Real Polynomials

Why

What are some simple functions? Here’s an answer: those that only involve addition and multiplication.¹

Definition

A real polynomial (or polynomial) of degree $d$ is a function $p: \R \to \R $ for which there exists a finite sequence $a = (a_0, a_1, \dots , a_d)$ so that

\[ p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_nx^d. \]

In particular, $q(x) = ax + b$ for $a, b \in \R $ is a polynomial of the first degree and $r(x) = ax^2 + bx + c$ for $a, b, c \in \R $ is a polynomial of the second degree.

In a sense, these are “simple” functions. We require addition (and substraction) and multiplication; but no division.

Properties

Let $p: \R \to \R $ be a polynomial of degree $d$. Then $p$ is continuous.

Let $p: \R \to \R $ be a polynomial of degree $d$. Then $p$ has derivatives of all orders. Every derivative of $p$ is a polynomial.

Let $p: \R \to \R $ be a polynomial of degree $d$. The every derivative of order $d+1$ or greater is the constant 0 function.

Future editions will modify and expand. ↩︎