Factor Theorem

Theorem: Polynomial $f(x)$ has a factor $(x-a)$ if and only if $f(a)=0$

Prerequisites:
Euclidean Polynomials Division (proof)
Polynomial Remainder Theorem (proof)

Proof:
Any polynomial can be written according to the Euclidean division as:

$\qquad\quad f(x) = q(x)g(x) + r(x)$

Here $q(x)$, $g(x)$ and $r(x)$ are the quotient, divisor and remainder respectively.

Putting $g(x) = (x-a)$, from the remainder theorem, $r(x) = f(a)$

$\therefore\quad\;\; \begin{equation} f(x) = q(x)(x-a) + f(a)\end{equation}\qquad\qquad\qquad\qquad\qquad\qquad \ldots (1)$

Now if $(x-a)$ is a factor of $f(x)$, then remainder must be $0$.

$\therefore\quad\;\; r(x) = f(a) = 0$

Conversely, if $f(a) = 0$, then from equation $(1)$;

$\qquad\quad f(x) = q(x)(x-a)$

Hence $(x-a)$ is the factor of $f(x)$.

Hence the theorem.


Recommended:
Factor Representation of a Polynomial
Polynomial Remainder Theorem