Find All Roots Calculator

Calculator

Polynomial coefficients

Enter from highest power to constant. Use commas, spaces, or lines.

Tolerance

Maximum iterations

Decimal places

Starting radius

Use 0 for automatic radius.

Starting phase degrees

Clean display

Treat tiny parts as zero

Example Data Table

Equation	Coefficient Input	Expected Roots	Note
x² - 5x + 6	1, -5, 6	2, 3	Two real roots
x² + 1	1, 0, 1	i, -i	Two complex roots
x⁴ - 5x² + 4	1, 0, -5, 0, 4	-2, -1, 1, 2	Zero placeholders are required
2x³ - 3x² - 11x + 6	2, -3, -11, 6	-2, 0.5, 3	Leading coefficient is supported

Formula Used

The calculator solves a polynomial written as:

P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

It applies the Durand Kerner update for all complex roots:

z_i,new = z_i - P(z_i) / Π(z_i - z_j), where j is not i.

The residual is:

Residual = |P(root)|

A smaller residual means the estimated root is closer to a true zero of the polynomial.

How to Use This Calculator

Write the polynomial coefficients from the highest power to the constant term.
Add zero for any missing power in the equation.
Set tolerance, maximum iterations, and decimal places.
Use automatic radius unless you need a custom starting circle.
Press the calculate button to find real and complex roots.
Check residual values before using the answer in reports.
Download CSV or PDF when you need a saved copy.

About the Find All Roots Calculator

Purpose

A find all roots calculator helps you solve polynomial equations. It accepts coefficients from the highest power down to the constant term. Then it searches for every real and complex solution. This is useful because many equations have roots that are not visible on a normal graph.

Method

The tool uses a complex root method. It starts with several trial points around a circle. Each point is improved again and again. The process stops when the change is smaller than your tolerance. It also reports residual error, so you can judge the result.

Root Meaning

Roots explain where a polynomial becomes zero. A simple linear equation has one root. A quadratic equation has two roots. A cubic equation has three roots. Higher degree equations continue this pattern when complex roots are counted. Repeated roots may appear more than once. Complex conjugate pairs often appear when all coefficients are real.

Advanced Options

This calculator is built for detailed checking. You can set tolerance, iterations, and decimal places. You can also choose whether tiny imaginary parts should be treated as zero. The output includes real part, imaginary part, magnitude, angle, and residual. These details help with engineering, algebra, modeling, and classroom review.

Input Quality

Use clean coefficient data for best accuracy. Avoid missing middle terms. Add zero when a power has no coefficient. For example, x^4 - 5x^2 + 4 should be entered as 1, 0, -5, 0, 4. That keeps every power in the correct position.

Reports

The download options are helpful for reports. The CSV file works well in spreadsheets. The PDF file gives a simple printable summary. The example table also shows how input order affects the equation. Always compare residual values after calculation. Smaller residuals usually mean a better root estimate.

Accuracy

Numerical root finding is powerful, but it is not magic. Very high degrees, repeated roots, and badly scaled coefficients can reduce accuracy. Increase iterations or relax tolerance when needed. You may also rescale coefficients. Good inputs, sensible settings, and residual checks make the results easier to trust.

Testing

Advanced users can compare several models quickly. Change one coefficient, run again, and review the movement of each root. This makes sensitivity testing easier. It can show when a small measurement error changes the behavior of a formula.

FAQs

1. What does this calculator find?

It finds every root of a polynomial, including real roots and complex roots. The number of returned roots should match the polynomial degree when repeated roots are counted.

2. How should I enter coefficients?

Enter coefficients from the highest power to the constant term. Separate them with commas, spaces, or new lines. Add zero for each missing power.

3. Can it solve complex roots?

Yes. The method works in the complex plane. Roots with imaginary parts are shown with real part, imaginary part, magnitude, angle, and residual.

4. What is a residual?

A residual is the value of |P(root)|. It shows how close the estimated root is to making the polynomial equal zero.

5. Why do I need zero placeholders?

Zero placeholders keep powers in the correct order. Without them, the calculator may read the equation as a different polynomial.

6. What tolerance should I use?

A small tolerance gives stricter convergence. Try 1e-10 for common work. Increase iterations if roots do not converge well.

7. Why are tiny imaginary parts shown?

Numerical methods can create tiny rounding noise. Enable the clean display option to treat very small real or imaginary parts as zero.

8. What can I download?

You can download a CSV spreadsheet file or a simple PDF report. Both include the polynomial, roots, convergence details, and residual values.