Properties of polynomial roots

In mathematics, a univariate polynomial is an expression of the form

where the $a i$ belong to some field, which, in this article, is always the field $\mathbb {C}$ of the complex numbers. The natural number $n$ is known as the degree of the polynomial.

In the following, $p$ will be used to represent the polynomial, so we have

A root of the polynomial $p$ is a solution of the equation $p = 0$ : that is, a complex number $a$ such that $p (a) = 0$ .

The fundamental theorem of algebra combined with the factor theorem states that the polynomial p has n roots in the complex plane, if they are counted with their multiplicities.

This article concerns various properties of the roots of $p$ , including their location in the complex plane.