BYTETOOLS

Polynomial Roots Calculator

Find every real and complex root of a polynomial from its coefficients using the Durand–Kerner method. Works for any degree from 1 to 10, instantly.

p(x) = x^3 − 6x^2 + 11x − 6 (degree 3)

Durand–Kerner iteration converged. Roots and residuals below.

All 3 roots

RootValueResidual |p(root)|
x13< 1e-14
x22< 1e-14
x31< 1e-14

What is the Polynomial Roots Calculator?

The ByteTools Polynomial Roots Calculator finds all roots — real and complex — of a polynomial from its list of coefficients.

  • All real and complex roots for degrees 1–10
  • Durand–Kerner simultaneous iteration in complex arithmetic
  • Residual |p(root)| shown for every root
  • Polynomial preview so you can verify your input
  • Handles decimal and negative coefficients
  • 100% private — runs entirely in your browser

How to use the Polynomial Roots Calculator

  1. 1

    Enter the polynomial's coefficients from the highest degree to the constant, separated by spaces or commas.

  2. 2

    Check the polynomial preview to confirm the tool read it correctly.

  3. 3

    Read the full list of roots — real values and complex a ± bi pairs.

  4. 4

    Review the residual column to see how precisely each root satisfies p(x) = 0.

  5. 5

    Copy all roots with one click.

About the Polynomial Roots Calculator

The ByteTools Polynomial Roots Calculator finds all roots — real and complex — of a polynomial from its list of coefficients. Enter the coefficients from the highest power down to the constant term, and the tool runs the Durand–Kerner (Weierstrass) iteration, a classic method that refines all root estimates simultaneously in complex arithmetic until they converge.

Because the algorithm works natively with complex numbers, it returns the complete set of roots for degrees from 1 up to 10, including complex-conjugate pairs, and reports the residual |p(root)| for each one so you can judge accuracy. Roots that are numerically real are displayed as plain real numbers.

Everything is computed with JavaScript, 100% locally in your browser. Your polynomial never leaves your device, and the full root list can be copied in one click.

Frequently asked questions

How many roots does a polynomial have?

By the fundamental theorem of algebra, a degree-n polynomial has exactly n roots when complex roots and multiplicities are counted. A degree-5 polynomial always has 5 roots, though some may coincide and some may be complex-conjugate pairs.

What is the Durand–Kerner method?

It is an iterative algorithm that starts with n rough complex guesses and improves all of them at once: each estimate is corrected by p(x) divided by the product of its differences from the other estimates. After a few dozen iterations the estimates converge to the actual roots simultaneously.

Why do complex roots come in conjugate pairs?

When a polynomial has only real coefficients, taking the complex conjugate of the equation p(x) = 0 shows that the conjugate of any root is also a root. That is why complex roots always appear as a + bi together with a − bi for real-coefficient polynomials like the ones this tool accepts.

How accurate are the computed roots?

The iteration runs until corrections drop below about 1e−12 or an iteration cap is reached, and each root is listed with its residual |p(root)|. Residuals near zero mean the root is accurate. Very high degrees or clustered repeated roots are harder numerically, which is why the tool caps the degree at 10.

How do I enter the polynomial x³ − 6x² + 11x − 6?

Type the coefficients from the highest power down: 1 -6 11 -6. The tool shows the polynomial it parsed so you can double-check, then returns the roots 1, 2 and 3. Missing powers must be entered as 0 — for x³ + 1 you would type 1 0 0 1.

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