A polynomial equation such as x³ − 6x² + 11x − 6 = 0 can be solved once the polynomial is factorised. The main tool is the factor theorem.
The factor theorem
If f(a) = 0, then (x − a) is a factor of f(x). To solve a cubic we test simple values (±1, ±2, ±3 and other factors of the constant term) until one root is found. The remainder theorem is related: the remainder when f(x) is divided by (x − a) is f(a).
Key idea
(x − a) is a factor of f(x) ⇔ f(a) = 0. A cubic equation has at most 3 real roots.
Solving a cubic
Steps: (1) find one root a by trial; (2) divide f(x) by (x − a) to get a quadratic; (3) factorise or use the quadratic formula on that quadratic; (4) list all roots.
Worked example
Solve x³ − 6x² + 11x − 6 = 0. Test x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor. Dividing gives x² − 5x + 6 = (x − 2)(x − 3). Hence x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3), and the roots are x = 1, 2, 3.
Remember
- Test factors of the constant term first.
- Factorise completely before writing the roots.