How to use Polynomial Root Finder

Enter comma-separated coefficients from the highest power to the constant.
Include zero coefficients for missing powers and calculate real candidates.
Review each normalized residual and the precision warning before using a root.

Example: Polynomial Root Finder

Polynomial Root Finder: 2 distinct approximate real root candidates.

You add

Coefficients, highest power first: 1, 0, -1

You get

2 distinct approximate real root candidates. -1 1 Repeated or close roots can be indistinguishable at floating-point precision. Results are numerical candidates, not a certified root count. -1 | 0 1 | 0

Options

Coefficient order: Enter coefficients from the highest power to the constant term. Keep zero coefficients for missing intermediate powers.
Residual: The normalized residual checks how nearly a returned candidate satisfies the entered polynomial at floating-point precision.

Supported inputs and limits

Real coefficients, degree 1–8, each within ±1,000,000 and a coefficient-derived bound within that range. Real candidates only, not complex roots. Floating-point residuals do not certify root count or multiplicity; close or repeated roots can be indistinguishable. Near-zero turning points are refused when repeated or clustered roots are numerically ambiguous.

Where your input is processed

This tool processes your input in this browser. Your text and files are not uploaded to UseFreeTools. Check this tool's limits for anything it may save on your device.

A small residual does not certify every root

This bounded finder reports approximate real candidates only. Close or repeated roots can be difficult to distinguish, and complex roots are outside its output. Numerically ambiguous turning points are refused rather than presented as a proven complete root list.

Questions about Polynomial Root Finder

Can this return complex roots?

No. This version searches for approximate real candidates.

What does a small residual prove?

It shows that substituting the candidate gives a small normalized polynomial value; it does not certify multiplicity or a complete root count.

Which coefficient goes first?

Enter the highest-power coefficient first and include zeros for missing powers.

Project manager: Tony Hines · Content updated 4 October 2026 · Report a problem