Roots of A Third Degree Polynomial Calculator

What is a Third Degree Polynomial?

A third degree polynomial is a polynomial equation of the form:

where:

• a, b, c, and d are coefficients
• a ≠ 0 (since it's a third degree polynomial)
• x is the variable

The roots of the polynomial are the values of x that satisfy the equation. A cubic polynomial can have either one real root and two complex conjugate roots, or three real roots (which may be repeated).

How to Find the Roots of a Third Degree Polynomial

Finding the roots of a cubic polynomial involves solving the equation ax³ + bx² + cx + d = 0. There are several methods to find the roots:

1. Factoring: If the polynomial can be factored, you can find the roots by setting each factor equal to zero.
2. Cardano's Formula: This is an analytical method that provides a solution to the cubic equation using radicals.
3. Numerical Methods: Approximation methods like Newton-Raphson can be used to find the roots numerically.

This calculator uses Cardano's formula to find the roots of the cubic polynomial.

The Formula for Finding Roots

Cardano's formula for finding the roots of a cubic polynomial ax³ + bx² + cx + d = 0 is:

This formula provides the roots of the cubic polynomial in terms of the coefficients a, b, c, and d.

Worked Example

Let's find the roots of the cubic polynomial x³ - 6x² + 11x - 6 = 0.

Using the calculator, we can find the roots of this polynomial. The roots are:

• x = 1
• x = 2
• x = 3

This shows that the polynomial can be factored as (x - 1)(x - 2)(x - 3) = 0, confirming the roots.