Online Polynomial Roots Calculator

Find the roots of any polynomial equation with our online polynomial roots calculator. Whether you're solving quadratic, cubic, or higher-degree polynomials, this tool provides accurate solutions with step-by-step results.

How to Use This Calculator

Using our polynomial roots calculator is simple:

Enter the coefficients of your polynomial in the input fields. For example, for the polynomial 3x³ + 2x² - 5x + 1, you would enter 3 for x³, 2 for x², -5 for x, and 1 for the constant term.
Click the "Calculate Roots" button to find the solutions.
Review the results, which include both real and complex roots when applicable.
Use the chart visualization to better understand the polynomial's behavior.

The calculator supports polynomials up to degree 6. For higher-degree polynomials, consider using specialized mathematical software.

What Are Polynomial Roots?

Polynomial roots are the values of x that satisfy the equation P(x) = 0, where P(x) is a polynomial function. These roots are also known as zeros or solutions of the polynomial equation.

For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3 because substituting these values makes the equation true.

Polynomial roots can be real or complex numbers. Complex roots always come in conjugate pairs for polynomials with real coefficients.

How to Find Polynomial Roots

There are several methods to find polynomial roots:

Factoring

For lower-degree polynomials, factoring is often the simplest method. You look for common factors or patterns that can be factored out.

Example: x² - 5x + 6 = (x - 2)(x - 3)

Quadratic Formula

For quadratic equations (degree 2), the quadratic formula provides a direct solution:

x = [-b ± √(b² - 4ac)] / (2a)

Numerical Methods

For higher-degree polynomials, numerical methods like Newton-Raphson or iterative approximation are often used.

Graphical Methods

Plotting the polynomial can help identify approximate roots by finding where the graph crosses the x-axis.

Example Calculations

Let's solve a few polynomial equations using our calculator:

Example 1: Quadratic Equation

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 x = [5 ± 1] / 2 Roots: x = 3 and x = 2

Example 2: Cubic Equation

Find the roots of x³ - 6x² + 11x - 6 = 0.

Using factoring:

(x - 1)(x - 2)(x - 3) = 0 Roots: x = 1, x = 2, x = 3

Example 3: Complex Roots

Find the roots of x² + 2x + 5 = 0.

Using the quadratic formula:

x = [-2 ± √(-16)] / 2 x = -1 ± 2i Roots: x = -1 + 2i and x = -1 - 2i

Frequently Asked Questions

What is the difference between a root and a solution?

In the context of polynomial equations, "root" and "solution" are often used interchangeably. Both refer to the values of x that satisfy the equation P(x) = 0.

Can this calculator solve any polynomial equation?

This calculator can solve polynomials up to degree 6. For higher-degree polynomials, specialized mathematical software is recommended.

What if my polynomial has complex roots?

The calculator will display both real and complex roots when applicable. Complex roots are shown in the form a + bi, where i is the imaginary unit.

How accurate are the results?

The calculator uses numerical methods to find roots, which are accurate to 15 decimal places. For most practical purposes, this level of precision is sufficient.