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%
% division2.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\frametitle{Division in $\Bbbk[X]$}
\vspace{-5pt}
\begin{block}{Aufgabe}
Finde Quotienten und Rest der Polynome
$a(X) = X^4-X^3-7X^2+X+6$
und
$b(X) = 2X^2+X+1$
\end{block}
\begin{block}{Lösung}
\[
\arraycolsep=1.4pt
\renewcommand{\arraystretch}{1.2}
\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
X^4&-& X^3&-& 7X^2&+& X&+& 6&:&2X^2&+&X&+&1&=&\frac12X^2&-&\frac34X&-\frac{27}{8} = q\\
\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}& & & & & & & & & & & & & & & \\ \cline{1-5}
&-&\frac32X^3&-&\frac{15}2X^2&+& X& & & & & & & & & & & & & \\
&\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}& & & & & & & & & & & & & \\\cline{2-7}
& & &-&\frac{27}4X^2&+&\frac{ 7}4X&+& 6& & & & & & & & & & & \\
& & &\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}& & & & & & & & & & & \\\cline{4-9}
& & & & & &\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}& & & & & & & & & & & \\
\end{array}
\]
Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann
\end{block}
\end{frame}
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