Polynomials

(from "A Programmer's introduction to Mathematics" - Jeremy Kun, 2018)

Definition

A single variable polynomial with real coefficients is a function $f$ that takes a real number as input, produces a real number as output, and has the form:

$$\displaystyle f(x) = an x^n + a{n-1} x^{n-1} + \cdots + a2 x^2 + a_1 x^1 + a_0 x^0 \ \qquad = a_n x^n + a{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0

where $$a_i$$ are real numbers, called the **coefficients** of $$f$$. The **polynomial's degree** of the polynomial is the integer $$n:n \in \mathbb{N}$$. $$\displaystyle f(x) = \sum_{i=0}^n a_i x^i

It is a summation of terms, enumerated with $i=(0..n)$, where each $i^{th}$ term consisting of a constant $a_i$ multiplied with the variable $x$ raised to the $i^{th}$ power, $x^i$.

The definition mentions 3 things: 1. a polynomial with real coefficients, i.e. a function $f$ 2. coefficients $a_i$, i.e. an array of reals $[a_0,\cdots,a_n]$ 3. a polynomial's degree, i.e. a natural number $n$

The concept of a polynomial is a bit more general: it is any function of a single numeric input that can be expressed using only addition and multiplication and constants.

Example If $g$ is the function name, $t$ is the input (variable) name, $b_i$ is an array of coefficients defined as $[2,0,4,−1]$ and $n=3$ is the degree of the polynomial. By definition, the polynomial $g$ has the form:

$$\displaystyle g(t) = 2 + 0t + 4t^2 + (−1)t^3 \ \quad = 2 + 4t^2 − t^3

$$

A polynomial, like a function, can be represented as a set of ordered pairs, sometimes called points - pairing each input with its output we can obtain a set of tuples which we can plot. A polynomial's graph, plotted as a curve in space, will reveal an interesting property: the number of curvey direction-changes depends on a polynomial's degree.