Evaluation homomorphism

Definition

Let $R$ be a commutative ring and let $c \in R$. The evaluation homomorphism at $c$ is the map $$\operatorname{ev}_c \colon R[x] \to R, \qquad \operatorname{ev}_c(f) = f(c),$$ where $f(c)$ is the value of $f$ at $c$. It is a ring homomorphism.

Remarks

The map $\operatorname{ev}_c$ is surjective, because $\operatorname{ev}_c(r) = r$ for each constant $r \in R$. Its kernel is the principal ideal $(x - c)$. This is a form of the factor theorem. The commutativity of $R$ is necessary: in general $(f g)(c) \neq f(c) g(c)$ when $c$ does not commute with the coefficients.

Explanations

L1 human

Evaluation at $c$ turns each polynomial into an element of $R$. It respects sums and products: to evaluate $f g$, you can evaluate $f$ and $g$ and multiply.