If I have a function $\frac{s^2-10s-171}{s^4-30s^3+260s^2-1010s+779}$ and I factor the numerator and denominator and set them equal to zero like so: $$s^4-30s^3+260s^2-1010s+779 = 0$$ $$s^2-10s-171 = 0$$ How come that when I solve for s and get 1, 19, 5+4j and 5-4j, that 19 is not a pole?
Also, when I solve for s to find zeros and get the following solutions: -9 and 19. Why is 19 not a zero here either?
It is $$\frac{(s+9)(s-19)}{(s-1)(s-19)(s^2-10s+41)}$$ the point $s=19$ is a hole and there exists a limit. The only pole is $s=1$.