Polynomial Calculator
Divide polynomials (with a synthetic-division table for linear divisors), find real and complex roots and the factored form, add, subtract and multiply, or evaluate P(x) with Horner’s method — with a graph and every step shown.
Written by Suhaib Hassan. The algorithms are standard, and every worked example is re-derived by hand. Our methodology.
It parses your expression into coefficients, divides with the classical long-division algorithm, tests rational roots, then refines the rest numerically (Durand–Kerner) to find every complex root.
It shows the working — the synthetic-division tableau, the remainder theorem check and the factored form — not just the answer.
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Synthetic division tableau
For a divisor x − c: bring down the first coefficient, multiply by c, add to the next coefficient, and repeat.
Graph of P(x)
Real roots are marked in green; the y-intercept in dark.
Step-by-step working
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Which method to use
Four standard tools for working with polynomials.
| Method | Divisor form | Cost | Best for |
|---|---|---|---|
| Long division | Any polynomial of lower degree | O(deg P · deg Q) | General division with remainder |
| Synthetic division (Ruffini) | x − c (monic linear) | O(n) | Fast division and the remainder theorem |
| Horner’s scheme | Evaluation at a point | n multiplications, n additions | Efficient, numerically stable evaluation |
| Rational root theorem | p/q, p | a₀, q | aₙ | Finite candidate list | Finding exact rational roots |
| Descartes’ rule of signs | Sign changes in P(x), P(−x) | Counting only | Bounding the number of positive and negative roots |
Worked examples
Synthetic division
Divide x³ − 6x² + 11x − 6 by x − 2.
1 → −6+2 = −4 → 11−8 = 3 → −6+6 = 0
quotient x² − 4x + 3, remainder 0
Remainder theorem
What is the remainder when P(x) = x³ − 2x² + x − 5 is divided by x − 3?
Factoring
Factor x³ − 6x² + 11x − 6 completely.
x² − 4x + 3 = (x − 1)(x − 3)
(x − 1)(x − 2)(x − 3)
What is a polynomial?
A polynomial is a sum of terms aₖxᵏ with non-negative integer powers, such as 3x⁴ − 2x + 7. Its degree is the highest power and the leading coefficient is the coefficient of that term. The Fundamental Theorem of Algebra says a degree-n polynomial has exactly n roots, counting multiplicity, in the complex numbers.
Key theorems
- Division algorithm: P(x) = Q(x)·D(x) + R(x) with deg R < deg D.
- Remainder theorem: dividing by x − c leaves remainder P(c).
- Factor theorem: (x − c) divides P(x) exactly when P(c) = 0.
- Rational root theorem: for integer coefficients, any rational root p/q has p dividing the constant term and q dividing the leading coefficient.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Horner, W. G. (1819). A new method of solving numerical equations of all orders, by continuous approximation. Philosophical Transactions of the Royal Society 109, 308–335.
- Gauss, C. F. (1799). Doctoral dissertation, Helmstedt — the first proof of the Fundamental Theorem of Algebra.
- NIST Digital Library of Mathematical Functions, section 1.11 (Zeros of polynomials).
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
When can I use synthetic division instead of long division?
When the divisor is linear, x − c (a monic linear factor). It needs only the coefficients and c, and is faster and less error-prone. For a divisor such as 2x − 6, divide the coefficients by 2 first (2x − 6 = 2(x − 3)) or use long division. For divisors of degree 2 or more, use long division.
How does Horner’s scheme reduce the number of operations?
It rewrites aₙxⁿ + … + a₀ as (…((aₙx + aₙ₋₁)x + aₙ₋₂)x + …)x + a₀, needing only n multiplications and n additions instead of computing each power separately. It is also more accurate in floating-point arithmetic.
What does the Fundamental Theorem of Algebra guarantee?
Every non-constant polynomial with complex coefficients has at least one complex root, and therefore exactly n roots counting multiplicity. Real polynomials may have complex roots, but they come in conjugate pairs a ± bi.
Why are complex roots always in pairs?
If a polynomial has real coefficients and a + bi is a root, its conjugate a − bi is also a root, because conjugation preserves real coefficients. So non-real roots of real polynomials appear in conjugate pairs.
Is there a formula for the roots of every polynomial?
Only up to degree 4 (the quadratic, cubic and quartic formulas). The Abel–Ruffini theorem shows there is no general algebraic formula for degree 5 or higher, so numerical methods — as used here — are needed.
How accurate are the numerical roots?
Roots are refined to about 12 significant digits. Repeated roots and very high degrees are less well conditioned, so their computed values can be slightly less accurate. Exact rational roots are found first, before any numerical work.
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