Polynomial Root Finder
Find every root of a polynomial of any degree using the Durand-Kerner method, with real and complex roots, multiplicity detection, and the residual at each root.
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Every polynomial of degree n has exactly n roots
The fundamental theorem of algebra guarantees it, provided the roots are counted with multiplicity and complex roots are allowed. A cubic always has three, even when only one of them is real. This is why a solver that reports only real roots is answering a narrower question than it appears to: the missing roots are not absent, they are complex, and in applications such as control systems and signal processing the complex ones carry the information about oscillation and stability.
There is no formula beyond degree four
Quadratics have the familiar formula, cubics and quartics have more elaborate ones, and the Abel-Ruffini theorem proves that no general formula in radicals exists for degree five and above. Roots of higher degree polynomials must be found numerically. The Durand-Kerner method used here starts with a set of complex guesses spread around a circle and refines them all simultaneously, converging on every root at once rather than finding one at a time and dividing it out.
Repeated roots converge slowly
A root of multiplicity two or more is genuinely harder to compute accurately, because the polynomial is flat near it and many nearby values are almost equally good answers. Iterative methods slow down and the achievable precision drops. Roots that come out very close together are therefore reported as a likely repeated root, with the caveat that distinguishing a true double root from two distinct roots very close together is not always possible numerically.
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Frequently Asked Questions
Why are there complex roots?
Because the fundamental theorem of algebra guarantees a degree n polynomial has exactly n roots when complex numbers are allowed and multiplicity is counted. A cubic with one visible crossing on the graph still has three roots; the other two form a complex conjugate pair.
Why is there no formula for degree five?
The Abel-Ruffini theorem proves that no general solution in radicals exists for degree five or above. Such polynomials have roots, and those roots simply cannot be written as a finite expression of the coefficients using arithmetic and roots.
What is the Durand-Kerner method?
An iterative method that starts with several complex guesses spread around a circle and refines all of them simultaneously, so every root converges at once. It avoids the accumulating error of finding one root and dividing it out repeatedly.
Why is a repeated root less accurate?
Because the polynomial is nearly flat around it, so many nearby values evaluate to almost zero and the iteration has little to steer by. Precision drops, and telling a true double root from two very close distinct roots is not always possible numerically.
What does the residual tell me?
The value of the polynomial at the computed root. It should be near zero. A large residual relative to the size of the coefficients means the root has not converged, which usually indicates a repeated root or a poorly scaled polynomial.
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How to Use
Enter the polynomial coefficients from the highest power down to find all its roots.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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