A polynomial function is a sum of monomials:

(46) |

The numbers are called the

This sum can be finite and terminate at some
(called the *degree* of the polynomial) or can (for certain series of coefficients
with ``nice'' properties) be infinite and converge to a well defined
function value. Everybody should be familiar with at least the
following forms:

(47) | |||

(48) | |||

(49) | |||

(50) |

where the first form is clearly independent of altogether.

Polynomial functions are a simple key to a huge amount of mathematics. For example, differential calculus. It is easy to derive:

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It is similarly simple to derive

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and we will derive both below to illustrate methodology and help students remember these two

Next we note that many continuous functions can be defined in terms of
their *power series* expansion. In fact *any* continuous
function can be expanded in the vicinity of a point as a power series,
and many of our favorite functions have well known power series that
serve as an alternative definition of the function. Although we will
not derive it here, one extremely general and powerful way to *compute* this expansion is via the *Taylor series*. Let us define
the Taylor series and its close friend and companion, the binomial
expansion.