Polynomial Function With Leading Coefficient 1 And Specified Roots

Finding the correct polynomial function that satisfies given conditions such as roots and leading coefficient requires a solid understanding of polynomial structure and its relationship with roots. In this detailed explanation, we will delve into the process of constructing such a polynomial, focusing on the critical aspects of leading coefficients, roots, and their multiplicities. We'll dissect the problem step by step, ensuring clarity and providing insights to help you confidently tackle similar problems. The core challenge here is to identify a polynomial function f(x) that not only has a leading coefficient of 1 but also possesses specific roots with defined multiplicities. The roots are -2 and 7, each with a multiplicity of 1, and 5 with a multiplicity of 2. This means the polynomial should have factors corresponding to these roots, raised to the power of their respective multiplicities. Understanding multiplicity is crucial because it determines how the graph of the polynomial behaves at the root. A multiplicity of 1 indicates that the graph crosses the x-axis at that root, while a multiplicity of 2 suggests the graph touches the x-axis and turns around. We need to construct our polynomial by incorporating these factors correctly, ensuring that the final expression expands to a polynomial with the desired properties. This involves not only writing the factors but also ensuring that the leading coefficient remains 1, as specified in the problem statement. The final polynomial will be a product of these factors, each contributing to the overall shape and behavior of the function. By carefully considering each root and its multiplicity, we can accurately determine the polynomial function that meets all the given criteria. Let’s explore this process in detail, providing you with a clear methodology to solve similar problems effectively.

Understanding Polynomial Roots and Multiplicities

To effectively construct a polynomial function, a fundamental understanding of polynomial roots and their multiplicities is essential. Roots, or zeros, of a polynomial are the values of x for which the function f(x) equals zero. Each root corresponds to a factor of the polynomial. For instance, if a polynomial has a root of a, then (x - a) is a factor of that polynomial. The multiplicity of a root indicates how many times that root appears as a solution of the polynomial equation. A root with a multiplicity of 1 means the corresponding factor appears once, while a multiplicity of 2 means the factor appears twice, and so on. This concept of multiplicity is crucial in determining the behavior of the polynomial graph at the x-intercepts. When a root has a multiplicity of 1, the graph of the polynomial crosses the x-axis at that point. However, if a root has a multiplicity of 2, the graph touches the x-axis at that point and turns around, creating a tangential intersection. Higher multiplicities can result in more complex behaviors, but the basic principle remains the same: the multiplicity influences how the graph interacts with the x-axis. In our specific problem, we have roots of -2 and 7, each with a multiplicity of 1, and a root of 5 with a multiplicity of 2. This tells us that (x + 2) and (x - 7) are factors that appear once, while (x - 5) appears twice. The polynomial function will thus include the factors (x + 2), (x - 7), and (x - 5)². Understanding these relationships between roots, multiplicities, and factors is the cornerstone of constructing the desired polynomial. By carefully considering each root and its multiplicity, we can build the polynomial expression step by step, ensuring that it satisfies all the given conditions.

Constructing the Polynomial Function

Constructing the polynomial function involves translating the information about the roots and their multiplicities into algebraic factors and then combining them to form the polynomial. We are given that the polynomial has roots -2 and 7, each with a multiplicity of 1, and a root 5 with a multiplicity of 2. This information allows us to build the factors of the polynomial. For the root -2, the corresponding factor is (x - (-2)), which simplifies to (x + 2). Since the multiplicity is 1, this factor appears once. Similarly, for the root 7, the corresponding factor is (x - 7), and it also appears once because its multiplicity is 1. The root 5 has a multiplicity of 2, meaning the factor corresponding to this root, (x - 5), appears twice. Thus, we have the factor (x - 5)². Combining these factors, we get the polynomial in factored form: f(x) = (x + 2)(x - 7)(x - 5)². However, we must also consider the leading coefficient. The problem specifies that the leading coefficient should be 1. In the factored form we have, the leading coefficient is indeed 1 because when we expand the polynomial, the highest power of x (which will be x⁴) will have a coefficient of 1. Therefore, no additional scaling factor is needed. Expanding this factored form will give us the polynomial in standard form, which can be useful for verifying the solution or for other purposes. However, the factored form itself is sufficient to identify the correct polynomial function from the given options. By carefully constructing the factors based on the roots and their multiplicities and ensuring the leading coefficient matches the requirement, we can confidently determine the polynomial function that meets all the specified conditions. The next step involves comparing our constructed polynomial with the provided options to identify the correct answer.

Identifying the Correct Option

Now that we have constructed the polynomial function f(x) = (x + 2)(x - 7)(x - 5)², the next step is to identify the correct option from the given choices. The key is to compare our constructed polynomial with the options provided, looking for an exact match in terms of factors and their multiplicities. Let's briefly recap the options:

A. f(x) = 2(x + 7)(x + 5)(x - 2) B. f(x) = 2(x - 7)(x - 5)(x + 2) C. f(x) = (x + 7)(x + 5)(x + 5)(x - 2) D. f(x) = (x + 2)(x - 7)(x - 5)²

Option A includes factors that do not match our roots. It has (x + 7), (x + 5), and (x - 2), which would correspond to roots of -7, -5, and 2, respectively. This does not match our required roots of -2, 7, and 5. Also, the factor of 2 as the leading coefficient does not comply with a leading coefficient of 1.

Option B also has incorrect roots. The factors (x - 7), (x - 5), and (x + 2) correspond to roots of 7, 5, and -2, which are the correct values, but it lacks the correct multiplicity for the root 5, and it also has a leading coefficient of 2 which is incorrect.

Option C has factors (x + 7), (x + 5), (x + 5), and (x - 2), which correspond to roots of -7, -5, -5, and 2. Again, these roots and their multiplicities do not match our required roots and multiplicities.

Option D, f(x) = (x + 2)(x - 7)(x - 5)², perfectly matches our constructed polynomial. It has the factors (x + 2), (x - 7), and (x - 5)², which correspond to roots of -2 (multiplicity 1), 7 (multiplicity 1), and 5 (multiplicity 2), exactly as specified in the problem. The leading coefficient is also 1, as required. Therefore, the correct option is D.

Conclusion

In conclusion, determining the correct polynomial function given specific roots, multiplicities, and a leading coefficient involves a systematic approach. First, understanding the relationship between roots and factors is crucial. Each root r corresponds to a factor of the form (x - r), and the multiplicity of the root determines the power to which the factor is raised. Second, these factors are combined to form the polynomial in factored form. Finally, the leading coefficient is adjusted to match the given requirement. In this case, we successfully constructed the polynomial f(x) = (x + 2)(x - 7)(x - 5)² that has roots -2 and 7 with multiplicity 1, and a root 5 with multiplicity 2, and a leading coefficient of 1. This systematic process ensures accuracy and efficiency in solving such problems. By carefully analyzing the given information and applying the principles of polynomial construction, we were able to confidently identify the correct answer from the provided options. This methodology can be applied to a wide range of polynomial problems, making it a valuable tool for mathematical problem-solving. Understanding these concepts thoroughly enhances your ability to tackle complex problems and reinforces your grasp of polynomial functions and their properties.