The given cubic equation is x^3 - 7x - 6. We need to find the root of this equation.

What is a root?
A root of an equation is a value that satisfies the equation when substituted for the variable. In other words, it is the value of x for which the equation becomes true.

Methods to find the root of a cubic equation:
There are various methods to find the root of a cubic equation, such as the Rational Root Theorem, synthetic division, and factoring. Let's use the Rational Root Theorem in this case.

Rational Root Theorem:
The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation, then p must be a factor of the constant term (in this case, -6), and q must be a factor of the leading coefficient (in this case, 1).

Factors of -6:
The factors of -6 are ±1, ±2, ±3, and ±6.

Factors of 1:
The factors of 1 are ±1.

Possible rational roots:
Combining the factors, the possible rational roots for the given equation are ±1, ±2, ±3, ±6.

Synthetic Division:
We can use synthetic division to test these possible roots one by one until we find a root that satisfies the equation.

Let's start with the possible root x = 1.

1 | 1 0 -7 -6
| 1 1 -6
|________
| 1 1 -6 0

Since the remainder is 0, x = 1 is a root of the equation.

Dividing by the root:
To find the other roots, we can divide the given equation by (x - 1) using either long division or synthetic division.

(x - 1)(x^2 + x - 6) = 0

Simplifying, we get:
x^2 + x - 6 = 0

Factoring:
Now, we can factor the quadratic equation.

(x + 3)(x - 2) = 0

Setting each factor equal to zero, we get:
x + 3 = 0 or x - 2 = 0

Solving these equations, we find the other two roots:
x = -3 or x = 2

Summary:
The roots of the given cubic equation x^3 - 7x - 6 are x = 1, x = -3, and x = 2. These values satisfy the equation when substituted for x, and they are obtained using the Rational Root Theorem, synthetic division, and factoring.