# Polynomial Roots Calculator

Find all real and complex roots of a polynomial from a coefficient list using a numerical root finder

> Canonical page: https://elysiatools.com/en/tools/polynomial-roots-calculator

- **Category:** Math & Numbers

- **Keywords:** polynomial, roots, zeros, complex roots, algebra, math

## Overview

The Polynomial Roots Calculator is a numerical math tool designed to find all real and complex roots of a polynomial equation. By simply entering a comma-separated list of coefficients, you can quickly calculate the zeros of cubic, quartic, or higher-degree polynomials without manual algebraic solving.

## Inputs

- **Coefficients** (text): e.g. 1, -6, 11, -6
- **Decimal Places** (number)
- **Maximum Iterations** (number)

## When to use

- Solving high-degree algebraic equations for math assignments or academic research.
- Finding the poles and zeros of transfer functions in control systems engineering.
- Calculating eigenvalues from characteristic polynomials in linear algebra.

## How it works

- Enter the polynomial coefficients in descending order of power, separated by commas (e.g., '1, -6, 11, -6' for x^3 - 6x^2 + 11x - 6).
- Adjust the decimal places to control the precision of the calculated roots.
- Modify the maximum iterations if dealing with highly complex polynomials that require deeper numerical solving.
- The tool processes the coefficients using a numerical root-finding algorithm and outputs all real and complex roots in JSON format.

## Use cases

- Engineering students analyzing the stability of linear time-invariant (LTI) systems.
- Mathematicians and researchers solving characteristic equations for matrix operations.
- Programmers needing a quick way to verify the output of custom root-finding algorithms.

## Frequently asked questions

### How should I format the coefficients?

Enter them as a comma-separated list in descending order of degree. Include zeros for any missing terms. For example, x^3 - 1 would be entered as '1, 0, 0, -1'.

### Can this calculator find complex roots?

Yes, the numerical root finder calculates both real and complex roots of the polynomial.

### What is the maximum degree polynomial I can solve?

The tool can handle high-degree polynomials, limited only by the numerical stability of the root-finding algorithm and the maximum iteration limit.

### What does the maximum iterations setting do?

It limits how many times the numerical algorithm attempts to converge on a root. Increase this value up to 500 if the tool struggles to find roots for complex polynomials.

### Why do I need to include zeros for missing terms?

The algorithm relies on the position of the coefficient in the list to determine its power. Omitting a zero changes the degree of all subsequent terms.

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