Polynomial Roots Calculator: Real and Complex Roots up to Degree 6
Type the coefficients from the highest power down, separated by spaces. You get all the roots, real and complex, with a graph of the polynomial.
- Nothing you type is stored
- Free
- No signup
Roots
1, 2, 3
Numeric roots- Real roots
- 3
- Complex roots
- 0
Roots are computed numerically (Aberth–Ehrlich) and rounded; simple fractions are shown exactly.
Show working
- p(x) = x³ − 6x² + 11x − 6
- n = 3
- Aberth–Ehrlich: zₖ ← zₖ − w, w = (p/p′) / (1 − (p/p′)·Σⱼ≠ₖ 1/(zₖ − zⱼ))
- max |p(xₖ)| = 0
Roots table
| Root | Real part | Imaginary part |
|---|---|---|
| x1 | 1 | 0 |
| x2 | 2 | 0 |
| x3 | 3 | 0 |
How to use
- Write the coefficients from the highest power to the constant, e.g. x³ − 6x² + 11x − 6 → “1 -6 11 -6”.
- Use 0 for any missing power.
- Read the roots and the table of real and imaginary parts.
Good to know
Method: the Aberth–Ehrlich iteration improves all n root estimates together until they stop moving. Repeated roots are detected and cleaned up, and roots that are simple fractions (like 1/3) are shown exactly.
A polynomial of degree n always has n roots when you count complex and repeated roots. Complex roots of a polynomial with real coefficients come in pairs p ± qi.
Results are numeric, usually correct to 8 or more significant digits. For degree 2 and 3 the quadratic and cubic calculators give exact surd forms.
Questions
How do I type 2x⁴ − 3x² + 1?
Type “2 0 -3 0 1”: coefficients of x⁴, x³, x², x and the constant.
Why do I see complex roots?
Some polynomials never cross the x-axis, e.g. x² + 1. Their roots are complex numbers such as ±i.
Why only up to degree 6?
It keeps results fast and reliable on any phone. Higher degrees are possible numerically but rarely needed in school or college work.
Are the answers exact?
They are accurate numerical values. Whole numbers and simple fractions are shown exactly; other roots as decimals.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.