Let $F$ be a field of characteristic $p$. Show that if $X^p-X-a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$.
I have no problem understanding it's solution except for one part. The solution assumes that it has one root $\alpha$ if it's reducible. Does reducible imply that polynomial has a root? For example:- $$(x+2)^4 - 2 = ((x+2)^2 - \sqrt 2)((x+2)^2 + \sqrt 2)$$ has two factors in $\Bbb Q[\sqrt 2]$ but has no root. Of course my example does not apply here.
How to show that $X^p-X-a$ has root in it's reducible in $F[X]$ for finite field? Does it apply to all reducible polynomials in polynomial rings over finite fields?