(x%2B6)(x%2B2).jpeg "(x-4)(x+6)(x+2)")
Exploring the Polynomial (x-4)(x+6)(x+2)
This article will delve into the fascinating world of the polynomial (x-4)(x+6)(x+2). We'll explore its key characteristics, including its roots, expanded form, and how to graph it.
Finding the Roots
The most straightforward way to find the roots of a polynomial in factored form is by setting each factor equal to zero and solving for x.
- (x-4) = 0 => x = 4
- (x+6) = 0 => x = -6
- (x+2) = 0 => x = -2
Therefore, the roots of the polynomial (x-4)(x+6)(x+2) are x = 4, x = -6, and x = -2. These are the points where the graph of the polynomial intersects the x-axis.
Expanding the Polynomial
To understand the behavior of the polynomial, we can expand it from its factored form:
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Start with the first two factors: (x-4)(x+6) = x² + 2x - 24
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Multiply the result by the third factor: (x² + 2x - 24)(x+2) = x³ + 4x² - 20x - 48
Therefore, the expanded form of the polynomial is x³ + 4x² - 20x - 48.
Graphing the Polynomial
The graph of this polynomial will be a cubic function. Its key features include:
- Roots: The graph will intersect the x-axis at the points x = 4, x = -6, and x = -2.
- End Behavior: Since the leading coefficient is positive (1 in the expanded form), the graph will rise to the right and fall to the left.
- Turning Points: Cubic functions can have up to two turning points, where the graph changes from increasing to decreasing or vice versa.
By plotting the roots and considering the end behavior, we can sketch a rough graph of the polynomial.
Additional Considerations
- Multiplicity of Roots: Each root appears only once in the factored form. This means that the graph will cross the x-axis at each root.
- Symmetry: This specific polynomial does not exhibit any symmetry.
By analyzing the factored form, expanding the polynomial, and understanding the key features of cubic functions, we gain a comprehensive understanding of the polynomial (x-4)(x+6)(x+2). This knowledge can be used to solve problems, analyze its behavior, and visualize its graph.