determine the locations of all the poles and zeros (including zeros at s = infinite). Make an S-Plane plot of the infinite poles and zeros Determine the locations of all the poles and zeros (including zeros at $S = \infty$). Make an $S$-Plane plot of the infinite poles and zeros.
$$G(s) = \dfrac{5S^2 + 20S + 15}{S(S + 3)(S^2 + 4S + 4)}$$
 A: We can find the non-infinite poles and zeros as follows:
First of all, factor the top and bottom, canceling out any common factors.  We have
$$
G(s) = \frac{5s^2 + 20s + 15}{s(s+3)(s^2 + 4s + 4)}
= \frac{5(s+1)(s+3)}{s(s+3)(s+2)^2}
= \frac{5(s+1)}{s(s+2)^2}
$$
The zeros are the values of $s$ for which $G(s)=0$.  $G(s)$ will be zero whenever our simplified numerator is zero. That is to say $s$ will be a zero if
$$
5(s+1) = 0
$$
The poles are the values of $s$ for which $G(s)$ has an infinite discontinuity.  $G(s)$ will be undefined whenever our simplified denominator is zero, so we have to find the zeros of the bottom, which is to say that $s$ will be a pole if
$$
s(s+2)^2 = 0
$$
Now, to find the behavior of $G$ at infinity, you need to find
$$
\lim_{s\to \infty} G(s) = 
\lim_{s \to \infty} \frac{5s^2 + 20s + 15}{s(s+3)(s^2 + 4s + 4)}
$$
If this limit is $0$, then $G$ has a zero at $\infty$.  If this limit is $\pm \infty$, then $G$ has a pole at $\infty$.  If this limit is any non-zero constant, then $G$ has neither a zero nor a pole at $\infty$.
