Where are the other polynomials without $\alpha$ as a root mapped?

Question

I'm reading Lang's Undergraduate Algebra, here:

Specifically here:

I am confused at how the homomorphism ($F[t]\to F(\alpha)$) is defined. It seems that if $\alpha$ is a root of some polinomial $p$, then it is mapped into $0$. But where are the other polinomials mapped? There are polynomials with roots $\beta,\gamma, etc$ and no $\alpha$ root, where are they mapped?