Polynomials and the factor theorem

A polynomial has terms in powers of x, like x³ − 2x² + 5. The factor theorem says (x − a) is a factor exactly when p(a) = 0, which lets you factorise and solve cubic equations.

Key facts

  • Remainder theorem: dividing p(x) by (x − a) leaves remainder p(a).
  • Factor theorem: if p(a) = 0, (x − a) is a factor.
  • Try factors of the constant term to find a first root.
  • Then divide (or compare coefficients) to find the quadratic factor.

Worked examples

Example 1

Factorise x³ − 2x² − 5x + 6 fully.

  1. Try x = 1. p(1) = 1 − 2 − 5 + 6 = 0
  2. So (x − 1) is a factor. Divide to get the quadratic. x² − x − 6
  3. Factorise it. (x − 1)(x + 2)(x − 3)

Answer: (x − 1)(x + 2)(x − 3)

Example 2

Find the remainder when 2x³ + x − 4 is divided by (x − 2).

  1. p(2) = 16 + 2 − 4 = 14

Answer: 14