🧮 Remainder Theorem Calculator – Find Polynomial Remainders Without Long Division
The Remainder Theorem is one of the most powerful shortcuts in algebra: when a polynomial P(x) is divided by a linear divisor (x − a), the remainder equals P(a). No long division needed — just substitute and evaluate. This calculator automates that process, handles multiple divisor formats, and optionally walks you through every substitution step.
What Is the Remainder Theorem?
Formally, if P(x) is a polynomial and you divide it by (x − a), then:
Here, Q(x) is the quotient polynomial and R is a constant — the remainder. Because R is always a constant (not another polynomial), you can find it instantly by evaluating the original polynomial at x = a.
Supported Divisor Forms
Not all linear divisors look like x − a. This calculator normalises three common forms before evaluation:
| Divisor Form | Evaluation Point | Example |
|---|---|---|
x − a | x = a | x − 3 → evaluate at x = 3 |
x + a | x = −a | x + 5 → evaluate at x = −5 |
bx − c | x = c/b | 2x − 6 → evaluate at x = 3 |
The Factor Theorem
The Factor Theorem is a special case of the Remainder Theorem. It states:
Whenever you compute a remainder and get zero, the divisor divides the polynomial exactly — it is a factor. This calculator shows a Factor Status badge in the results to immediately communicate whether the divisor qualifies as a factor.
How to Use This Calculator
There are two input modes available:
- Expression Mode — type the polynomial using standard algebraic notation. Use
^for exponents and*for multiplication (e.g.x^3 - 6*x^2 + 11*x - 6). - Coefficient Mode — enter a comma-separated list of coefficients in descending order of degree (e.g.
1, -6, 11, -6for the same cubic above). The calculator pads any missing powers with zero automatically.
Then enter your linear divisor (e.g. x - 2, x + 3, or 2x - 6), choose your variable name, set decimal precision, and click Calculate.
Step-by-Step Explanation
Enable the Show Steps toggle to see every term of the polynomial evaluated individually at x = a. The table shows each monomial and its numeric value, making it easy to identify arithmetic mistakes and understand exactly how the remainder accumulates from each term.
Worked Examples
Example 1 — Zero remainder (factor found):
Evaluate at x = 2: P(2) = 8 − 24 + 22 − 6 = 0
Remainder = 0 → (x − 2) is a factor
Example 2 — Non-zero remainder:
Evaluation point: x = 6/2 = 3
P(3) = 9 + 12 + 1 = 22
Remainder = 22 → (2x − 6) is NOT a factor
Example 3 — Negative evaluation point:
Evaluation point: x = −3
P(−3) = 2(−27) + (−3) − 5 = −54 − 3 − 5 = −62
Remainder = −62
Applications in Mathematics
The Remainder Theorem and Factor Theorem are foundational tools in algebra with wide-ranging applications:
- Root finding: Test candidate roots of a polynomial without full division. A zero remainder confirms an exact root.
- Polynomial factorisation: Identify linear factors one by one, progressively reducing the degree of the polynomial.
- Synthetic division preparation: The Remainder Theorem tells you the final value in a synthetic division table before you carry out any arithmetic.
- Interpolation and error estimation: In numerical analysis, the remainder term of polynomial interpolation mirrors this structure.
Tips for Accurate Input
- Use explicit multiplication: write
3*x^2, not3x^2(some parsers require the asterisk). - For negative leading terms, include the sign:
-2*x^3 + x - 1. - In Coefficient Mode, include zeros for any missing degrees:
1, 0, -4, 3representsx³ − 4x + 3. - The divisor must be linear (degree 1). Entering a quadratic or higher-degree divisor will produce an error.