Polynomials

Why

Definition

Let $(A, +, \cdot )$ be a ring.

A polynomial in $A$ of degree $d$ is a function $p: A \to A$ for which there exists a finite sequence $c = (c_0, c_1, \dots , c_{d-1}, c_d) \in A^{d+1}$ satisfying

\[ p(a) = c_0 + c_1a^1 + c_2a^2 + \dots + c_da^d, \]

for all $a \in A$. We call the sequence $c$ the polynomial coefficients, and call the $c_i$ the coefficients of $p$. We call $d+1$ the order of the polynomial.

Clearly, to every polynomial in $A$ of degree $d$ there corresponds a sequence in $A$ of length $d+1$, and vice versa. For this reason, we can identify polynomials by their coefficients.

Examples

The function $f: A \to A$ is a polynomial of degree 0 and order 1 if there exists $c_0$ so that

\[ f(a) = c_0 \]

for all $a \in A$.

The function $g: A \to A$ is a polynomial of degree 1 and order 2 if there exists $c_0$ and $c_1$ so that

\[ g(a) = c_0 + c_1a \]

The function $h: A \to A$ is a polynomial of degree 2 and order 3 if there exists $c_0$ and $c_1$ so that

\[ h(a) = c_0 + c_1a + c_2a^2. \]

In other words, a second degree polynomial is a quadratic.

Future editions will include, and most likely will build on quadratics and an appeal to the simplicity of the “natural” algebraic operations. ↩︎