The Factor Theorem
Simple Explanation
The Factor Theorem is a direct consequence of the Remainder Theorem: (x − a) is a factor of P(x) — meaning it divides P(x) exactly, with remainder 0 — exactly when P(a) = 0, i.e. when a is a root of P(x).
Why Do We Need It?
This turns "is this a factor?" into a simple substitution-and-check — the essential first step for factoring any polynomial of degree 3 or higher.
Formula
The Factor Theorem
(x − a) is a factor of P(x) ⇔ P(a) = 0
A linear expression (x − a) divides a polynomial P(x) exactly (with zero remainder) exactly when a is a root of P(x) — substituting a into P(x) gives zero.
- P(x)
- — the polynomial being tested
- a
- — a candidate root — the constant in the linear factor (x − a)
When to use it: Whenever you need to test whether a specific value is a root of a polynomial, or to find factors of a polynomial by testing candidate values.
Worked Example
Test whether an expression is a factor
Is (x − 1) a factor of P(x) = x³ − 6x² + 11x − 6?
Why Does This Work?
By the remainder theorem, dividing P(x) by (x − a) leaves remainder P(a). "(x−a) is a factor" means that division leaves remainder exactly 0 — combining these two facts directly gives: (x−a) is a factor of P(x) if and only if P(a) = 0.
Real-Life Example
Checking a proposed solution to a design equation
An engineer proposes a specific value as a solution to a polynomial design equation and wants to confirm it exactly.
Testing that value with the factor theorem (checking whether P(a)=0) instantly confirms or rules out the proposed solution.
Practice
Is (x + 2) a factor of P(x) = x³ + 3x² − x − 6?
MediumCommon mistake
Testing the wrong value of a — for a candidate factor (x + 3), the correct test value is a = −3 (since x+3 = x−(−3)), not a = 3.
Quick Review
- (x − a) is a factor of P(x) if and only if P(a) = 0.
- A direct consequence of the remainder theorem: remainder 0 means exact division.
- The standard first test when trying to factor a higher-degree polynomial.