The remainder theorem
When a polynomial f(x) is divided by (x − a), the remainder is simply f(a). You do not need long division — just substitute x = a.
Key idea
Remainder theorem: dividing f(x) by (x − a) leaves remainder f(a).
Factor theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.
The factor theorem
The factor theorem is the special case where the remainder is zero. If f(a) = 0, then (x − a) divides f(x) exactly, so it is a factor. This is the main tool for factorising cubics.
Worked example
Show that (x − 1) is a factor of f(x) = x³ − 2x² − 5x + 6. Substitute x = 1: f(1) = 1 − 2 − 5 + 6 = 0. Since f(1) = 0, by the factor theorem (x − 1) is a factor.
Remember
- Remainder on dividing by (x − a) is f(a).
- For a divisor (ax − b), use x = b⁄a.
- (x − a) is a factor exactly when f(a) = 0.