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% division2.tex
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% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
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\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}
\uncover<2->{%
\begin{block}{Lösung}
\vspace{-15pt}
\[
\arraycolsep=1.4pt
\renewcommand{\arraystretch}{1.2}
\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
\llap{$($}X^4&-&       X^3&-&         7X^2&+&          X&+&           6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\
\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+&   \frac12X^2\rlap{$)$}}& &           & &            & &    & & & & & &          & &        &             \\
   &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+&          X}& &            & &    & & & & & &          & &        &             \\
   &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& &            & &    & & & & & &          & &        &             \\
   & &          &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+&           6}& &    & & & & & &          & &        &             \\
   & &          &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& &    & & & & & &          & &        &             \\
   & &          & &             & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& &    & & & & & &          & &        &             \\
\end{array}
\]
Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann
\end{block}}

\end{frame}