Complex roots of a ploynomial

Question

Is there a way to find all the complex root for the polynomial $P(x)=(1+x+x^{2})^{n}$?

There must be $2n$ different complex roots given that the trinomial inside the brackets is of degree 2, am I right?

Answer 1

Hint:

To solve $ (1+x+x^2)^n= $ $ =\underbrace{(1+x+x^2)\cdot(1+x+x^2)\cdots(1+x+x^2)}_{n}=0 $

is the same as solve $1+x+x^2=0$ for $n$ times (product cancellation law).

Answer 2

No, you are not right. There are only two roots ($-\frac12\pm\frac{\sqrt3}2i$), and each of them has multiplicity $n$.