Finding The Roots Of F(x)=(x-6)^2(x+2)^2 With Multiplicity
Find the roots of f(x)=(x-6)^2(x+2)^2 and determine their multiplicities.
Understanding the Problem: Roots and Multiplicity
To find the roots of the given function, f(x) = (x-6)2(x+2)2, it is essential to understand what roots are and the concept of multiplicity. In essence, the roots of a function are the values of x for which f(x) equals zero. These values represent the points where the graph of the function intersects the x-axis. The multiplicity of a root refers to the number of times a particular root appears as a solution of the polynomial equation. This concept is pivotal because it influences the behavior of the graph near the root, specifically whether the graph crosses the x-axis or simply touches it and turns back.
When dealing with polynomial functions, each factor corresponds to a potential root. For example, the factor (x - a) indicates that x = a is a root. The exponent of the factor determines the multiplicity of that root. A factor raised to the power of n implies that the corresponding root has a multiplicity of n. Understanding this relationship between factors, roots, and multiplicity is the first key step in finding the roots of f(x) = (x-6)2(x+2)2. In our specific case, we can observe two distinct factors: (x - 6) and (x + 2), each raised to the power of 2. This observation suggests that we will have two roots, and their multiplicities will play a significant role in the graph's behavior around those roots.
Now, let's dive deeper into how we can extract the roots and their multiplicities from the given function. By setting the function to zero, we can identify the values of x that satisfy the equation. This process involves understanding that a product of factors equals zero if and only if at least one of the factors is zero. Hence, we will set each factor equal to zero and solve for x. This methodical approach ensures that we capture all possible roots of the function, accounting for their respective multiplicities. Recognizing the multiplicities is crucial because it affects how the function behaves near the roots – a root with even multiplicity results in the graph touching the x-axis and turning back, while a root with odd multiplicity causes the graph to cross the x-axis. The following sections will meticulously walk through the steps to identify the roots and their multiplicities for our given function.
Step-by-Step Solution
Let's delve into the step-by-step process of finding the roots of the function f(x)=(x-6)2(x+2)2. The fundamental principle here is that the roots of a function are the values of x for which f(x) = 0. Given that our function is already factored, this task becomes significantly more straightforward. We can directly set each factor equal to zero and solve for x. This approach leverages the property that if a product of terms equals zero, then at least one of those terms must be zero. Therefore, we'll analyze each factor individually to identify the roots and their corresponding multiplicities.
The first factor we consider is (x - 6)^2. Setting this equal to zero gives us (x - 6)^2 = 0. To solve for x, we take the square root of both sides, which leads to x - 6 = 0. Adding 6 to both sides, we find x = 6. Since the factor (x - 6) is raised to the power of 2, this indicates that the root x = 6 has a multiplicity of 2. Multiplicity is a critical concept as it tells us about the behavior of the graph near the root. A root with a multiplicity of 2 means that the graph will touch the x-axis at x = 6 but will not cross it; instead, it will turn back. This is because the function approaches the x-axis from one side, touches it, and then retreats in the same direction.
Next, we analyze the second factor, (x + 2)^2. Setting this factor equal to zero yields (x + 2)^2 = 0. Taking the square root of both sides gives x + 2 = 0. Subtracting 2 from both sides, we find x = -2. Similar to the previous case, the factor (x + 2) is raised to the power of 2, which means the root x = -2 also has a multiplicity of 2. Consequently, the graph of the function will touch the x-axis at x = -2 and turn back, exhibiting the same behavior as at x = 6. This detailed analysis of each factor allows us to precisely determine the roots and their multiplicities, which are crucial for understanding the overall behavior and graph of the function.
Identifying the Roots and Their Multiplicities
After meticulously analyzing the factored form of the function, f(x) = (x-6)2(x+2)2, we can now definitively identify the roots and their multiplicities. This step is crucial for understanding the function's behavior and sketching its graph. By setting each factor equal to zero, we isolated the potential roots, and by observing the exponents, we determined their multiplicities. Let's recap and clearly state our findings.
The first root we identified is x = 6. This root arises from the factor (x - 6)^2. As we discussed earlier, the exponent of the factor indicates the multiplicity of the corresponding root. In this case, the exponent is 2, which means that the root x = 6 has a multiplicity of 2. A root with a multiplicity of 2 is significant because it implies that the graph of the function touches the x-axis at x = 6 but does not cross it. Instead, the graph will approach the x-axis, touch it, and then turn back in the same direction. This behavior is characteristic of roots with even multiplicities, providing a key visual cue about the function's graph.
The second root we found is x = -2. This root comes from the factor (x + 2)^2. Similar to the previous case, the exponent of this factor is 2, indicating that the root x = -2 also has a multiplicity of 2. Therefore, the graph of the function will exhibit the same behavior at x = -2 as it does at x = 6. It will touch the x-axis and turn back, without crossing it. Understanding the multiplicity of roots is fundamental in polynomial functions, as it directly influences the graph's interaction with the x-axis. In this particular function, both roots have an even multiplicity, leading to a distinct graphical characteristic: the graph touches the x-axis at these points but does not cross it. This knowledge allows us to sketch a more accurate representation of the function, including its critical points and overall shape.
Conclusion: The Roots and Their Significance
In conclusion, after a detailed examination of the function f(x) = (x-6)2(x+2)2, we have successfully identified its roots and their multiplicities. This process involved setting each factor of the function to zero and solving for x, while carefully considering the exponent of each factor to determine the multiplicity of the corresponding root. Our analysis revealed two distinct roots: x = 6 and x = -2, both with a multiplicity of 2. These findings are crucial for understanding the behavior of the function and its graphical representation.
The significance of these roots and their multiplicities extends beyond just solving the equation f(x) = 0. The roots represent the points where the graph of the function intersects the x-axis, providing key anchors for sketching the curve. The multiplicity, on the other hand, adds another layer of information, dictating how the graph interacts with the x-axis at these points. A multiplicity of 2, as we found for both roots, indicates that the graph touches the x-axis at these points but does not cross it. This behavior is a direct consequence of the even multiplicity, causing the graph to “bounce” off the x-axis rather than pass through it.
Moreover, understanding the roots and their multiplicities allows us to gain deeper insights into the polynomial's nature. For instance, the fact that both roots have even multiplicities suggests that the function will have the same sign on both sides of each root. This means that the function will be non-negative (or non-positive) in the intervals surrounding the roots, depending on the leading coefficient of the polynomial. In the case of our function, since the leading coefficient is positive, the function will be non-negative. This level of understanding is essential for various mathematical applications, including solving inequalities, analyzing function behavior, and optimizing mathematical models. Therefore, the process of finding roots and understanding their multiplicities is a fundamental skill in algebra and calculus, providing a solid foundation for more advanced mathematical concepts.
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