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\begin{document}
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{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{2s-8}{s^2-5s+6}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{4}{s-2}-\frac{2}{s-3}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[Y(s)](t)=4e^{2t}-2e^{3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{Q(s)}{P(s)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{2s-8}{s^2-5s+6}=\frac{A}{s-2}+\frac{B}{s-3}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[Y(s)](t)=Ae^{2t}+Be^{3t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{A}{s-2}+\frac{B}{s-3}=\frac{A(s-3)+B(s-2)}{(s-2)(s-3)}
=\frac{(A+B)s+(-3A-2B)}{s^2-5s+6}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{2s-8}{s^2-5s+6}=\frac{(A+B)s+(-3A-2B)}{s^2-5s+6}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}2=A+B \hspace{.2in} -8=-3A -2B
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(s)=(s-a_1)(s-a_2)\cdots(s-a_n),
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{Q(s)}{P(s)}=\frac{A_1}{s-a_1}+\frac{A_2}{s-a_2}
+\cdots+\frac{A_n}{s-a_n}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[\frac{Q(s)}{P(s)}](t)=A_1e^{a_1t}+A_2e^{a_2t}+\cdots
A_ne^{a_nt}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{s+1}{(s+2)^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{s+1}{(s+2)^2}=\frac{A}{s+2}+\frac{B}{(s+2)^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[Y(s)](t)=Ae^{-2t}+Bte^{-2t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{3s+9}{s^2+4s+5}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}s^2+4s+5=s^2+4s+4+1=(s+2)^2+1
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{3s+9}{(s+2)^2+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{3s+9}{(s+2)^2+1}=\frac{A(s+2)+B}{(s+2)^2+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{3s+9}{(s+2)^2+1}=A\frac{s+2}{(s+2)^2+1}+B\frac{1}{(s+2)^2+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}y(t)=L^{-1}[Y(s)](t)=Ae^{-2t}\cos t+Be^{-2t}\sin t
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{3s+9}{(s+2)^2+1}=\frac{A(s+2)+B}{(s+2)^2+1}
=\frac{As+2A+B}{(s+2)^2+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}Y(s)=\frac{s^2+9s+3}{(s-1)(s^2+4s+5)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{s^2+9s+3}{(s-1)(s^2+4s+5)}=\frac{A}{s-1}+{Bs+C}{s^2+4s+5}
\end{displaymath}
}

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\end{document}
