(x%2B2)(x-6).jpeg "(x-5)(x+2)(x-6)")
Factoring and Solving the Cubic Equation: (x-5)(x+2)(x-6)
The expression (x-5)(x+2)(x-6) represents a cubic equation in factored form. This factored form is incredibly useful for understanding the equation's properties and solving for its roots. Let's explore its key features:
Understanding the Factored Form
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Roots: The factored form directly reveals the roots (or solutions) of the equation. Each factor represents a linear term that equals zero when the variable x takes on a specific value. In this case, the roots are:
- x = 5
- x = -2
- x = 6
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Zero Product Property: The principle behind solving a factored equation is the zero product property. This states that if the product of several factors equals zero, at least one of the factors must be zero.
Expanding the Expression
To see the equation in its standard polynomial form, we can expand the product:
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Start with any two factors: (x-5)(x+2) = x² - 3x - 10
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Multiply the result by the remaining factor: (x² - 3x - 10)(x-6) = x³ - 9x² + 18x - 10x² + 30x + 60
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Simplify: x³ - 19x² + 48x + 60
Therefore, the expanded form of the equation is x³ - 19x² + 48x + 60 = 0.
Graphing the Cubic Function
The graph of the function y = (x-5)(x+2)(x-6) will intersect the x-axis at the points corresponding to the roots (5, 0), (-2, 0), and (6, 0). Since it's a cubic function, the graph will have a general "S" shape, changing direction twice.
Applications of Factored Form
The factored form of an equation offers significant advantages:
- Solving for roots: It directly provides the roots of the equation.
- Identifying intercepts: The roots correspond to the x-intercepts of the function's graph.
- Understanding behavior: The factored form helps visualize how the function changes its value as x varies.
In conclusion, understanding the factored form (x-5)(x+2)(x-6) offers a clear way to analyze the roots, behavior, and graph of the cubic equation. This representation significantly simplifies the process of working with this type of polynomial.