% m211oh72.tex - The Generalized Remainder Formula
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\begin{center}\tcr{\rmH The Generalized Remainder Formula} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcr{Theorem 9}] \tcg{(The Generalized Substitution Method):}\\
Given polynomials \mc{f(t)} and
\begin{equation}
 \tcb{g(t) = (t - \lm_{\scr1})^{\scr m_1}\ldots (t - \lm_{\scr s})^{\scr m_s}}, \label{1}
\end{equation} 
there is a unique polynomial \mc{r(t)} of degree \mc{\le m - 1} which satisfies the following \tcg{system of equations}:
\tcb{\beq  
 r^{\scr(k)}(\lm_{\scr i}) = f^{\scr(k)}(\lm_{\scr i}),\;\quad 
\parbox{5.5cm}{$1 \le i\le s,$\\ $0 \le k \le m_{\scr i}-1.$}
\eeq}
Moreover, we then have that 
$$
  \tcb{r(t) = \txt{rem}(f,g).}
$$ 
\item[\tcr{Theorem 10}] \tcg{(The Generalized Remainder Formula):}\\
Let \mc{g(t)} be as in equation \tcb{\sref{1}} above. Then there are unique polynomials 
\beq
\tcb{e_{ik}(t), \quad  \parbox{5.6cm}{$1 \le i\le s,$\\ $0 \le k \le m_{\scr i}-1,$}}
\eeq
of degree \mc{\le m-1} such that for any \mc{f \in \C[t]} we have:
\beq
\tcb{\txt{rem}(f, g) = \sum_{\scr i=1}^{\scr s} \sum_{\scr k=0}^{m_i-1} f^{\scr(k)}(\lm_{\scr i})e_{ik}(t).}
\eeq
\end{description}

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