Polynomial Function With Given Roots And Multiplicities Explained

Determining the correct polynomial function given its roots and multiplicities involves understanding the fundamental relationship between roots and factors of a polynomial. This article will delve into the process of constructing a polynomial function from its given characteristics, including roots, multiplicities, and the leading coefficient. We will explore the underlying principles and step-by-step methods to arrive at the correct solution. Specifically, we aim to identify the polynomial function with a leading coefficient of 1, roots -2 and 7 each with multiplicity 1, and a root 5 with multiplicity 2.

Understanding Polynomial Functions

Polynomial functions are fundamental in algebra and calculus, and they can be expressed in the general form:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$

where:

• $a_n, a_{n-1}, ..., a_1, a_0$ are coefficients, which are constants.
• $x$ is the variable.
• $n$ is a non-negative integer representing the degree of the polynomial.
• $a_n$ is the leading coefficient, and if $a_n = 1$, the polynomial is called a monic polynomial.

A root of a polynomial function $f(x)$ is a value $x$ for which $f(x) = 0$. Roots are also known as zeros of the polynomial. Each root corresponds to a factor of the polynomial. For instance, if $r$ is a root of $f(x)$, then $(x - r)$ is a factor of $f(x)$.

Multiplicity refers to the number of times a particular root appears in the factorization of the polynomial. If a root $r$ has a multiplicity of $m$, then the factor $(x - r)$ appears $m$ times in the factored form of the polynomial. For example, if a root $r$ has multiplicity 2, the factor $(x - r)^2$ will be part of the polynomial.

Understanding these concepts is crucial for constructing a polynomial function from its roots and multiplicities. The roots tell us the values of $x$ that make the function equal to zero, and the multiplicities tell us how many times each corresponding factor appears in the polynomial. The leading coefficient scales the entire polynomial, ensuring it fits the specified form.

Constructing Polynomial Functions from Roots and Multiplicities

To construct a polynomial function from its roots and multiplicities, follow these key steps:

1. Identify the roots and their multiplicities: Begin by listing all the roots of the polynomial and their respective multiplicities. This information is essential for forming the factors of the polynomial.

2. Form the factors: For each root $r$ with multiplicity $m$, create a factor of the form $(x - r)^m$. This step translates the roots and their multiplicities into algebraic expressions that will form the building blocks of the polynomial.

3. Multiply the factors: Multiply all the factors together. This product gives you a polynomial function with the specified roots and multiplicities. If the roots are $r_1, r_2, ..., r_k$ with multiplicities $m_1, m_2, ..., m_k$ respectively, the polynomial will have the form:

   $$f(x) = a(x - r_1)^{m_1} (x - r_2)^{m_2} ... (x - r_k)^{m_k}$$

   where $a$ is the leading coefficient.

4. Adjust the leading coefficient: If the leading coefficient is specified, adjust the constant factor $a$ to match the required leading coefficient. This ensures that the polynomial has the correct scaling factor.

By following these steps, you can construct a polynomial function that satisfies the given conditions. This method is particularly useful in various mathematical contexts, including curve fitting, equation solving, and analysis of polynomial behavior.

Applying the Concepts to the Problem

In the given problem, we are asked to find the polynomial function with a leading coefficient of 1, roots -2 and 7 with multiplicity 1, and a root 5 with multiplicity 2. Let’s apply the concepts we've discussed to solve this problem.

1. Identify the roots and their multiplicities:

   • Root -2 with multiplicity 1
   • Root 7 with multiplicity 1
   • Root 5 with multiplicity 2

2. Form the factors:

   • For root -2, the factor is $(x - (-2)) = (x + 2)$
   • For root 7, the factor is $(x - 7)$
   • For root 5 with multiplicity 2, the factor is $(x - 5)^2 = (x - 5)(x - 5)$

3. Multiply the factors:

   $$f(x) = a(x + 2)(x - 7)(x - 5)(x - 5)$$

4. Adjust the leading coefficient:

   Since the leading coefficient is 1, $a = 1$. Therefore,

   $$f(x) = (x + 2)(x - 7)(x - 5)(x - 5)$$

Now, let's compare this result with the given 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. $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$

Comparing our constructed polynomial with the options, we find that option D matches our result. The polynomial function $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$ has the specified roots and multiplicities, with a leading coefficient of 1.

Detailed Analysis of the Correct Option

The correct option, D, $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$, accurately represents the polynomial function with the given characteristics. Let's break down why this is the correct answer:

1. Roots:

   • The factor $(x + 2)$ corresponds to the root $x = -2$.
   • The factor $(x - 7)$ corresponds to the root $x = 7$.
   • The repeated factor $(x - 5)(x - 5)$ corresponds to the root $x = 5$ with multiplicity 2.

2. Multiplicities:

   • The root -2 appears once, so it has multiplicity 1.
   • The root 7 appears once, so it has multiplicity 1.
   • The root 5 appears twice, so it has multiplicity 2.

3. Leading Coefficient:

   To determine the leading coefficient, we need to consider the terms with the highest power of $x$ in each factor. When we expand the polynomial, the highest power term will be the product of the $x$ terms in each factor:

   $$x imes x imes x imes x = x^4$$

The coefficient of this $x^4$ term will be the leading coefficient. In this case, the leading coefficient is 1, as there are no constant multipliers in front of the factors. This aligns with the requirement that the leading coefficient should be 1.

Why Other Options are Incorrect

To further solidify our understanding, let’s examine why the other options are incorrect:

• Option A: $f(x) = 2(x + 7)(x + 5)(x - 2)$

  • This option has a leading coefficient of 2, which does not match the required leading coefficient of 1.
  • The roots are -7, -5, and 2, which do not match the specified roots of -2, 7, and 5 (with multiplicity 2).

• Option B: $f(x) = 2(x - 7)(x - 5)(x + 2)$

  • This option also has a leading coefficient of 2, which is incorrect.
  • The roots are 7, 5, and -2, but the root 5 only appears once, whereas it should appear twice.

• Option C: $f(x) = (x + 7)(x + 5)(x + 5)(x - 2)$

  • The leading coefficient is 1, which is correct.
  • The roots are -7, -5 (with multiplicity 2), and 2, which do not match the specified roots.

By analyzing each option, we can clearly see that only option D correctly represents the polynomial function with the given roots, multiplicities, and leading coefficient.

Conclusion

In summary, the polynomial function with a leading coefficient of 1, roots -2 and 7 with multiplicity 1, and root 5 with multiplicity 2 is $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$. This conclusion was reached by systematically constructing the polynomial from its roots and multiplicities, and then verifying the result against the provided options.

Understanding how to construct polynomial functions from their roots and multiplicities is a valuable skill in algebra and higher mathematics. It allows for the analysis and manipulation of polynomial equations and functions, which are essential in various applications, including engineering, physics, and computer science. The process involves identifying the roots and their multiplicities, forming the corresponding factors, multiplying these factors together, and adjusting the leading coefficient as necessary. By mastering this technique, one can confidently tackle a wide range of polynomial-related problems.

Therefore, the correct answer is D. $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$.