# Polynomial

Expressions of the form $x_{2}+2x+1$ are called polynomials.

## Notation

The general polynomial in the variable may be written as $P(x)=a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{0}$

- The numbers are called
**coefficients**. - The
**degree**of $P$ is $n$ (assuming that $a_{n}=0$). - The coefficient $a_{n}$ is called the
**leading coefficient**.

We use the notation $R[x]$ to denote the set of all polynomials with coefficients in $R$.

Also see Monomial Basis.

## Properties

**Lemma 1: Degree of a Product (DP)**
For all non-zero polynomials $f(x)$ and $g(x)$ in $R[x]$, we have
$gf(x)g(x)=gf(x)+gg(x)$

### Polynomial Divisibility

**Transitivity of Divisibility for Polynomials (TDP)**
For all polynomials $f(x)$, $g(x)$, $h(x)$ in $R[x]$, if $f(x)∣g(x)$ and $g(x)∣h(x)$, then $f(x)∣h(x)$.

**Divisibility of Polynomial Combinations (DPC)**
For all polynomials $f(x)$, $g(x)$, $h(x)$ in $R[x]$, if $f(x)∣g(x)$ and $f(x)∣h(x)$, then $f(x)∣(a(x)g(x)+b(x)h(x))$ for all polynomials $a(x)$, $b(x)$ in $R[x]$.

### Division Algorithm for Polynomials

**Division Algorithm for Polynomials (DAP)**
For all polynomials $f(x)$ and $g(x)$ in $R[x]$ with $g(x)$ not the zero polynomial, there exist unique polynomials $q(x)$ and $r(x)$ in $R[x]$ such that $f(x)=q(x)g(x)+r(x)$, where $r(x)$ is the zero polynomial, or $gr(x)<gg(x)$.

**Remainder Theorem (RT)**
For all polynomials $f(x)∈R[x]$ and all $c∈R$, the remainder polynomial when $f(x)$ is divided by $x−c$ is the constant polynomial $f(c)$.

**Factor Theorem (FT)**
For all polynomials $f(x)∈R[x]$ and all $c∈R$, the linear polynomial $x−c$ is a factor of the polynomial $f(x)$ if and only if $f(c)=0$ (equivalently, $c$ is a root of the polynomial $f(x)$).

### Polynomial Factorization

**Factorization Into Irreducible Polynomials (FIIP)**
Every polynomial in $R[x]$ of positive degree can be written as a product of irreducible polynomials.