Editing Partial fraction decomposition (section)

{{Short description|Rational fractions as sums of simple terms}}
{{more footnotes|date=September 2012}}
In [[algebra]], the '''partial fraction decomposition''' or '''partial fraction expansion''' of a [[rational fraction]] (that is, a [[fraction (mathematics)|fraction]] such that the numerator and the denominator are both [[polynomial]]s) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.<ref>{{cite book |last1=Larson |first1=Ron |title=Algebra & Trigonometry |date=2016 |publisher=Cengage Learning |isbn=9781337271172 |url=https://books.google.com/books?id=Ft-5DQAAQBAJ&q=partial+fraction%27&pg=PA662 |language=en}}</ref>

The importance of the partial fraction decomposition lies in the fact that it provides [[algorithm]]s for various computations with [[rational function]]s, including the explicit computation of [[antiderivative]]s,<ref>Horowitz, Ellis. "[https://ftp.cs.wisc.edu/pub/techreports/1970/TR91.pdf Algorithms for partial fraction decomposition and rational function integration]." Proceedings of the second ACM symposium on Symbolic and algebraic manipulation. ACM, 1971.</ref> [[Taylor series| Taylor series expansions]], [[Z-transform|inverse Z-transform]]s, and [[Laplace transform|inverse Laplace transform]]s. The concept was discovered independently in 1702 by both [[Johann Bernoulli]] and [[Gottfried Leibniz]].<ref>{{cite book |last=Grosholz |first=Emily |date=2000 |title=The Growth of Mathematical Knowledge |publisher=Kluwer Academic Publilshers |page=179 |isbn=978-90-481-5391-6 }}</ref>

In symbols, the ''partial fraction decomposition'' of a rational fraction of the form <math display="inline"> \frac{f(x)}{g(x)}, </math> where {{math|''f''}} and {{math|''g''}} are polynomials, is the expression of the rational fraction as

<math display="block">\frac{f(x)}{g(x)}=p(x) + \sum_j \frac{f_j(x)}{g_j(x)} </math>

where
{{math|''p''(''x'')}} is a polynomial, and, for each {{mvar|j}},
the [[denominator]] {{math|''g''<sub>''j''</sub> (''x'')}} is a [[Exponentiation|power]] of an [[irreducible polynomial]] (i.e. not factorizable into polynomials of positive degrees), and
the [[numerator]] {{math|''f''<sub>''j''</sub> (''x'')}} is a polynomial of a smaller degree than the degree of this irreducible polynomial.

When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "[[square-free polynomial]]" in the description of the outcome. This allows replacing [[polynomial factorization]] by the much easier-to-compute [[square-free factorization]]. This is sufficient for most applications, and avoids introducing [[irrational number|irrational coefficients]] when the coefficients of the input polynomials are [[integer]]s or [[rational number]]s.