Are these polynomials irreducible in $\mathbb{Q}[x]$?

Question

$$x^3+x^2+x+1$$

$$x^5+x^3+x^2+1$$

$$x^5+x^3+x+1$$

I've tried applying Eisenstein criterion but I can't figure it out. Thanks in advance.

Answer 1

$x^3 + x^2 + x + 1$ has a root $-1$, and so does the second polynomial.

Answer 2

$$x^3+x^2+x+1=(x+1)(x^2+1)$$ $$x^5+x^3+x^2+1=(x^2+1)(x^3+1)$$

For $f(x)=x^5+x^3+x+1$, we have $f(2)=43$, which is prime, so by Cohn's criterion, $f(x)$ is irreducible over $\mathbb{Z}$.