Factor Theorem

$x - c$ is a factor of $f(x)$ exactly when $f(c) = 0$ .

The Factor Theorem is the zero-remainder case of the Remainder Theorem: $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$ , i.e. $c$ is a zero of $f$ . It turns "confirm a candidate root" into "check one division comes out even."

Examples

Since $f(2) = 0$ for $f(x) = x^2 - 4$ , the Factor Theorem guarantees $x - 2$ is a factor.