Polynomial Roots Calculator
Find every real and complex root of a polynomial of degree 1 to 12. Enter coefficients from the highest-degree term down to the constant, and get exact formulas for degree ≤ 2 or Durand–Kerner iteration for higher degrees.
Input
Comma- or space-separated coefficients, from the highest-degree term down to the constant term (degree 1 to 12). Decimals are supported.
Output
| Root | Real part | Imaginary part |
|---|---|---|
| No data yet | ||
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/polynomial-roots-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"coefficients": "1, -5"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `polynomial-roots-calculator` tool (Polynomial Roots Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/polynomial-roots-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Polynomial Roots Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|
Also available with
Guides
Finding every root of a polynomial by hand only works cleanly up to degree 2 — beyond that there's no general algebraic formula, and hunting for real or complex roots by trial substitution gets unreliable fast. This calculator solves for all of them at once: real, irrational, and complex, for any polynomial from degree 1 up to 12.
How to use it
- Enter the polynomial's coefficients, highest degree first — for x³ − 6x² + 11x − 6, enter
1, -6, 11, -6(a missing term is a zero, and it matters: don't skip it). - Read off the parsed equation (to confirm it matches what you meant), the full roots table, and which method solved it.
Decimals are supported — coefficients don't need to be whole numbers.
The method
- Degree 1: solved directly, x = −b⁄a.
- Degree 2: the quadratic formula, x = (−b ± √(b² − 4ac)) ⁄ 2a — a positive discriminant gives two real roots, zero gives a repeated real root, and a negative discriminant gives a complex conjugate pair.
- Degree 3–12: Durand–Kerner iteration, a numerical method that refines all n roots simultaneously each pass — r ← r − f(r) ⁄ Π(r − rⱼ) over every other current root guess — starting from points spread around a circle sized to bound every root (Cauchy's bound). It converges to the complete set of real and complex roots together, which is why it's the standard approach once no closed-form formula exists.
Tiny floating-point residue (far below any digit shown) is snapped to exactly 0, so a root that's actually real doesn't display a stray imaginary part like 1e-14i.
FAQ
Why does a cubic sometimes show one real root and a complex pair, and other times three real roots?
That's determined by the polynomial itself, not the method — a real cubic always has at least one real root, and either two more real roots or a complex conjugate pair. The calculator finds whichever the coefficients actually produce.
How precise are the roots?
Durand–Kerner iterates until the roots stop moving by more than 1e-13, then results are shown to 6 significant figures — accurate enough for any practical use, though it's a numerical approximation rather than an exact closed form for degree 3 and up.
What's the largest polynomial this handles?
Degree 12 (13 coefficients). Beyond that, root-finding becomes numerically unstable for a general-purpose calculator — the coefficients of a high-degree polynomial can be extremely sensitive to its roots' exact positions.
Related tools
For a rational-roots-only search on integer-coefficient polynomials — with exact fractions and the factored form spelled out — see the Polynomial Factoring Calculator. For a quadratic specifically, with a full step-by-step derivation, see the Quadratic Formula Calculator. To divide a polynomial by a known root and check the remainder, see the Synthetic Division Calculator.
Privacy
All calculations run entirely in your browser — the coefficients you enter are never sent to a server.