Show $ x^4 + x^2 + 1 $ has no integer roots, but that it has a root modulo 3 and factorize it

Question

Show that the following polynomial $ x^4 + x^2 + 1 $ has no integer roots, but that it has roots modulo 3, and factorize it over $ℤ_3$.

I'm not sure how to go about this problem. Thank you for your help!

Answer 1

$x^4+x^2+1$ is positive since even powers of reals are non-negative and $1$ is positive.

To prove it has a root $\bmod 3$ it suffices to check the three congruence classes.