Roots of 3rd Degree Polynomial Calculator

What is a 3rd Degree Polynomial?

A third-degree polynomial is a mathematical expression that can be written in the form:

Where a, b, c, and d are constants, and a ≠ 0. The roots of this polynomial are the values of x that satisfy the equation. Finding these roots is essential in many areas of mathematics, engineering, and science.

Unlike quadratic equations (which have at most two real roots), cubic equations can have one real root and two complex roots, or three real roots. This calculator uses numerical methods to approximate the roots of any cubic equation.

How to Use This Calculator

Using this calculator is straightforward. Follow these steps:

1. Enter the coefficients of your cubic equation in the input fields provided.
2. Click the "Calculate" button to find the roots.
3. Review the results, which will show the approximate roots of your equation.
4. If needed, you can reset the calculator to enter a new equation.

The calculator will display the roots in a clear format, and you can also visualize the polynomial and its roots using the chart.

How the Calculator Works

This calculator uses numerical methods to approximate the roots of a cubic equation. The most common method for solving cubic equations is Cardano's formula, which provides exact solutions when they exist. However, for more general cases, numerical methods like Newton-Raphson are used to find approximate roots.

The calculator follows these steps:

1. Accepts the coefficients a, b, c, and d of the cubic equation.
2. Applies numerical methods to find the roots.
3. Displays the roots in a clear format.
4. Generates a chart to visualize the polynomial and its roots.

Worked Example

Let's solve the cubic equation x³ - 6x² + 11x - 6 = 0.

Using the calculator:

1. Enter a = 1, b = -6, c = 11, d = -6.
2. Click "Calculate".
3. The calculator will display the roots: x = 1, x = 2, x = 3.

These are the exact roots of the equation, which can be verified by factoring: (x - 1)(x - 2)(x - 3) = 0.

The chart will show the polynomial curve crossing the x-axis at these points.