Factor theorem

Statement

Let $R$ be a commutative ring, let $f \in R[x]$ and let $c \in R$. Then $x - c$ divides $f$ in $R[x]$ if and only if $c$ is a root of $f$.

Remarks

The proof uses the remainder theorem: $f = (x - c) q + f(c)$. So $x - c$ divides $f$ exactly when $f(c) = 0$. For example, $1$ is a root of $x^3 - 1$, and $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The theorem gives the bound on the number of roots of a polynomial over a field.

Explanations

L1 human

Each root gives a linear factor. If $f(c) = 0$, then you can write $f$ as $(x - c)$ times another polynomial. Conversely, a factor $x - c$ makes $f(c) = 0$.