Polynomial series are one of the simplest and most useful mathematical functions, and have two important uses in statistics.

- Fitting regression curves, as described in Unit 14, and,
- Approximating complex frequency distributions using simpler ones.

However, whilst polynomials are reasonably good at interpolation, they are notoriously unreliable for extrapolation. Also, because it is the same a which appears in each term, the terms within a polynomial tend to be related.

Curiously enough, where y=0, it is known as a **polynomial equation**. Also, because x^{0}=1 and x^{1}=x, the general form is often simplified to

y = c_{0} + c_{1}x + c_{2}x^{2} ... + c_{i}x^{i}
The highest power ( i ) of x is known as its **degree** or **order**. So a 'third order polynomial' should contain 3 terms. However, if any of the constants equal zero, they may be omitted. So, for example, if c_{0}, c_{1,} c_{2} & c_{3} = 0 but c_{4} = 1,
then a fourth-order polynomial (in powers of x) simplifies to x^{4}

More importantly, if you stop calculating a polynomial after the first i terms, (in principle at least) all of the trailing terms can be combined to form a remainder, R_{i} - the relative magnitude of which provides a measure of how good an approximation is provided by the leading i-term polynomial.