Factorise a cubic (no calculator solver needed)
A standard method from More about polynomials: the steps and worked examples.
Steps
- Try x = ±1, ±2, ±3, … (factors of the constant term) until f(a) = 0.
- Then (x − a) is a factor.
- Divide f(x) by (x − a) to get a quadratic.
- Factorise the quadratic (or use the formula if it won't factorise).
Worked examples
Example 1: 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 2: 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.