Polynomial Root Finder
Find all real roots of a polynomial up to degree 8 — multiple roots included. Enter coefficients from the highest degree down to the constant term.
Polynomial
Highest degree first, comma separated.
1, -6, 11, -6 means x³ − 6x² + 11x − 6. · 1, 0, -4 means x² − 4.
How it works
Between any two consecutive critical points (roots of the derivative) a polynomial is monotone, so each interval contains at most one root. The finder locates the critical points recursively, brackets sign changes, and refines each root by bisection to about 1e-9. Even-multiplicity roots (where the graph just touches the axis) are caught by midpoint checks. Each reported root is verified by evaluating p(x) at it.
Frequently asked questions
What about complex roots?
Only real roots are reported; complex conjugate pairs are skipped. If your polynomial has no real roots, the tool says so rather than inventing complex ones.
Why is the degree capped at 8?
High-degree polynomials are extremely sensitive to coefficient rounding (the Wilkinson effect) — beyond degree 8, double precision itself becomes the limit, not the algorithm. Scale or factor such polynomials first.
How are multiple roots handled?
Repeated roots don't change sign, so sign-change bracketing alone would miss them. The finder also checks interval midpoints, which catches even-multiplicity roots where the graph just touches the axis. The p(x) residual next to each root confirms the refinement.
Is my data sent to a server?
No. Every computation runs in JavaScript in your own browser. Nothing you type ever leaves your device.