Roots of Degree 4 Polynomial Calculator

How to Use This Calculator

To find the roots of a quartic polynomial:

1. Enter the coefficients for each term of your polynomial in the form: ax⁴ + bx³ + cx² + dx + e = 0
2. Click "Calculate Roots"
3. View the results showing all real and complex roots
4. Use the chart to visualize the polynomial and its roots

The calculator will display all roots, including complex ones, and show them on a graph for better understanding.

Formula Used

The calculator uses numerical methods to approximate the roots of the quartic equation:

For polynomials that can be factored into simpler quadratics, exact solutions can be found using the quadratic formula. When exact solutions are not possible, the calculator uses the Newton-Raphson method to approximate the roots.

Worked Examples

Example 1: Simple Quartic

Find the roots of x⁴ - 5x² + 4 = 0

This can be factored as (x² - 4)(x² - 1) = 0, giving roots at x = ±2 and x = ±1.

Example 2: Complex Roots

Find the roots of x⁴ + 2x³ + 5x² + 4x + 4 = 0

The calculator will find approximate complex roots using numerical methods.

Interpreting Results

The calculator displays all roots in the form:

• Real roots (x = value)
• Complex roots (x = a ± bi)

The graph visualization helps understand where the polynomial crosses or touches the x-axis.