# Polynomial

Expressions of the form are called polynomials.

## Notation

The general polynomial in the variable may be written as

• The numbers are called coefficients.
• The degree of is (assuming that ).
• The coefficient is called the leading coefficient.

We use the notation to denote the set of all polynomials with coefficients in .

Also see Monomial Basis.

## Properties

Lemma 1: Degree of a Product (DP) For all non-zero polynomials and in , we have

### Polynomial Divisibility

Transitivity of Divisibility for Polynomials (TDP) For all polynomials , , in , if and , then .

Divisibility of Polynomial Combinations (DPC) For all polynomials , , in , if and , then for all polynomials , in .

### Division Algorithm for Polynomials

Division Algorithm for Polynomials (DAP) For all polynomials and in with not the zero polynomial, there exist unique polynomials and in such that , where is the zero polynomial, or .

Remainder Theorem (RT) For all polynomials and all , the remainder polynomial when is divided by is the constant polynomial .

Factor Theorem (FT) For all polynomials and all , the linear polynomial is a factor of the polynomial if and only if (equivalently, is a root of the polynomial ).

### Polynomial Factorization

Factorization Into Irreducible Polynomials (FIIP) Every polynomial in of positive degree can be written as a product of irreducible polynomials.