Algebra
Remainder & factor theorems
If P(a) = 0, then (x − a) is a factor. The fastest factor check there is.
Two related theorems for polynomials.
Walk through
Step 1 of 5
The question
Is (x − 2) a factor of P(x) = x³ − 8?
The two theorems
- Remainder theorem: P(x) ÷ (x − a) leaves remainder P(a).
- Factor theorem: (x − a) is a factor ⇔ P(a) = 0 (the special case where the remainder is zero).
- For a divisor like (2x − 1), use a = 1/2 — set the bracket to zero and solve.
Your turn
Find the remainder when x³ + 2x − 1 is divided by (x − 1).
Hunting for a factor of a cubic? Test small values like ±1, ±2, ±(factors of the constant) — the first one giving P(a) = 0 hands you a factor.
The theorems
- Remainder: when P(x) is divided by (x − a), the remainder is P(a).
- Factor: if P(a) = 0, then (x − a) is a factor of P(x).
Try it
Is (x − 2) a factor of x³ − 8?
P(2) = 8 − 8 = 0. Yes — it's a factor.