Polynomial

A polynomial is any function which obtains its result solely by some combination of: multiplying its inputs together, adding them together, or rescaling them.

Examples

The volume of a cube is a polynomial in its side length.

Every linear transformation is uniquely associated to a polynomial called its characteristic polynomial.

A computer program consisting of some number of for loops (and no recursion or gotos or other loops), where each for loop runs exactly $n$ times, will take a number of steps to run which is given by some polynomial in $n$. In fact, any polynomial at all can be realized in this way.

Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers).

A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc.

Piecewise polynomial functions (a type of spline) are used in computer graphics to make things look smooth.

Rational functions [TODO move to its own page]

There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.