A rational function can be written as

$$R(q)=\frac{N(q)}{D(q)},$$

where $N$ and $D$ are polynomials and $q$ may be real or complex.

A zero is a value of $q$ that makes the numerator vanish. A pole is a value that makes the denominator vanish, after any exact common factors have been cancelled.

Factoring exposes these locations:

$$R(q)=K\frac{\prod_i(q-z_i)}{\prod_j(q-p_j)},$$

where $z_i$ are zeros and $p_j$ are poles.

Near a zero, the function tends toward zero. Near a pole, its magnitude can grow without bound.

For rational transfer functions, zeros and poles become a compact way to describe how a system behaves. Their dynamical meaning depends on the transform variable: continuous-time systems use the $s$-plane, while discrete-time systems use the $z$-plane.