(x-2)(x-3)%3D0.jpeg "(x-1)(x-2)(x-3)=0")
Solving the Cubic Equation: (x-1)(x-2)(x-3) = 0
This equation represents a cubic function set equal to zero. Let's break down how to solve it and understand its implications.
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 case, we have three factors: (x-1), (x-2), and (x-3). Therefore, to make the entire product equal to zero, at least one of these factors must equal zero.
Finding the Solutions
Let's examine each factor:
- (x-1) = 0: Solving for x, we get x = 1
- (x-2) = 0: Solving for x, we get x = 2
- (x-3) = 0: Solving for x, we get x = 3
Therefore, the solutions to the equation (x-1)(x-2)(x-3) = 0 are x = 1, x = 2, and x = 3.
Graphical Representation
The equation (x-1)(x-2)(x-3) = 0 represents a cubic function. When graphed, this function will cross the x-axis at the points x = 1, x = 2, and x = 3. These points are the roots or zeros of the function.
Understanding the Solutions
The solutions we found represent the values of x where the function equals zero. In a real-world context, these solutions could represent critical points, such as break-even points in business or specific times when a physical quantity reaches zero.
Conclusion
Solving the equation (x-1)(x-2)(x-3) = 0 utilizes the Zero Product Property to find the three solutions: x = 1, x = 2, and x = 3. These solutions represent the roots or zeros of the corresponding cubic function and have various interpretations depending on the real-world application.