Algebra
Polynomial Solver
Enter polynomial coefficients and receive all complex roots using the Durand–Kerner method.
Enter coefficients (highest degree first) to approximate complex roots.
General polynomial
The calculator normalizes the polynomial a₀xⁿ + a₁xⁿ⁻¹ + … + aₙ and iteratively refines the roots until convergence.
How to use
- List coefficients from highest degree to constant term.
- Separate coefficients with commas or spaces.
- Review each root along with its complex form.
Example
Input: Coefficients = 1, −6, 11, −6
Output: Roots: x = 1, 2, 3
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Polynomial Solver: polynomial equation solver, polynomial root calculator, system of equations calculator, simultaneous equations solver. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Polynomial Solver: Finds approximate roots (real or complex) for higher-degree polynomials. It's built around polynomial, roots, equation solver, so you can go from a raw question to a checked answer without switching tools.
The math behind it: The calculator normalizes the polynomial a₀xⁿ + a₁xⁿ⁻¹ + … + aₙ and iteratively refines the roots until convergence.
To use it well: (1) List coefficients from highest degree to constant term. (2) Separate coefficients with commas or spaces. (3) Review each root along with its complex form. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering Coefficients = 1, −6, 11, −6 returns Roots: x = 1, 2, 3. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
How accurate are the roots?
Roots converge to double-precision accuracy for well-conditioned polynomials. The output rounds to eight decimals for readability.
Does the order of coefficients matter?
Yes. Always supply coefficients starting with the highest power of x down to the constant term.
What formula does the Polynomial Solver use?
The calculator normalizes the polynomial a₀xⁿ + a₁xⁿ⁻¹ + … + aₙ and iteratively refines the roots until convergence.
How do I use the Polynomial Solver?
List coefficients from highest degree to constant term. Separate coefficients with commas or spaces. Review each root along with its complex form.