Different types of Polynomial Equations

For a non-negative integer $n$, a polynomial of degree $n$ in one variable $x$ is an expression

where the coefficients $a_r$ ($r=0,1,\ldots,n$) are constants and the leading coefficient $a_n \ne 0$. The variable $x$ may be real or complex. When every $a_r$ is real we say "$P$ is a polynomial over $\mathbb R$"; the same phrasing is used for $\mathbb Q$ (rationals), $\mathbb Z$ (integers) and $\mathbb C$ (complex numbers). The term $a_nx^n$ is the leading term, and a polynomial with $a_n=1$ is called monic.

The corresponding polynomial equation is $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$. If a number $c$ satisfies $a_nc^n+a_{n-1}c^{n-1}+\cdots+a_1c+a_0=0$, then $c$ is called a zero of the polynomial and a root (or solution) of the equation — the two words describe the exact same number, viewed either as "where $P$ vanishes" or "what satisfies the equation," and this chapter uses both freely.