Polynomial Root Finder (any degree, numeric)

Understanding the polynomial root finder — concept and principle

This tool finds all roots, real and complex, of a polynomial of any degree. You enter the coefficients from the highest power down to the constant, and it returns n roots for a degree-n polynomial. Beyond degree 4 there is no general formula in radicals (Abel-Ruffini theorem), so numerical methods are the standard way to solve such equations in engineering, control theory, signal processing and physics.

How it is calculated

1. Leading zeros are removed and every coefficient is divided by the leading one, so the polynomial is monic: p(z) = zⁿ + c₁zⁿ⁻¹ + … + cₙ.
2. The method starts from n different complex guesses spread around a circle.
3. Durand-Kerner (Weierstrass) iteration updates every guess at once: zᵢ ← zᵢ − p(zᵢ) ÷ ∏(zᵢ − zⱼ), the product taken over all j ≠ i. p(zᵢ) is evaluated with Horner's method.
4. After 500 rounds the roots are shown to 8 significant digits; an imaginary part below 10⁻⁹ is treated as zero and the root is shown as real.

The principle

A monic polynomial of degree n factors completely as p(z) = (z − r₁)(z − r₂)…(z − rₙ) (fundamental theorem of algebra). If all guesses except zᵢ were already exact roots, then p(zᵢ) ÷ ∏(zᵢ − zⱼ) would equal exactly zᵢ − rᵢ, so one step would land on the root. Applying this correction to all guesses together pulls each one toward a different root, and the denominator keeps them from collapsing onto the same root. Near simple roots convergence is quadratic: the number of correct digits roughly doubles each round.

Worked example

1. Input: 1 0 −7 6, meaning p(x) = x³ − 7x + 6 (the x² coefficient is 0).
2. The polynomial is already monic, so three starting guesses are iterated.
3. The guesses converge to 2, −3 and 1.
4. The calculator sorts the roots from smallest to largest. Result: −3, 1, 2.
5. Check: p(1) = 1 − 7 + 6 = 0, so (x − 1) is a factor; dividing gives x² + x − 6 = (x − 2)(x + 3).

How to read the result

Roots appear in the order the iteration settles, not sorted. Complex roots of a real polynomial come in conjugate pairs a ± bi. Vieta's formulas give a quick check: the roots sum to −(second coefficient ÷ first), here 2 − 3 + 1 = 0.

Common mistakes

• Leaving out zero coefficients, which shifts every power. x³ − 7x + 6 must be entered as 1 0 −7 6.
• Entering coefficients from the constant upward.
• Expecting exact values such as √2; the output is a decimal approximation.

Frequently asked questions

Why does a degree-n polynomial always have n roots?

By the fundamental theorem of algebra, counted with multiplicity and allowing complex numbers.

Is there a formula for quintic equations?

Not in general. The Abel-Ruffini theorem shows that no formula using only arithmetic and radicals solves every polynomial of degree 5 or higher.

How do I find only the real roots?

Take the roots shown without an i term.

Sources: E. Durand, Solutions numériques des équations algébriques, Masson, 1960 · I. O. Kerner, Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen, Numerische Mathematik 8 (1966)