What does the factor theorem show?

A polynomial p(x) can be written as p(x) = (x - z)q(x) + r(x)

If we evaluate P at z, that is, find p(z) then we have,

p(z) = (z - z)q(z) + r(z)

= 0q(z) + r(z)

= r(z)

Now, if p(z) = 0 then r(z) = 0 - we have no (zero) remainder.

This means that (z - a) is a factor of p(x) - that is p(x) = (x - z)q(x).

Example:

p(x) = x^2 - 4x - 12 [1]

Chosing (x + 3) as our divisor, then,

p(x) = (x +3)q(x) + r(x)

Using [1]: p(-3) = 9 => r(x) = 9, which is NOT zero, and therefore (x +3) is NOT a factor of p(x)

For interest's sake, in this case p(x) = (x - 3)(x - 7) + 9

However, from [1]: p(6) = 36 - 24 - 12 = 0 = r(x) - ZERO remainder => (x - 6) is a factor of p(x).

So we know that p(x) = (x - 6)q(x)

In summary, considering the Remainder Theorem, if we guess a value of z that evaluates p(z) to zero, then we know that (x - z) is a factor of p(x): the Factor Theorem.