I'm trying to understand the last step in a proof from Dummit and Foote on page 513. The theorem states that if $p(x)$ is irreducible in $F[x]$ (where $F$ is a field), then there exists a field $K$ containing an isomorphic copy of $F$ in which $p(x)$ has a root. 

The general argument makes sense. I'm getting hung up on the last part though. If we restrict the canonical projection $\pi:F[x]\rightarrow F[x]/(p(x))$ to $F$, it follows that $K$ contains an isomorphic copy of $F$ (I'm skipping the details here). Let $\overline{x}=\pi(x)$ denote the image of $x$ in the quotient $K$. 

The next line states $p(\overline{x})=\overline{p(x)}$ because $\pi$ is a homomorphism. Why does this follow?