| # | Root | Multiplicity | Modulus | Argument (rad) | Residual |
|---|
| Example | Polynomial | Coefficient Notes | What to Expect |
|---|---|---|---|
| 1 | z4 + 1 | Leading coefficient 1, constant 1, middle terms zero. | Four roots on the unit circle at quarter-turn offsets. |
| 2 | z3 - (2 + i)z2 + (1 + 3i)z - (1 + i) | Complex coefficients entered through separate real and imaginary inputs. | Three complex roots with a direct linear-factor decomposition. |
| 3 | 2z4 - 3z2 + 5 | Even-power structure can create symmetric root patterns. | Scalar leading factor times four complex linear factors. |
A degree n polynomial is written as P(z) = anzn + an-1zn-1 + ... + a0, where each coefficient may be real or complex.
Over the complex field, the factorization target is P(z) = anΠ(z - rk). Each rk is a complex root, and repeated roots are grouped into multiplicities when their numerical distance is below the selected tolerance.
This calculator finds the roots numerically using the Durand–Kerner iteration:
zk,new = zk - P(zk) / Π(zk - zj) for j ≠ k
After iteration converges, the page reports the linear-factor form, root modulus, root argument, and a residual check |P(r)| for each computed root.
- Choose the polynomial degree. The page will generate one coefficient card for every power from zn down to the constant term.
- Enter each coefficient using separate real and imaginary boxes. For real-only values, keep the imaginary box at zero.
- Select display precision, tolerance, and iteration limit. Smaller tolerances produce stricter verification but may require more iterations.
- Press Factor Polynomial. The result block appears below the header and above the form, showing the polynomial, factorized form, summary metrics, and root table.
- Use Download CSV for a spreadsheet-friendly root export or Download PDF for a printable factorization report.
Convergence and Numerical Stability
Complex polynomial factorization is sensitive to scaling, repeated roots, and coefficient magnitude. This calculator limits degree, exposes tolerance, and reports residuals so users can distinguish a stable factorization from a merely plausible numerical approximation. When the leading coefficient is large or tiny, normalization improves iteration balance and reduces drift.
Root Distribution Patterns
Many polynomials reveal geometric structure in the complex plane. Even polynomials often produce symmetric roots, conjugate coefficients can generate mirrored patterns, and cyclotomic-style expressions place roots near circles. The Plotly graph helps users inspect clustering, spacing, and argument changes that are hard to interpret from factor text alone.
Multiplicity Detection
Repeated or nearly repeated roots make factorization harder because update denominators shrink during Durand–Kerner iteration. The calculator groups roots when their separation falls below a practical threshold tied to tolerance. That approach helps convert noisy numerical outputs into readable factors while still preserving a useful residual check for validation.
Interpretation of Residuals
Residual values measure how well a computed root satisfies the original polynomial. A residual close to zero supports the factorization, while a larger value signals rounding pressure, slow convergence, or an ill-conditioned input. In instructional settings, residual reporting is especially valuable because it connects symbolic expectations with numerical evidence.
Use in Teaching and Verification
This calculator supports algebra classes, complex analysis exercises, and independent checking of manually derived factors. Students can enter coefficients term by term, compare exact expectations with approximate roots, and visualize why high-degree expressions rarely factor cleanly without numerical methods. Instructors can also use exported tables for worked examples or assignments.
Why Visualization Matters
A factor list communicates exact structure, but a graph communicates behavior. Seeing roots spread across quadrants, align on circles, or cluster near one point helps users explain multiplicity, symmetry, and sensitivity more clearly. Combined with CSV and PDF exports, the visual layer turns one computation into a reusable analysis record. For advanced review, users should compare factor strings, plotted locations, and residual magnitudes together rather than relying on any single display. That combined reading reduces interpretation errors, especially when coefficients contain imaginary parts, roots are tightly packed, or the polynomial has repeated factors that visually overlap on the complex plane during classroom demonstrations and audits.
1. Does the calculator work with real-only coefficients?
Yes. Leave every imaginary input at zero. The solver still factors the polynomial over the complex field, so real polynomials may return both real roots and complex conjugate pairs.
2. Why do I see tiny residual values instead of exact zeros?
The method is numerical, so outputs are approximations. Small residuals usually indicate a strong result, while larger residuals suggest insufficient iterations, tight clustering, or unstable scaling.
3. What does multiplicity mean here?
Multiplicity tells you how many times the same root repeats. The calculator groups roots whose numerical distance is smaller than the effective grouping threshold.
4. Why is the leading coefficient shown separately in the factorization?
A polynomial factors into its leading coefficient multiplied by linear root factors. Keeping that scalar visible preserves the original polynomial exactly during reconstruction.
5. When should I increase the iteration limit?
Increase iterations when residuals remain high, roots appear unstable, or repeated roots are suspected. Harder polynomials may need more updates before convergence becomes acceptable.
6. What does the Plotly graph add?
The graph shows root placement on the complex plane. It helps you spot symmetry, clustering, conjugate behavior, and repeated-root patterns faster than reading factor text alone.