Concept preview

What you will be able to do

1. 01Use the remainder theorem to evaluate and the factor theorem to test factors
2. 02Apply the rational root theorem and division to factor higher-degree polynomials
3. 03Solve polynomial equations completely, including irreducible quadratic factors

Dividing a polynomial $P(x)$ by a linear $(x - a)$ leaves a remainder, and the remainder theorem says that remainder is exactly $P(a)$ — you never have to do the division to find it, just substitute. Formally $P(x) = (x - a)Q(x) + P(a)$. This turns a division problem into a single evaluation, which is why it is the workhorse of everything that follows.