Polynomial Functions

A polynomial function is a sum of monomials:

$$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_n x^n + \ldots$$ (46)

The numbers $a_0,a_1,\ldots,a_n,\ldots$ are called the coefficients of the polynomial.

This sum can be finite and terminate at some $n$ (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:

$$f(x) = a_0 \quad{\rm (0th\ degree,\ constant)}$$ (47)

$$f(x) = a_0 + a_1 x \quad{\rm (1st\ degree,\ linear)}$$ (48)

$$f(x) = a_0 + a_1 x + a_2 x^2 \quad{\rm (2nd\ degree,\ quadratic)}$$ (49)

$$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \quad{\rm (3rd\ degree,\ cubic)}$$ (50)

where the first form is clearly independent of $x$ altogether.

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

$$\frac{d x^n}{d x} = n x^{n-1}$$ (51)

It is similarly simple to derive

$$\int x^n dx = \frac{1}{n+1} x^{n+1} + {\rm constant}$$ (52)

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

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.