The Remainder and Factor Theorems
7 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
The remainder theorem is a shortcut: when you divide a polynomial f(x) by (x − c), the remainder is simply f(c). You can find a remainder by substitution instead of long division. Its close relative, the factor theorem, tells you when (x − c) divides a polynomial exactly.
The remainder theorem
If a polynomial f(x) is divided by (x − c), the remainder equals f(c). Example: divide f(x) = x³ − 4x² + 2x + 5 by (x − 2). Instead of dividing, substitute x = 2: f(2) = 8 − 16 + 4 + 5 = 1. The remainder is 1.
Why it works: division gives f(x) = (x − c)·q(x) + r. Putting x = c makes the first term zero, leaving f(c) = r.
The factor theorem
(x − c) is a factor of f(x) exactly when f(c) = 0. For f(x) = x³ − 6x² + 11x − 6: f(3) = 27 − 54 + 33 − 6 = 0, so (x − 3) is a factor. Testing f(1) and f(2) also gives zero, so the roots are 1, 2 and 3 and f(x) = (x − 1)(x − 2)(x − 3).
To find candidates, try the factors of the constant term (±1, ±2, ±3, ±6 here). This is the rational root idea.
Synthetic division
A fast way to divide by (x − c). For x³ − 6x² + 11x − 6 ÷ (x − 3), write the coefficients 1, −6, 11, −6 and c = 3. Bring down 1. Multiply by 3 and add: −6 + 3 = −3. Multiply by 3 and add: 11 − 9 = 2. Multiply by 3 and add: −6 + 6 = 0. The last number is the remainder (0), and the others give the quotient x² − 3x + 2. That factors as (x − 1)(x − 2).
A remainder that is not zero
Divide 2x³ + 3x² − 5x + 4 by (x − 1) with c = 1: bring down 2; 3 + 2 = 5; −5 + 5 = 0; 4 + 0 = 4. The quotient is 2x² + 5x and the remainder is 4, which matches f(1) = 2 + 3 − 5 + 4 = 4.
When to use each idea
- Just need the remainder? Substitute (remainder theorem).
- Testing whether (x − c) is a factor? Check whether f(c) = 0.
- Need the quotient? Use synthetic or long division.
For dividing by things like (2x + 1), evaluate at x = −1/2 instead.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Solve x² − 7x + 10 = 0.
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x₁: 5 · x₂: 2
- Find the roots of f(x) = x² − 6x + 8.
Show answer
Function: f(x) = x^2 - 6x + 8
Frequently asked questions
What is the remainder theorem?
When f(x) is divided by (x − c), the remainder is f(c).
What is the factor theorem?
(x − c) is a factor of f(x) if and only if f(c) = 0.
How do I find the remainder when dividing by (x + 3)?
Rewrite x + 3 as x − (−3) and evaluate f(−3).
What is synthetic division?
A shortcut for dividing a polynomial by a linear factor using only the coefficients.
Try it yourself
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