Overview
The factor and remainder theorems let you factorise and solve cubic (and higher) polynomials.
The theorems
- Remainder theorem: when a polynomial f(x) is divided by (x − a), the remainder
is f(a).
- Factor theorem: if f(a) = 0, then (x − a) is a factor of f(x).
Method for a cubic
- Trial values (factors of the constant term) until you find one giving f(a) = 0.
- That gives one factor (x − a).
- Divide (or compare coefficients) to get the remaining quadratic.
- Factorise the quadratic and solve.
Worked example
f(x) = x³ − 6x² + 11x − 6. Try x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor. Dividing gives (x − 1)(x² − 5x + 6) = (x − 1)(x − 2)(x − 3). Roots: 1, 2, 3.
Common mistakes
- Using (x + a) when f(a) = 0 — the factor is (x − a).
Exam tips
- Test the factors of the constant term first when hunting for a root.