(x3-8)(x-5)(2x+1)=0

Jasonbradley

2 min read

·

Jun 17, 2024

19

Share

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.