Finding Roots Multiplicities Of Polynomial Function F(x)=(x-6)^2(x+2)^2
In the fascinating world of mathematics, polynomial functions hold a prominent position. Understanding their behavior, especially identifying their roots, is crucial in various fields, from engineering to economics. Roots, also known as zeros, are the x-values where the function intersects the x-axis, making the function's value equal to zero. This article delves into the process of finding the roots of the polynomial function f(x) = (x - 6)^2(x + 2)^2 and determining their respective multiplicities. Multiplicity refers to the number of times a particular root appears as a factor in the polynomial. Understanding multiplicity is essential because it provides insights into the function's behavior near the root, such as whether the graph crosses the x-axis or simply touches it and turns around. This comprehensive guide will not only provide the solution but also explain the underlying concepts and methodologies, ensuring a thorough understanding of root-finding techniques.
The process of finding the roots of a polynomial function is a fundamental concept in algebra, with far-reaching implications in various branches of mathematics and its applications. In essence, the roots of a polynomial function are the values of the variable (typically denoted as x) for which the function evaluates to zero. These roots provide critical information about the behavior of the function, including where it intersects the x-axis, the intervals where the function is positive or negative, and the overall shape of its graph. Understanding the concept of roots is pivotal for solving equations, modeling real-world phenomena, and making predictions based on mathematical models. When we talk about finding the roots, we're essentially asking the question: "For what values of x does the function f(x) equal zero?" The answer to this question unlocks a deeper understanding of the polynomial's characteristics and its role in mathematical problem-solving. The journey to finding roots often involves a combination of algebraic techniques, graphical analysis, and, for higher-degree polynomials, numerical methods. Each root carries with it a piece of the puzzle that, when assembled, reveals the complete picture of the polynomial's behavior.
To begin our exploration, let's closely examine the given polynomial function: f(x) = (x - 6)^2(x + 2)^2. This function is presented in its factored form, which is a significant advantage for root finding. The factored form directly reveals the roots of the polynomial and their multiplicities. By observing the structure of the polynomial, we can identify the factors that contribute to the roots. Each factor of the form (x - a) indicates a root at x = a, because when x equals a, that factor becomes zero, causing the entire function to become zero. The exponent associated with each factor, in this case, the power of 2 for both factors, signifies the multiplicity of the corresponding root. This multiplicity tells us how many times that particular root is repeated. For instance, a factor raised to the power of 2 indicates a root with a multiplicity of 2, meaning that the root appears twice. Recognizing these key aspects of the factored form allows us to quickly determine the roots and their multiplicities without the need for complex calculations. The factored form is a powerful tool in the analysis of polynomial functions, providing a clear and concise representation of the roots and their behavior.
The cornerstone of finding roots lies in setting the function equal to zero. This is because the roots are precisely the x-values that make the function's value zero. Therefore, we start by setting f(x) = 0: (x - 6)^2(x + 2)^2 = 0. This equation represents the fundamental condition for finding the roots. It states that the product of the factors (x - 6)^2 and (x + 2)^2 must equal zero. In mathematics, a product of factors is zero if and only if at least one of the factors is zero. This principle is the key to unlocking the roots of the polynomial. It allows us to break down the complex equation into simpler equations, each corresponding to a single factor. By focusing on each factor individually and finding the x-values that make that factor zero, we can systematically identify all the roots of the polynomial. This approach transforms the problem of finding roots into a series of manageable steps, making the solution process more straightforward and accessible. The equation (x - 6)^2(x + 2)^2 = 0 serves as the starting point for our root-finding journey, guiding us towards the specific x-values that satisfy this condition.
Now, let's proceed to solve the equation (x - 6)^2(x + 2)^2 = 0. Following the principle that a product is zero if at least one factor is zero, we set each factor equal to zero separately. This gives us two equations: (x - 6)^2 = 0 and (x + 2)^2 = 0. Each of these equations corresponds to a potential root of the polynomial. To solve (x - 6)^2 = 0, we take the square root of both sides, which yields x - 6 = 0. Adding 6 to both sides, we find x = 6. Similarly, for (x + 2)^2 = 0, taking the square root of both sides gives us x + 2 = 0. Subtracting 2 from both sides, we obtain x = -2. These two values, 6 and -2, are the roots of the polynomial function. They are the x-values where the graph of the function intersects the x-axis. However, to fully understand the behavior of the function near these roots, we must also consider their multiplicities. The multiplicity of a root provides insights into how the graph of the function interacts with the x-axis at that point, whether it crosses the axis or simply touches it and turns around.
Delving into the concept of multiplicity, we observe that the factor (x - 6) is squared, indicating that the root x = 6 has a multiplicity of 2. This means that the root 6 appears twice in the factored form of the polynomial. Similarly, the factor (x + 2) is also squared, implying that the root x = -2 has a multiplicity of 2 as well. The multiplicity of a root profoundly affects the behavior of the graph of the polynomial function near that root. Specifically, when a root has an even multiplicity, such as 2, the graph of the function touches the x-axis at that point but does not cross it. Instead, the graph bounces off the x-axis, changing direction without passing through. This behavior is characteristic of roots with even multiplicities. In contrast, if a root has an odd multiplicity, the graph of the function crosses the x-axis at that point. The multiplicity acts as a guide, revealing the nature of the function's interaction with the x-axis at each root. In this case, both roots, 6 and -2, have a multiplicity of 2, indicating that the graph of f(x) will touch the x-axis at these points and then turn around, without crossing it. This understanding of multiplicity is crucial for sketching the graph of the polynomial and gaining a comprehensive view of its behavior.
Therefore, the roots of the function f(x) = (x - 6)^2(x + 2)^2 are x = 6 with a multiplicity of 2 and x = -2 with a multiplicity of 2. This means that the graph of the function touches the x-axis at x = 6 and x = -2, but it does not cross the x-axis at these points. The multiplicities provide critical information about the function's behavior near these roots. A multiplicity of 2 indicates that the graph has a parabolic shape near the root, either opening upwards or downwards, but not passing through the x-axis. This understanding of roots and multiplicities is essential for sketching the graph of the polynomial function and gaining insights into its overall characteristics. The roots are the foundation upon which we build our understanding of the polynomial's behavior, and the multiplicities add another layer of detail, revealing how the function interacts with the x-axis. In this case, the roots 6 and -2, both with a multiplicity of 2, paint a clear picture of the function's graph touching the x-axis at these points and then turning around, a behavior that is characteristic of roots with even multiplicities.
In conclusion, by setting the function to zero and analyzing the factored form, we successfully identified the roots of f(x) = (x - 6)^2(x + 2)^2 as x = 6 with multiplicity 2 and x = -2 with multiplicity 2. This exercise highlights the importance of understanding roots and their multiplicities in polynomial functions. The roots provide the foundation for understanding where the function intersects the x-axis, while the multiplicities reveal the behavior of the graph near these intersections. This knowledge is invaluable for sketching the graph of the function and gaining insights into its overall characteristics. Furthermore, the techniques used in this analysis are applicable to a wide range of polynomial functions, making them a fundamental tool in mathematical analysis. The ability to find roots and determine their multiplicities is a cornerstone of algebra and calculus, with applications in various fields, including engineering, physics, and economics. Mastering these concepts empowers us to model and understand real-world phenomena using polynomial functions.
Based on our analysis, the correct options are:
- D. 6 with multiplicity 2
- F. -2 with multiplicity 2
