {{math|''b'' {{=}} 0}}.}}

The uniqueness can be proved as follows. Let {{math|1=''d'' = max(1 + deg ''f'', deg ''g'')}}. All together, {{math|''b''}} and the {{math|''a''<sub>''ij''</sub>}} have {{mvar|d}} coefficients. The shape of the decomposition defines a [[linear map]] from coefficient vectors to polynomials {{mvar|f}} of degree less than {{mvar|d}}. The existence proof means that this map is [[surjective]]. As the two [[vector space]]s have the same dimension, the map is also [[injective]], which means uniqueness of the decomposition. By the way, this proof induces an algorithm for computing the decomposition through [[linear algebra]].

If {{math|''K''}} is the field of [[complex number]]s, the [[fundamental theorem of algebra]] implies that all  {{math|''p''<sub>''i''</sub>}} have degree one, and all numerators <math>a_{ij}</math> are constants. When {{math|''K''}} is the field of [[real number]]s, some of the {{math|''p''<sub>''i''</sub>}} may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur.

In the preceding theorem, one may replace "distinct irreducible polynomials" by "[[pairwise coprime]] polynomials that are coprime with their derivative". For example, the {{math|''p''<sub>''i''</sub>}} may be the factors of the [[square-free factorization]] of {{math|''g''}}. When {{math|''K''}} is the field of [[rational number]]s, as it is typically the case in [[computer algebra]], this allows to replace factorization by [[polynomial greatest common divisor|greatest common divisor]] computation for computing a partial fraction decomposition.