ID020 · Algebraic identities · Classes 9–11
Factor theorem
The factor theorem: (x − r) is a factor of p(x) exactly when p(r) = 0. On the graph, r is where the curve crosses the x-axis.
p(r) = 0Factor theorem
📐 Formula
p(r) = 0 ⇔ (x − r) divides p(x)
✏️ Worked example
p(x) = x² − 5x + 6: p(2) = 4 − 10 + 6 = 0, so (x − 2) is a factor: p(x) = (x − 2)(x − 3).
🌍 Where it is used
Factorising cubics and finding roots.
❓ Quick check
Is (x − 1) a factor of x² + x − 2?
🏋️ Practice
EasyIs (x + 1) a factor of x² − 1?
Answer: Yes
MediumIs (x − 2) a factor of x² + 4?
Answer: No (p(2) = 8)
ChallengeFind a factor of x³ − 6x² + 11x − 6.
Answer: (x − 1), since p(1) = 0