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{\bf Pr{\'e}pa CAPES UPMC 2008-2009} \hfill Emmanuel Ferrand, Laurent Koelblen, Matthieu Romagny.
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\Large
Jeudi 8 janvier 2009\\
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Fractions rationnelles
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D\'ecomposer en \'el\'ements simples dans $\RM(X)$ les fractions rationnelles\\
$ \frac{1}{(X-1)^{3}} \ ,\ \frac{X}{X^{2}-1} \ ,\ 
\frac{n!}{X(X+1)\cdots (X+n)}\ ,\ \frac{1}{(X^{2}+X+1)^{3}} \ ,\ \frac{X^{3}}{(X^{2}+X+1)^{3}} \ ,\
\frac{1}{X^{2n}-1}\ $.

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D\'ecomposer en \'el\'ements simples dans $\RM(X)$ les fractions rationnelles\\
$\frac{X^{2}}{(X+1)^{3}(X-1)^{2}} \ ,\  \frac{X^{n}}{(X+1)^{3}(X-1)^{2}}\ ,\
\frac{X^{4}-1}{(X-1)^4(X^2+1)^2}\ ,\ \frac{X^{n}-1}{(X-1)^4(X^2+1)^2}\ ,\
\frac{1}{(x-a)^{n}(x-b)^{n}}\ $.

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