The factor theorem
A polynomial has factor x − a exactly when its value at a is zero.
Evaluate the polynomial first. If f(a) = 0, then x − a is a factor. Divide by x − a, then investigate the lower-degree polynomial.
For a factor px − q, use a = q/p. Substitution with rational values can be done exactly using fractions. The related remainder theorem says the remainder on division by x − a is f(a), even when that value is not zero.
Worked example
Factorise x³ − 2x² − 5x + 6.
- f(1) = 1 − 2 − 5 + 6 = 0, so x − 1 is a factor.
- Division gives x² − x − 6.
- Factor the quadratic as (x − 3)(x + 2).
Answer: (x − 1)(x − 3)(x + 2)
Revise first: Expanding and factorising.
Practise the factor theorem
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 2.11: Factor theorem
- 9FM0 · core · Assumed A-level Mathematics knowledge: Factor theorem
- 7367 · core · Assumed A-level Mathematics knowledge: Factor theorem
- H245 · core · Assumed A-level Mathematics knowledge: Factor theorem
- H645 · core · Assumed A-level Mathematics knowledge: Factor theorem
- 7M20 · core · A2.1: Factor theorem
- 6993 · core · AL4: Factor theorem
- 4PM1 · core · 3B: Factor theorem
- 8365 · core · 2.11: A cubic with a rational root and complete equivalent product
- 9FM0 · core · Assumed A-level Mathematics knowledge: A cubic with a rational root and complete equivalent product
- 7367 · core · Assumed A-level Mathematics knowledge: A cubic with a rational root and complete equivalent product
- H245 · core · Assumed A-level Mathematics knowledge: A cubic with a rational root and complete equivalent product
- H645 · core · Assumed A-level Mathematics knowledge: A cubic with a rational root and complete equivalent product
- 7M20 · core · A2.1: A cubic with a rational root and complete equivalent product
- 6993 · core · AL4: A cubic with a rational root and complete equivalent product
- 4PM1 · core · 3B: A cubic with a rational root and complete equivalent product
Next practice: Solving quadratic equations.