Wolfram Polynomial Root Calculator

What is a Polynomial Root Calculator?

A polynomial root calculator finds the values of x that satisfy the equation P(x) = 0 for a given polynomial P(x). These roots are also called zeros of the polynomial. The calculator uses advanced numerical methods to find both real and complex roots.

Polynomials can be of any degree, from linear (degree 1) to higher-degree equations. The roots provide important information about the polynomial's behavior, including its graph intersections with the x-axis.

How to Use This Calculator

1. Enter the coefficients of your polynomial in the input fields. For example, for the polynomial 3x² + 2x - 5, you would enter coefficients: 3, 2, -5.
2. Select the degree of your polynomial (the highest power of x).
3. Click "Calculate Roots" to find the roots.
4. Review the results, which will show both real and complex roots when applicable.
5. Use the visualization to better understand the polynomial's behavior.

The calculator will display the roots in both decimal and exact form when possible, along with a graphical representation of the polynomial.

Formula Used

The calculator uses numerical methods to approximate roots for polynomials of degree 5 or higher. For lower-degree polynomials, exact solutions are calculated using algebraic methods.

Exact solutions exist for polynomials of degree 4 or lower, while higher-degree polynomials typically require numerical approximation methods.

Worked Examples

Example 1: Quadratic Polynomial

Find the roots of 2x² - 5x + 3 = 0.

Using the quadratic formula: x = [5 ± √(25 - 24)] / 4 = [5 ± 1]/4.

Roots: x = 1.5 and x = 1.

Example 2: Cubic Polynomial

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

This factors as (x-1)(x-2)(x-3) = 0, giving roots x = 1, x = 2, and x = 3.