Remainder Theorem Calculator
Dividing x³ − x² + x − 2 by x − 3 the long way takes most of a page. The remainder theorem gets the same remainder in one line: substitute 3, and you have it. That shortcut is the whole idea, and the remainder theorem calculator above applies it to any polynomial and any linear divisor — including the ones like 2x − 1 that most tools quietly refuse.
What Is the Remainder Theorem?
For a polynomial f(x) divided by a linear divisor x − c:
remainder = f(c)
That is the entire theorem. The remainder of the division is the polynomial evaluated at the divisor's root — no division required. It is sometimes called the polynomial remainder theorem, and occasionally Bézout's little theorem, after the mathematician who studied Euclidean division of polynomials.
Why it holds is worth one sentence. Any division can be written f(x) = (x − c)·q(x) + r. Put x = c into both sides: the first term vanishes because (c − c) = 0, leaving f(c) = r.
Finding the Root of the Divisor
Everything depends on getting c right, and this is where marks are lost. The root is the value that makes the divisor zero, so the sign flips from the one written in the bracket.
| Divisor | Set it to zero | Root c |
|---|---|---|
| x − 3 | x − 3 = 0 | 3 |
| x + 2 | x + 2 = 0 | −2 |
| 2x − 1 | 2x = 1 | ½ |
| 3x + 2 | 3x = −2 | −⅔ |
| x | x = 0 | 0 |
How to Find the Remainder Step by Step
Take f(x) = x³ − x² + x − 2 divided by x − 3, the example the calculator opens with.
Step 1 — find the root. x − 3 = 0 gives c = 3.
Step 2 — substitute. f(3) = 27 − 9 + 3 − 2 = 19.
That is the remainder. Two lines, and no division was performed. As a check, the long division gives the quotient x² + 2x + 7 with remainder 19 — the same number, after considerably more work.
Synthetic Division: Remainder and Quotient Together
Substitution gives the remainder alone. When the question also wants the quotient, synthetic division gets both in one pass, and it is the method the calculator draws for you.
Write the coefficients in order, filling any missing power with 0. Put the root on the left. Bring the first coefficient down, multiply it by the root, add it to the next coefficient, and repeat. For x³ − x² + x − 2 with root 3:
1 → 1×3 = 3, so −1 + 3 = 2 → 2×3 = 6, so 1 + 6 = 7 → 7×3 = 21, so −2 + 21 = 19
The numbers along the bottom are 1, 2, 7 and 19: the quotient is x² + 2x + 7 and the last value is the remainder. The zero-filling matters — in x⁴ − 10x + 1 the coefficient row is 1, 0, 0, −10, 1, and skipping those zeros is the most common mistake with this method.
The Factor Theorem: When the Remainder Is Zero
The factor theorem is the remainder theorem read one step further. If f(c) = 0, then the remainder is zero, which means x − c divides f(x) exactly — so x − c is a factor, and c is a root of the polynomial.
Try f(x) = x³ − 6x² + 11x − 6 divided by x − 1. Substituting: 1 − 6 + 11 − 6 = 0. The remainder is zero, so (x − 1) is a factor, and synthetic division hands you the rest: f(x) = (x − 1)(x² − 5x + 6).
That is how the theorem is normally used in practice — not to find a remainder for its own sake, but to test candidate factors and peel a polynomial apart one factor at a time. The calculator states the verdict on every calculation, so a zero remainder is never something you have to notice yourself.
Dividing by 2x − 1 and Other Non-Monic Divisors
Exam questions do not stop at x − c. Divide 2x³ + 3x² − 8x + 3 by 2x − 1 and two things change.
The root is a fraction. 2x − 1 = 0 gives x = ½, so the remainder is f(½) — which here comes out as 0, making 2x − 1 a factor.
The quotient needs adjusting. Synthetic division works with x − ½, not 2x − 1, so the quotient it produces must be divided by 2. Here it gives 2x² + 4x − 6, and halving that gives the true quotient x² + 2x − 3. The remainder is unaffected.
This is also where decimals betray you. Testing ½ as 0.5 is harmless, but a root like −⅔ becomes 0.6667 and the arithmetic drifts — you end up with a remainder near zero and no way to know whether it should have been zero. This calculator works in exact fractions from start to finish, so x² + 1 divided by 2x − 1 returns exactly 5/4, never 1.25 or 1.249999.
How to Use This Remainder Theorem Calculator
Two boxes, written as you see them. The polynomial goes in the first — any term order, brackets, fractional coefficients, powers as ^ or pasted as x². The divisor goes in the second: x − 3, x + 2, 2x − 1, or even a bare number like 3, which is read as (x − 3) and says so.
The keypad serves whichever box you tapped last, and that box is highlighted so the target is never in doubt. Each field has its own keyboard toggle if you would rather type.
Press = and read the band. The remainder appears large, with a badge saying whether the divisor is a factor. Below it, the synthetic division grid is drawn exactly as it is written on paper — root cell, coefficient row, product row, and the sums underlined with the remainder boxed at the end.
Open "Quotient & full division" for the quotient and the identity f(x) = (divisor)(quotient) + remainder written out. The step panel below repeats the argument in order, finishing with the factor verdict.
Worked Examples
| Division | Root | Remainder | Quotient |
|---|---|---|---|
| x³ − x² + x − 2 ÷ (x − 3) | 3 | 19 | x² + 2x + 7 |
| x² + 3x − 2 ÷ (x − 2) | 2 | 8 | x + 5 |
| x⁴ − 10x + 1 ÷ (x + 3) | −3 | 112 | x³ − 3x² + 9x − 37 |
| x³ − 6x² + 11x − 6 ÷ (x − 1) | 1 | 0 — a factor | x² − 5x + 6 |
| x³ + 2x² − 5x − 6 ÷ (x + 1) | −1 | 0 — a factor | x² + x − 6 |
| 2x³ + 3x² − 8x + 3 ÷ (2x − 1) | ½ | 0 — a factor | x² + 2x − 3 |
| x² + 1 ÷ (2x − 1) | ½ | 5/4 | ½x + ¼ |
Every row can be tapped straight into the calculator. The last two are the ones worth trying: a fractional root that stays exact, and a fractional remainder that never turns into a decimal.
Frequently Asked Questions
What is the remainder theorem?
What is the remainder theorem formula?
How do you find the root of the divisor?
What is the difference between the remainder theorem and the factor theorem?
Can you use the remainder theorem with 2x − 1?
Does the remainder theorem work for quadratic divisors?
What if the polynomial is missing a term?
Is synthetic division the same as the remainder theorem?
Why is it called Bézout's little theorem?
Is this remainder theorem calculator free?
The remainder theorem turns a page of long division into a single substitution, and the factor theorem turns that same number into a yes-or-no answer about factors. Find the root, substitute it, and read the result — or run synthetic division when the quotient is wanted too. The calculator at the top does both, keeps fractional roots exact, and draws the grid the way you would write it. When a whole polynomial needs factoring, the rational zeros theorem calculator supplies the candidates to test, and the rest of the free toolkit lives on the Monkza calculators page.