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Just a note

Polynomial series are one of the simplest and most useful mathematical functions, and have two important uses in statistics.
  1. Fitting regression curves, as described in Unit 14, and,
  2. 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 x0=1 and x1=x, the general form is often simplified to

y = c0 + c1x + c2x2 ... + cixi

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 c0, c1, c2 & c3 = 0 but c4 = 1, then a fourth-order polynomial (in powers of x) simplifies to x4

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, Ri - the relative magnitude of which provides a measure of how good an approximation is provided by the leading i-term polynomial.