The Reducible and Irreducible Polynomials

Differences between Reducible and Irreducible Polynomial

To distinguish between reducible and irreducible polynomial we use the properties
of both polynomials which are:

Irreducible polynomial

(1)if $deg(f(x))=1$, then $f(x)$ is irreducible over F
(2)the special case where F=c, suppose that $f(x)\in{c[x]}$
then $f(x)$ is irreducible over c if and only if $deg(f(x))=1$.
(3)The special case when $F=\mathbb{R}$, assume that $f(x)\in\mathbb{R}[x]$ then

$f(x)$ is irreducible over $\mathbb{R}$ if and only if either $deg(f(x))=1$ or $deg(f(x))=2$ and the discriminant of $f(x)$ is negative.

Reducible Polynomial

(1)if $deg(f(x))=2,3$ then $f(x)$ is reducible over F if and only if there exists an
element $a\in{F}$ such that $f(a)=0F$
(2)Suppose that $char(F)\neq{2}$. suppose $f(x)=ax^2+bx+c$,where $a,b,c\in{F}$ and
$a\neq{0F}$,
then $f(x)$ is reducible over F, if and only if $b^2-4ac$
of F is called the "discriminant" of the polynomial $f(x)$.
(3)The special case where $F=\mathbb{Q}$, suppose that $f(x)\in\mathbb{Z}[x]$ and that
$f(0)\neq{0}$. then we write $f(x)=\sum^n_{j=1}a_jx^j$ where $a_0\neq{0}$
and $a_n\neq{0}$. suppose that $r\in\mathbb{Q}$ and $f(r)=0$ then we can write r in
reduced form: $r=\frac{a}{b}$, where $a,b\in\mathbb{Z}$, and $b>0$.