Find the remainder when $x^3 - 4x^2 + x + 6$ is divided by $(x - 1)$.

By the remainder theorem the remainder is $f(1) = 1 - 4 + 1 + 6 = 4$. A remainder of zero would mean $(x-1)$ is a factor, which it is not; the factors of this cubic are $(x-2)$, $(x+1)$ and $(x-3)$.