Solving the Cubic Equation: (x+4)(x+2)(x1) = 0
This equation represents a cubic polynomial, which means it has a highest power of x equal to 3. Solving it involves finding the values of x that make the equation true.
The Zero Product Property
The key to solving this equation lies in the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
In our equation, we have three factors:
 (x + 4)
 (x + 2)
 (x  1)
To make the product equal to zero, at least one of these factors must equal zero.
Finding the Solutions
Therefore, we set each factor equal to zero and solve for x:

(x + 4) = 0 Subtracting 4 from both sides, we get x = 4

(x + 2) = 0 Subtracting 2 from both sides, we get x = 2

(x  1) = 0 Adding 1 to both sides, we get x = 1
The Solution Set
The solutions to the equation (x + 4)(x + 2)(x  1) = 0 are x = 4, x = 2, and x = 1. These are the values of x that make the equation true.
Visualization
This cubic equation represents a curve that intersects the xaxis at the points x = 4, x = 2, and x = 1. These points are called the roots or zeros of the equation.
Conclusion
By using the Zero Product Property, we were able to efficiently solve the cubic equation (x + 4)(x + 2)(x  1) = 0. This method allows us to identify the values of x that satisfy the equation and represent the points where the curve intersects the xaxis.