How to find the roots of a cubic polynomial?

To find the roots of a cubic polynomial, we can use the factor theorem and synthetic division.

First, we need to write the cubic polynomial in the form of (x-a)(x-b)(x-c), where a, b, and c are the roots of the polynomial. To do this, we can use the factor theorem, which states that if f(a) = 0, then (x-a) is a factor of f(x).

Next, we can use synthetic division to find the remaining quadratic factor. We divide the cubic polynomial by (x-a), (x-b), or (x-c) one at a time, and the resulting quadratic factor will have the remaining two roots.

For example, let's find the roots of the cubic polynomial f(x) = x^3 - 6x^2 + 11x - 6.

First, we can try to factor out (x-1) using the factor theorem. We find that f(1) = 1 - 6 + 11 - 6 = 0, so (x-1) is a factor of f(x). Using synthetic division, we get:

1 | 1 -6 11 -6
| 1 -5 6
|__________
1 -5 6 0

So the remaining quadratic factor is x^2 - 5x + 6, which can be factored as (x-2)(x-3). Therefore, the roots of f(x) are 1, 2, and 3.

In summary, to find the roots of a cubic polynomial, we can use the factor theorem and synthetic division to factor the polynomial into linear and quadratic factors, and then solve for the roots.