Polynomials

Definition: Polynomial

A polynomial over a commutative ring $R$ in the variables $x_1,\cdots ,x_n$ is an expression which can be written as a finite sum of monomials in $x_1,\cdots ,x_n$.

NOTATION

We usually use capital Latin letters such as $P$ and $Q$ to denote polynomials. If we want to be explicit about the variables, we write $P(x_1,\cdots ,x_n)$.

Definition: Leading Coefficient

The leading coefficient of a polynomial is the coefficient of its highest-degree monomial.

Degree

Definition: Degree of a Polynomial

The degree of a nonzero polynomial $P$ is the highest degree amongst its monomials.

NOTATION

$$\mathrm{deg}(P)$$

EXAMPLE

$$\mathrm{deg}(3x^{3}y+4z^5{x}^3{y}^2)=5$$

Evaluation

Definition: Evaluation of a Polynomial

Let $P(x_1,\cdots ,x_n)$ be a polynomial over a commutative ring $R$.

The value or evaluation of $P$ at an $n$-Tuples $(r_1,\cdots ,r_n)$, where $r_1,\cdots ,r_n\in R$, is obtained by replacing $x_i$ with $r_i$ and carrying out all ring operations.

NOTATION

$$P(r_1,\cdots ,r_n)$$

Roots

Definition: Root of a Polynomial

The roots of a polynomial $P$ are the solutions to the polynomial equation $P=0$.

Equality

DEFINITION

Two polynomials are equal if they are built from the same monomials.

NOTE

The order of the monomials is irrelevant.

EXAMPLE

$$4x^{2}y{z}^4-3x^{3}y+7z^2=7z^2+4x^{2}y{z}^4-3x^{3}y$$

Operations

Definition: Polynomial Addition

TODO

Definition: Polynomial Multiplication

TODO