Remainder theorem

Statement

Let $R$ be a commutative ring, let $f \in R[x]$ and let $c \in R$. Then the remainder of the division of $f$ by $x - c$ is the constant polynomial $f(c)$. That is, there is $q \in R[x]$ with $f = (x - c) q + f(c)$.

Remarks

The proof: division by the monic polynomial $x - c$ gives $f = (x - c) q + r$ with $r$ a constant. Evaluation at $c$ gives $r = f(c)$. For example, the remainder of $x^3 - 2x + 5$ divided by $x - 2$ is $2^3 - 4 + 5 = 9$.

Explanations

L1 human

You do not need long division to find the remainder of $f$ divided by $x - c$. You only compute $f(c)$.