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Partial fraction decomposition
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== Basic principles == Let <math display="block">R(x) = \frac FG</math> be a [[rational fraction]], where {{mvar|F}} and {{mvar|G}} are [[univariate polynomial]]s in the [[Indeterminate (variable)|indeterminate]] {{math|''x''}} over a field. The existence of the partial fraction can be proved by applying inductively the following reduction steps. ===Polynomial part=== There exist two polynomials {{mvar|E}} and {{math|''F''{{sub|1}}}} such that <math display="block">\frac FG=E+\frac{F_1}G,</math> and <math display="block">\deg F_1 <\deg G,</math> where <math>\deg P</math> denotes the [[degree of a polynomial|degree]] of the polynomial {{mvar|P}}. This results immediately from the [[Euclidean division of polynomials|Euclidean division]] of {{mvar|F}} by {{mvar|G}}, which asserts the existence of {{mvar|E}} and {{math|''F''{{sub|1}}}} such that <math>F = EG + F_1</math> and <math>\deg F_1 < \deg G.</math> This allows supposing in the next steps that <math>\deg F <\deg G.</math> ===Factors of the denominator=== If <math>\deg F < \deg G,</math> and <math display="block">G = G_1 G_2,</math> where {{math|''G''{{sub|1}}}} and {{math|''G''{{sub|2}}}} are [[coprime polynomials]], then there exist polynomials <math>F_1</math> and <math>F_2</math> such that <math display="block">\frac FG=\frac{F_1}{G_1}+\frac{F_2}{G_2},</math> and <math display="block">\deg F_1 < \deg G_1\quad\text{and}\quad\deg F_2 < \deg G_2.</math> This can be proved as follows. [[Bézout's identity for polynomials|Bézout's identity]] asserts the existence of polynomials {{math|''C''}} and {{math|''D''}} such that <math display="block">CG_1 + DG_2 = 1</math> (by hypothesis, {{math|1}} is a [[Polynomial greatest common divisor|greatest common divisor]] of {{math|''G''{{sub|1}}}} and {{math|''G''{{sub|2}}}}). Let <math>DF=G_1Q+F_1</math> with <math>\deg F_1 < \deg G_1</math> be the [[Euclidean division of polynomials|Euclidean division]] of {{mvar|DF}} by <math>G_1.</math> Setting <math>F_2=CF+QG_2,</math> one gets <math display="block">\begin{align} \frac FG&=\frac{F(CG_1 + DG_2)}{G_1G_2} =\frac{D F}{G_1}+\frac{CF}{G_2}\\ &=\frac{F_1+G_1Q}{G_1}+\frac{F_2-G_2Q}{G_2}\\ &=\frac{F_1}{G_1} + Q + \frac{F_2}{G_2} - Q\\ &=\frac{F_1}{G_1}+\frac{F_2}{G_2}. \end{align}</math> It remains to show that <math>\deg F_2 < \deg G_2.</math> By reducing the last sum of fractions to a common denominator, one gets <math>F=F_2G_1+F_1G_2,</math> and thus <math display="block">\begin{align} \deg F_2 &=\deg(F-F_1G_2)-\deg G_1 \le \max(\deg F,\deg (F_1G_2))-\deg G_1\\ &< \max(\deg G,\deg(G_1G_2))-\deg G_1= \deg G_2 \end{align}</math> ===Powers in the denominator=== Using the preceding decomposition inductively one gets fractions of the form <math>\frac F {G^k},</math> with <math>\deg F < \deg G^k= k\deg G,</math> where {{mvar|G}} is an [[irreducible polynomial]]. If {{math|''k'' > 1}}, one can decompose further, by using that an irreducible polynomial is a [[square-free polynomial]], that is, <math>1</math> is a [[Polynomial greatest common divisor|greatest common divisor]] of the polynomial and its [[derivative]]. If <math>G'</math> is the derivative of {{mvar|G}}, [[Polynomial greatest common divisor#Bézout's identity and extended GCD algorithm|Bézout's identity]] provides polynomials {{mvar|C}} and {{mvar|D}} such that <math>CG + DG' = 1</math> and thus <math>F=FCG+FDG'.</math> Euclidean division of <math>FDG'</math> by <math>G</math> gives polynomials <math>H_k</math> and <math>Q</math> such that <math>FDG' = QG + H_k</math> and <math>\deg H_k < \deg G.</math> Setting <math>F_{k-1}=FC+Q,</math> one gets <math display="block">\frac F {G^k} = \frac{H_k}{G^k}+\frac{F_{k-1}}{G^{k-1}},</math> with <math>\deg H_k <\deg G.</math> Iterating this process with <math>\frac{F_{k-1}}{G^{k-1}}</math> in place of <math>\frac F{G^k}</math> leads eventually to the following theorem. ===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.
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