---
tag: 0GJ7T0NK49
kind: definition
name: Polynomial evaluation
msc: [13F20]
defines: 02MAKWDKGT
uses: [0Q9EZS90GY]
sources:
  - url: https://en.wikipedia.org/wiki/Polynomial_ring
    locator: Polynomial evaluation
---

## Definition

Let $R$ be a [[ring|0894J980RK]], 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|0125QVNCFQ]].
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 $f(2) = 5$.