Descomposició en fraccions parcials: diferència entre les revisions
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La decisió de quins polinomis són irreductibles depèn de quin [[camp (matemàtiques)|camp]] d'[[escalar (matemàtiques)|escalar]]s s'adopti. Si es treballa amb [[nombre real|nombres reals]], llavors els polinomis irreductibles són de grau 1 o 2. Si es permeten [[nombre complex|nombres complexos]], només els polinomis de primer grau són irreductibles. Finalment, si es permeten només [[nombre racional|nombres racionals]], o un [[camp finit]], llavors els graus dels polinomis irreductibles poden ser més elevats.
== Principis bàsics ==
Els principis bàsics relacionats amb aquest procediment són força simples.
Si s'assumeix una funció racional ''R''(''x'') = ''ƒ''(''x'')/''g''(''x'') en una [[indeterminada (variable)|indeterminada]] ''x'', té un denominador que factoritza com:
:<math> g(x) = P(x) \cdot Q(x) \, </math>
over a [[field (mathematics)|field]] ''K'' (we can take this to be [[real number]]s, or [[complex number]]s). If ''P'' and ''Q'' have no common factor, then ''R'' may be written as
:<math> \frac{A}{P} + \frac{B}{Q}</math>
for some polynomials ''A''(''x'') and ''B''(''x'') over ''K''. The ''existence'' of such a decomposition is a consequence of the fact that the [[polynomial ring]] over ''K'' is a [[principal ideal domain]], so that
:<math>CP + DQ = 1 \, </math>
for some polynomials ''C''(''x'') and ''D''(''x'') (see [[Bézout's identity]]).
Using this idea inductively we can write ''R''(''x'') as a sum with denominators powers of [[irreducible polynomial]]s. To take this further, if required, write:
:<math>\frac {G(x)}{F(x)^n}</math>
as a sum with denominators powers of ''F'' and [[numerator]]s of degree less than ''F'', plus a possible extra polynomial. This can be done by the [[Euclidean algorithm]], polynomial case. The result is the following [[theorem]]:
{{quotation|1=Let ''ƒ'' and ''g'' be nonzero polynomials over a field ''K''. Write ''g'' as a product of powers of distinct irreducible polynomials :
: <math>g=\prod_{i=1}^k p_i^{n_i}.</math>
There are (unique) polynomials ''b'' and ''a''<sub> ''ij''</sub> with deg ''a''<sub> ''ij''</sub> < deg ''p''<sub> ''i''</sub> such that
: <math>\frac{f}{g}=b+\sum_{i=1}^k\sum_{j=1}^{n_i}\frac{a_{ij}}{p_i^j}.</math>
If deg ''ƒ'' < deg ''g'', then ''b'' = 0.}}
Therefore when the field ''K'' is the complex numbers, we can assume that each ''p''<sub>''i''</sub> has degree 1 (by the [[fundamental theorem of algebra]]) the numerators will be constant. When ''K'' is the real numbers, some of the ''p''<sub>''i''</sub> might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur.
In the preceding theorem, one may replace "distinct irreducible polynomials" by "[[pairwise coprime]] polynomials that are coprime with their derivative". For example, the ''p''<sub>''i''</sub> may be the factors of the [[square-free factorization]] of ''g''. When ''K'' is the field of the rational numbers, as it is typically the case in [[computer algebra]], this allows to replace factorization by [[polynomial greatest common divisor|greatest common divisor]] to compute the partial fraction decomposition.
== Vegeu també ==
* [[Integració per fraccions parcials]]
[[Categoria:Àlgebra]]
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