Polynomial Root Finder (Cubic, Quartic & Beyond)

Type a polynomial as you would write it — x^3 - 6x^2 + 11x - 6, (x-1)^2(x+3), 3(x+0.5)^2 - 12 — or as a list of coefficients from the highest power down, and this finds every root, real and complex, for any degree up to 30. Beyond the quadratic formula there are closed forms for cubics and quartics, but they are unwieldy and numerically fragile, and from degree five there are none at all (Abel–Ruffini), so this uses the Aberth–Ehrlich method: all roots are refined together from a circle of starting points until none of them moves by more than a few units in the last place. Each root is listed with its residual — |p(z)| relative to the size of the terms, so a value near 10⁻¹⁶ means the root is as exact as double precision allows — and the polynomial is written back in factored form over the reals, with a quadratic factor for each pair of complex conjugate roots. Sum and product of the roots are compared with the coefficients (Vieta's formulas) as a further check. Repeated roots deserve a warning. A double root can only be located to about 10⁻⁸ and a triple root to about 10⁻⁵ in floating point, because the polynomial is flat there; the tool recognises roots that coincide to within that accuracy, reports them once with their multiplicity, and recovers nearly full precision by refining the value as a simple root of the (m − 1)th derivative. Some polynomials are simply ill-conditioned: in Wilkinson's famous example, the product of (x − k) for k = 1 to 20, changing the x¹⁹ coefficient from −210 by just 2⁻²³ — about one part in two billion — turns ten of its real roots into five complex pairs, some with imaginary parts near 3. Coefficients are real numbers; complex coefficients are not supported.