Factor theorem
A standard method from More about polynomials: the steps and worked examples.
Steps
(x − a) is a factor of f(x) if and only if f(a) = 0.
- Find the value that makes the divisor zero (x − a gives a).
- Work out f(a).
- If f(a) = 0, (x − a) is a factor. If not, it is not a factor, and f(a) is the remainder.
Worked examples
Example 1: Remainder theorem: find the remainder when x³ − 2x² + 4x − 7 is divided by x − 2.
f(2) = 8 − 8 + 8 − 7 = 1. (Long division gives quotient x² + 4, remainder 1.)
Example 2: Factor theorem: factorise the cubic f(x) = x³ − 4x² + x + 6.
f(−1) = −1 − 4 − 1 + 6 = 0, so (x + 1) is a factor
f(x) ÷ (x + 1) = x² − 5x + 6
f(x) = (x + 1)(x − 2)(x − 3)Example 3: Factorise x³ − 8.
f(2) = 0, so x − 2 is a factor: (x − 2)(x² + 2x + 4).
For the whole topic, read More about polynomials: key points, common mistakes and how the DSE tests it.