Proving irreducible over $\mathbb{R}[x]$?

Question

I was given the polynomial $f(x) = x^3 + 3x^2 -8$. I have successfully proved it is irreducible over $\mathbb{Q}[x]$. Does this imply it is also irreducible over $\mathbb{R}[x]$?

Answer 1

No. $f\in\mathbb{Q}[x]$ irreducible does not imply $f\in\mathbb{R}[x]$ irreducible. For instance $f(x)=x^2-2$. This has no roots in $\mathbb{Q}$, but roots in $\mathbb{R}$.

Since your polynomial has degree 3 it follows easyly from the intermediate value theorem, that is has at least one root.

Answer 2

It must be reducible over $\mathbb R $. It is a 3rd degree polynomial and so must have a real root.

Answer 3

Real $3$. degree polynomial is always reducible in $\mathbb{R}$.

Answer 4

used fundadamental theorem of algebra.....every odd degree polynomial has atleast one root in $\mathbb{R}$