where the root of $f(x) $ belong?

Question

let $f(x) \in \mathbb{Z[x]}$ be a monic polynomial.then root of $f $

choose the correct options

$1.$ can belong to $\mathbb{Z}$

$2.$ always belong to ($\mathbb R$ \ $\mathbb{Q} )\cup \mathbb{Z}$

$3.$always belong to ($\mathbb C$ \ $\mathbb{Q} )\cup \mathbb{Z}$

$4.$ can belong to ($\mathbb{Q}$ \ $\mathbb{Z})$

My attempts : i take $f(x) = x^2 -1$ then root will belong to $\mathbb{Z}$,so option 1 is obiously true

here im confusing about the other options

Any Hints/solution will appreciated

thanks in advance

Answer 1

It follows from the rational root theorem that all rational roots of such a polynomial are in fact integers. So, the third option is true and the fourth option is false. And $x^2+1$ shows that the second option is false too.

Answer 2

Hint:

A consequence of the rational roots theorem is that a rational root of a monic polynomial in $\mathbf{Z}[X]$ is necessarily an integer.