Polynomial Root Solver - Find All Real & Complex Roots up to Degree 6
Polynomial Root Solver
Cubic, quartic and higher: every real and complex root
Degrees 1 to 6
From a straight line to a sixth-degree polynomial; a polynomial of degree n always has exactly n roots, counted with multiplicity.
Real and Complex
Complex roots of real polynomials come in conjugate pairs a ± bi, and both are listed.
Exact Rational Roots
With integer coefficients, roots such as 3 or 1/2 are confirmed exactly with integer arithmetic.
Private
The solver runs entirely in your browser.
How the Roots Are Found
There is a formula for quadratics, and longer ones for cubics and quartics, but none for degree five and above (the Abel-Ruffini theorem). So the solver uses the Durand-Kerner method, which refines guesses for all n roots at once in the complex plane, then sharpens each one with Newton's method. The result is accurate to about 15 significant digits for simple roots.
A repeated root, such as x = 1 in (x − 1)3, is numerically harder: it comes back as a tight cluster, which is merged and reported once with its multiplicity. Two genuinely different roots closer together than about one part in ten thousand can be merged the same way. Example: x3 − 6x2 + 11x − 6 = (x − 1)(x − 2)(x − 3), so the roots are exactly 1, 2 and 3.
Key Takeaways
- All roots at once: Real and complex roots are found together, with their multiplicity.
- Zero roots handled: A missing constant term means x = 0 is a root; it is factored out first.
- Exact when possible: Rational roots of integer polynomials are shown as exact fractions.