Final answer:
The zero of the function G(s) is located at s=1, and the poles are located at s=0 and two complex conjugate poles at s = -0.5 + j1.3229 and s = -0.5 - j1.3229 on the s-plane.
Step-by-step explanation:
The given function G(s) is G(s) = (s-1) / s(s² + s + 2) e⁻¹. To find the pole and zero locations, we examine the factors in the numerator and denominator of G(s), excluding the exponential term which does not affect the locations of poles or zeros.
Zeros of G(s) are found where the numerator equals zero, which in this case occurs at s = 1. Therefore, there is one zero at s = 1.
Poles of G(s) are found where the denominator equals zero. First, there is a simple pole at s = 0 from the s term. The quadratic term s² + s + 2 can be factored or solved for zeros using the quadratic formula, but it has no real roots because the discriminant (1 - 4*2) is negative. This indicates that there are two complex conjugate poles. For illustration, if we were to find the exact complex poles, they would be at s = -0.5 + j1.3229 and s = -0.5 - j1.3229, where j is the imaginary unit.
On the s-plane, the zero is represented by a circle at (1,0), and the poles are represented by crosses: one on the origin and two symmetrically about the real axis at (-0.5, ±1.3229).