Polynomial evaluation

Definition

Let $R$ be a ring, let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$ and let $c \in R$. The value of $f$ at $c$ is the element $$f(c) = a_0 + a_1 c + a_2 c^2 + \dots + a_n c^n \in R.$$

Remarks

The value $f(c)$ does not depend on the choice of $n$, because the extra coefficients are $0$. For $f = x^2 - 2$ in $\mathbb{Z}[x]$, $f(3) = 7$. If $R$ is commutative, then $(f g)(c) = f(c) g(c)$ and $(f + g)(c) = f(c) + g(c)$. This gives the evaluation homomorphism. Horner's rule computes $f(c) = a_0 + c(a_1 + c(a_2 + \dots + c a_n))$ with $n$ multiplications.

Explanations

L1 human

To evaluate $f$ at $c$, you put $c$ in place of $x$ and compute. For $f = x^2 + 1$ and $c = 2$, you get $f(2) = 5$.