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 x0, you can use long division (or if you are lazy, a computer program...) to divide the polynomial with (x - x0) to give a polynomial of one smaller degree, where the other solutions will be easier to find.
For example, the solutions to the equation x3 - 2x2 - 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 x3 - 2x2 - 2x + 4 = (x - 2)(x2 - 2), so that the additional solutions +/- sqrt(2) are apparent.