$11.$ Partial fraction expansion $(10$ points$)$

a$)$ i$)$ Find roots of the denominator, ii$)$ identify cases and iii$)$ write the form of the partial fraction expan$-$

sion $($no need to determine coefficients$)$ for the fractions below:

A$)$ F$($s$)=$ s$2+1$

$($s$2+7$s $+12)($s$216)$

B$)$ F$($s$)=$ s $+1$

$($s$2+4$s $+13)($s$2+9)2$

b$)$ Find the partial fraction expansion of the following fraction with following steps i$)$ write down the

form of simple fractions and ii$)$ determine coefficients.

F$($s$)=3$s$3+19$s$2+(43+$ A$)$s $+34+5$ A

$($s$2+7$s $+10)($s$2+4$s $+4)$ where A is a real constant