Polynomial roots calculator
Type the coefficients from the highest power to the constant, for example 1, −6, 11, −6 for x³ − 6x² + 11x − 6. The calculator returns every root, with complex roots written as a + bi.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Enter the coefficients, highest power first.
- Check the equation line to confirm the polynomial was read as intended.
- Read the roots. Complex roots appear as pairs.
How it works
Leading zeros are dropped and trailing zeros become roots at 0. The remaining polynomial is made monic and solved with the Durand–Kerner iteration in complex arithmetic, then polished with Newton steps on the original polynomial. Tiny imaginary or real parts, below one part in a billion, are set to 0.
Worked example
x³ − 6x² + 11x − 6 = 0 has coefficients 1, −6, 11, −6.
- It factors as (x − 1)(x − 2)(x − 3).
- The roots are 1, 2 and 3, all real.
For x² + 2x + 5 = 0 (coefficients 1, 2, 5) the roots are −1 − 2i and −1 + 2i.
Tips
- For degree 2, the quadratic equation solver also shows the discriminant and the steps.
- Substituting a root back into the polynomial should give almost 0.
- The sum of the roots equals −(second coefficient / first coefficient).
Frequently asked questions
How do I enter a polynomial?
List the coefficients from the highest power down to the constant term, separated by commas. For 2x³ − 4x + 7 enter 2, 0, -4, 7, with 0 standing in for the missing x² term.
What is the Durand–Kerner method?
It starts with one guess per root in the complex plane and improves them all at once, each guess moving by p(z) divided by the product of its distances to the other guesses. The iteration stops when the guesses no longer move, followed by two Newton refinement steps.
Why are some roots complex?
A polynomial of degree n has n roots counting repeats, but they need not be real numbers. A real polynomial that has a complex root also has its conjugate, so complex roots come in pairs like −1 + 2i and −1 − 2i.
Why is there no formula for degree 5?
The Abel–Ruffini theorem proves that general quintic equations cannot be solved with radicals. Numerical methods like this one still find the roots to high precision.
How accurate are repeated roots?
Simple roots are accurate to about 8 significant digits. A repeated root, as in (x − 1)², is the hard case for this method and may be off in the sixth or seventh digit, so treat it as approximate.