(x%2B3)(x-5).jpeg "(x-2)(x+3)(x-5)")
Factoring and Solving the Equation (x-2)(x+3)(x-5) = 0
The expression (x-2)(x+3)(x-5) represents a cubic polynomial in factored form. This form allows us to quickly identify the roots or solutions to the equation (x-2)(x+3)(x-5) = 0.
Understanding 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. This property is fundamental to solving polynomial equations.
Finding the Solutions
To find the solutions to (x-2)(x+3)(x-5) = 0, we apply the Zero Product Property:
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Set each factor equal to zero:
- x - 2 = 0
- x + 3 = 0
- x - 5 = 0
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Solve for x in each equation:
- x = 2
- x = -3
- x = 5
Therefore, the solutions to the equation (x-2)(x+3)(x-5) = 0 are x = 2, x = -3, and x = 5.
Graphical Interpretation
The solutions (x = 2, x = -3, and x = 5) represent the x-intercepts of the graph of the cubic polynomial y = (x-2)(x+3)(x-5). At these points, the graph crosses the x-axis.
Expanding the Expression
While the factored form is useful for finding the solutions, we can also expand the expression to obtain the standard form of the cubic polynomial:
(x-2)(x+3)(x-5) = (x² + x - 6)(x-5) = x³ - 4x² - 11x + 30
This expanded form is helpful for understanding the behavior of the polynomial and for performing other operations.
Summary
The equation (x-2)(x+3)(x-5) = 0 represents a cubic polynomial in factored form, which allows us to easily identify its solutions. The solutions are x = 2, x = -3, and x = 5. These solutions correspond to the x-intercepts of the graph of the polynomial. We can also expand the expression to obtain its standard form, which provides another way to analyze the polynomial.