Editing Partial fraction decomposition (section)

===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>