## Solving the Cubic Equation: (x³ - 8)(x - 5)(2x + 1) = 0

This equation represents a cubic function, meaning it has a highest power of x equal to 3. To find the solutions (also known as roots or zeros), we can utilize the **Zero Product Property**. This property states that if the product of several factors equals zero, then at least one of the factors must be zero.

Let's break down the equation:

**(x³ - 8)(x - 5)(2x + 1) = 0**

We have three factors:

**(x³ - 8)****(x - 5)****(2x + 1)**

To find the solutions, we set each factor equal to zero and solve:

**1. (x³ - 8) = 0**

- This can be factored as a difference of cubes: (x - 2)(x² + 2x + 4) = 0
- Solving (x - 2) = 0 gives us
**x = 2** - The quadratic factor (x² + 2x + 4) = 0 does not factor easily and requires the quadratic formula to solve. However, it has complex solutions, which are outside the scope of this example.

**2. (x - 5) = 0**

- Solving this directly gives us
**x = 5**

**3. (2x + 1) = 0**

- Solving for x gives us
**x = -1/2**

Therefore, the solutions to the cubic equation (x³ - 8)(x - 5)(2x + 1) = 0 are:

**x = 2****x = 5****x = -1/2**

These solutions represent the x-intercepts of the graph of the function. The graph would intersect the x-axis at these three points.