**Theorem:** Given a polynomial *P(x)*, *(x - a)* divides *P* (is a factor of *P*) iff *P(a) = 0*.

**Proof:** The factor theorem follows from a more general statement: the remainder when dividing a polynomial *P(x)* with *(x - a)* is *P(a)*. To see why this is so, assume that the quotient is *Q(x)* and the remainder *R*, that is:

*P(x) = (x - a)Q(x) + R*

Note that every polynomial can be written on this form. But now

*P(a) = (a - a)Q(x) + R = 0*Q(x) + R = R*, as required. The remainder will be zero, and

*(x - a)* a factor,

iff *a* is a zero of

*P*. ∎

The factor theorem is useful when solving polynomial equations, because once you have found a solution *x*_{0}, you can use long division (or if you are lazy, a computer program...) to divide the polynomial with *(x - x*_{0}) to give a polynomial of one smaller degree, where the other solutions will be easier to find.

For example, the solutions to the equation *x*^{3} - 2x^{2} - 2x + 4 = 0 are not immediatly obvious. Through a bit of trial and error, we find that *x = 2* is a solution. We can then divide by *(x - 2)*, to find that *x*^{3} - 2x^{2} - 2x + 4 = (x - 2)(x^{2} - 2), so that the additional solutions +/- sqrt(2) are apparent.