All z-transforms in the above examples are rational, i.e., they can
be written as a ratio of polynomials of variable in the general form

where is the numerator polynomial of order with roots , and is the denominator polynomial of order with roots . In general, we assume the order of the numerator polynomial is lower than that of the denominator polynomial, i.e., . If this is not the case, we can always expand into multiple terms so that is true for each of terms.

The *zeros* and *poles* of a rational
are defined as:

**Zero**: Each of the roots of the numerator polynomial for which is a*zero*of .If the order of exceeds that of (i.e., ), then , i.e., there is a zero at infinity:

**Pole**: Each of the roots of the denominator polynomial for which is a*pole*of .If the order of exceeds that of (i.e., ), then , i.e, there is a pole at infinity: