Analyze real and imaginary roots from polynomial coefficients. See magnitude, angle, and multiplicity patterns clearly. Download clean tables and plots for study, teaching, reporting.
Enter a polynomial degree and its real coefficients. The solver returns every complex root, then plots each result on the complex plane.
| Polynomial | Degree | Coefficients | Expected Root Pattern |
|---|---|---|---|
| x2 + 1 = 0 | 2 | a2 = 1, a1 = 0, a0 = 1 | Pure imaginary pair: +i and -i |
| x2 - 4x + 13 = 0 | 2 | a2 = 1, a1 = -4, a0 = 13 | Conjugate pair: 2 ± 3i |
| x3 - 1 = 0 | 3 | a3 = 1, a2 = 0, a1 = 0, a0 = -1 | One real root and two complex companions |
| x4 + 1 = 0 | 4 | a4 = 1, a3 = 0, a2 = 0, a1 = 0, a0 = 1 | Four unit roots at odd eighth-angle positions |
This calculator solves P(z) = anzn + an-1zn-1 + ... + a1z + a0 = 0.
For degree two, the roots also agree with the quadratic identity z = [-b ± √(b² - 4ac)] / 2a. For higher degrees, the solver uses the Durand-Kerner iteration:
zᵢ(k+1) = zᵢ(k) - P(zᵢ(k)) / ∏(zᵢ(k) - zⱼ(k)), for every j ≠ i.
Each displayed root also includes:
|z| = √(Re(z)² + Im(z)²)arg(z) = atan2(Im(z), Re(z))|P(z)|, which measures numerical accuracyA complex root is a solution containing real and imaginary parts. It satisfies the polynomial exactly, even when the graph never crosses the horizontal axis.
It solves polynomial equations with real coefficients from degree 1 through degree 6. Examples include quadratic, cubic, quartic, quintic, and sextic forms.
Polynomials with real coefficients usually produce non-real roots in conjugate pairs. If one root is a + bi, another root is a - bi.
The residual is |P(z)| after substitution. Smaller values mean the computed root satisfies the equation more closely. Very small residuals indicate strong numerical accuracy.
Repeated roots sit very close together numerically. Iterative methods can converge more slowly there, so multiplicity estimates should be interpreted alongside the residual values.
The Argand plot places real parts on the horizontal axis and imaginary parts on the vertical axis. This makes symmetry, spacing, and conjugate structure immediately visible.
Scaling can improve numerical stability when coefficients vary wildly in size. The roots remain related, but the solver often behaves better after reasonable normalization.
Yes. Enter the published coefficients, compare the displayed roots, inspect the angles and moduli, then confirm accuracy through the residual column and plot.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.