How to calculate the roots of a cubic polynomial?

To calculate the roots of a cubic polynomial, we can use the cubic formula or factor theorem.

The cubic formula is a general formula that can be used to find the roots of any cubic polynomial. However, it can be quite complex and time-consuming to use. The formula is:

x = (-b ± √(b^2 - 4ac - 3b^3) ) / (2a)

where a, b, and c are the coefficients of the cubic polynomial ax^3 + bx^2 + cx + d.

Alternatively, we can use the factor theorem to find the roots of a cubic polynomial. This involves factoring the polynomial into linear and quadratic factors, and then solving for the roots of each factor. The factor theorem states that if a polynomial f(x) has a factor (x - r), then f(r) = 0.

For example, consider the cubic polynomial x^3 - 6x^2 + 11x - 6. We can factor this polynomial as (x - 1)(x - 2)(x - 3), using either long division or synthetic division. Therefore, the roots of the polynomial are x = 1, x = 2, and x = 3.

In some cases, the polynomial may not factor easily, in which case the cubic formula may be the best option. However, if the polynomial does have rational roots, the factor theorem can be a quicker and simpler method.