Question

How can I tell, when doing synthetic division, if the binomial is a FACTOR of the dividend?

Answer 1

If you get 0 as the last value in the bottom row, then the binomial is a factor of the dividend.

Let's say the binomial is of the form (x-k) and it multiplies with some other polynomial q(x) to get p(x), so,

p(x) = (x-k)*q(x)

If you plug in x = k, then,

p(k) = (k-k)*q(k)

p(k) = 0

The input x = k leads to the output y = 0. Therefore, if (x-k) is a factor of p(x), then x = k is a root of p(x).

It turns out that the last value in the bottom row of a synthetic division table is the remainder after long division. By the remainder theorem, p(k) = r where r is the remainder after dividing p(x) by (x-k). If r = 0, then (x-k) is a factor, p(k) = 0, and x = k is a root.