Online Tool Store
Polynomial Root Finder
Polynomial Root Finder handles the arithmetic exactly where a closed form exists and numerically where none does — the precision note tells you which you got.
🔒 This tool runs entirely in your browser. Your files are never uploaded to a server.
Math & Science
Polynomial Root Finder
Frontend preview — no upload or external service.
Roots
x^3 - 6x^2 + 11x - 6 has roots 1, 2 and 3, each with multiplicity 1, factoring cleanly as (x-1)(x-2)(x-3).
How the Polynomial Root Finder works
- Enter the coefficients from the highest power down, including zeros for missing terms.
- Choose whether complex roots are shown — a cubic always has three roots, but some may be complex.
- Check the multiplicity column, since a repeated root behaves very differently from two distinct roots close together.
The method
A degree n polynomial has exactly n roots counted with multiplicity, over the complex numbers.
for ax^2 + bx + c: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Closed-form solutions exist up to degree four; beyond that, roots are found numerically, which is why higher-degree answers are approximations.
FAQ
Why are some roots complex?
Because the polynomial does not cross the x-axis there. Real coefficients guarantee complex roots come in conjugate pairs, which is why a cubic always has at least one real root but a quartic may have none.
Is there a formula for degree five?
No general one in radicals — that is the Abel-Ruffini theorem. Degree five and above are solved numerically, which is why those answers carry a precision limit rather than being exact.
What does multiplicity mean?
How many times a root repeats. A root of multiplicity two touches the axis without crossing it; multiplicity three crosses with an inflection. It changes the graph's shape at that point.
How we compare
| Feature | Online Tool Store | A graphing calculator | A stats package |
|---|---|---|---|
| Complex roots shown | ✓ | ✗ | ✓ |
| Multiplicity reported | ✓ | ✗ | Sometimes |
| Nothing uploaded | ✓ | ✓ | ✗ |
| No install | ✓ | ✗ | ✓ |
Polynomial Root Finder handles the arithmetic exactly where a closed form exists and numerically where none does — the precision note tells you which you got.