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Solving the Cubic Equation: (x+2)(x-1)(x-7) = 0
This equation represents a cubic function, meaning it has a highest power of 3. To solve it, we can use 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 equal to zero.
Let's break down the equation:
- (x + 2)(x - 1)(x - 7) = 0
Applying the Zero Product Property, we can set each factor equal to zero and solve for x:
- x + 2 = 0
- x - 1 = 0
- x - 7 = 0
Solving each equation:
- x = -2
- x = 1
- x = 7
Therefore, the solutions to the equation (x+2)(x-1)(x-7) = 0 are x = -2, x = 1, and x = 7.
This means that the graph of the cubic function will intersect the x-axis at these three points.
Understanding the Concept
The Zero Product Property allows us to easily find the roots of a polynomial equation when it is expressed in factored form. By setting each factor equal to zero, we isolate the variables and obtain the values that make the entire product equal to zero.
This method is crucial for understanding polynomial equations and their solutions. It simplifies the process of finding roots and helps us understand the relationship between the factors of a polynomial and its x-intercepts.