Math

Polynomial Root Calculator

Find All Roots of Any Polynomial

Enter any polynomial expression and find all its roots -- real and complex. This calculator uses exact algebraic methods for degree 3 or less, and robust numerical methods (Newton-Raphson + bisection) for higher degrees. See roots plotted on an interactive graph.

Quick Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Quadratic Equation

Solve a classic quadratic with two real roots.

Key values: x^2 - 5x + 6 = 0 · Two real roots

Linear Equation

Solve a simple linear equation for x.

Key values: 3x + 7 = 22 · Single root

Transcendental Equation

Find roots of sin(x) = x/3 using numerical methods.

Key values: sin(x) = x/3 · Numerical solution

Cubic with Complex Roots

Solve a cubic equation and show complex roots.

Key values: x^3 + x + 1 = 0 · Complex roots enabled

Documentation

This calculator is also known as Polynomial Root Calculator.

Read the complete guide

The Fundamental Theorem of Algebra

Every polynomial of degree n has exactly n roots (counting multiplicity) in the complex numbers. This calculator finds all of them -- both the real roots (shown on the graph) and the complex roots (shown in the solutions panel).

Examples

Finding Roots of a Quartic Polynomial

Find all roots of x^4 - 5x^2 + 4 = 0.

This is a bi-quadratic equation. Substituting u = x^2 gives u^2 - 5u + 4 = 0, which factors as (u-1)(u-4) = 0. So x = 1, -1, 2, -2.

Key takeaway: Even higher-degree polynomials can sometimes be solved by recognizing special structures.

Working with Polynomial Roots

Key concepts for polynomial root analysis:

By the Fundamental Theorem of Algebra, a degree-n polynomial has exactly n roots (counting multiplicity)
Complex roots of polynomials with real coefficients always come in conjugate pairs
Use Descartes' Rule of Signs to predict the number of positive and negative real roots
The Rational Root Theorem can help identify rational roots of polynomials with integer coefficients

Frequently Asked Questions about Polynomial Root Calculator

Can this find complex roots?

Yes. Enable "Show complex roots" in the Advanced Settings to see all complex roots. Complex roots of polynomials with real coefficients always come in conjugate pairs.

What is the maximum degree supported?

There is no hard limit. Polynomials of degree 3 or less are solved exactly. For degree 4 and above, numerical methods find real roots in the specified domain.