(x-2)(x-3)(x-4)(x-5)%3D0.jpeg "(x-1)(x-2)(x-3)(x-4)(x-5)=0")
Solving the Equation: (x-1)(x-2)(x-3)(x-4)(x-5) = 0
This equation, (x-1)(x-2)(x-3)(x-4)(x-5) = 0, is a polynomial equation in factored form. To solve for x, we can use the Zero Product Property:
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Applying this to our equation, we can see that for the product to equal zero, at least one of the following must be true:
- x - 1 = 0
- x - 2 = 0
- x - 3 = 0
- x - 4 = 0
- x - 5 = 0
Solving each of these individual equations, we get:
- x = 1
- x = 2
- x = 3
- x = 4
- x = 5
Therefore, the solutions to the equation (x-1)(x-2)(x-3)(x-4)(x-5) = 0 are x = 1, x = 2, x = 3, x = 4, and x = 5.
Understanding the Solutions
Each solution represents a point where the graph of the polynomial function y = (x-1)(x-2)(x-3)(x-4)(x-5) intersects the x-axis. This is because at these points, the value of y (the function's output) is zero.
Visualization
Imagine the graph of this equation. It would be a curve that crosses the x-axis at the points x=1, x=2, x=3, x=4, and x=5. This is because at these specific x-values, the function's output is zero, meaning the graph touches the x-axis.
Conclusion
The equation (x-1)(x-2)(x-3)(x-4)(x-5) = 0 provides a simple example of how the Zero Product Property can be used to solve polynomial equations. The solutions to this equation represent the x-intercepts of the corresponding polynomial function.