Poles and zeros of system function

As discussed before the system function $H\left(z\right)$ of a rational LTI system has a numerator and a denominator polynomial. $$\begin{array}{ccc}H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}=\frac{{\sum }_{k=0}^M{b}_k{z}^{-k}}{1-{\sum }_{k=1}^N{a}_k{z}^{-k}}& =& b_0\frac{{\prod }_{k=1}^M(1-{\beta }_k{z}^{-1})}{{\prod }_{k=1}^N(1-{\alpha }_k{z}^{-1})}\end{array}$$ Both numerator and denominator polynomials can be split as a product of first order terms. Therefore, the system function $H\left(z\right)$ of a rational LTI system is defined, to within a scale factor $b_0$, by the location of the poles ${\alpha }_k$, and zeros ${\beta }_k$. Note that each term in the numerator $1-{\beta }_k{z}^{-1}$ contributes a zero to the system function at $z={\beta }_k$ and a pole at $z=0$. Similarly, each term in the denominator $1-{\alpha }_k{z}^{-1}$ contributes a pole at $z={\alpha }_k$ and a zero at $z=0$. On the one hand, an FIR system consists, besides poles at $z=0$, of only zeros. On the other hand an IIR system consists of both poles and zeros. If the impulse response $h\left[n\right]$ is real-valued, the system function $H\left(z\right)$ is a conjugate symmetric function of $z$, thus $H\left(z\right)=H^{\ast }\left(z^{\ast }\right)$ and the complex poles and zeros occur in conjugate symmetric pairs. Thus if there is a complex pole (zero) at $z=z_0$, there is also a pole (zero) at $z=z_0^{\ast }$.