Polynomial Function With Leading Coefficient 2, Root -4 (Multiplicity 3) And Root 10 (Multiplicity 1)

In the realm of polynomial functions, understanding the relationship between roots, multiplicities, and leading coefficients is crucial. This article delves into the process of constructing a polynomial function given its roots, their multiplicities, and the leading coefficient. Specifically, we aim to identify the polynomial function that has a leading coefficient of 2, a root of -4 with a multiplicity of 3, and a root of 10 with a multiplicity of 1. This exploration will not only clarify the fundamental concepts but also provide a step-by-step approach to solving similar problems. Polynomial functions are fundamental in algebra and calculus, and mastering their construction is essential for various applications in mathematics, science, and engineering. We'll break down each component, from roots and multiplicities to the leading coefficient, and demonstrate how they come together to define a unique polynomial. This understanding is pivotal for anyone delving deeper into mathematical analysis and problem-solving.

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial function is given by:

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

where $a_n, a_{n-1}, ..., a_1, a_0$ are coefficients and n is a non-negative integer representing the degree of the polynomial. Each term $a_i x^i$ is a monomial. The term $a_n x^n$ with the highest power of x determines the degree of the polynomial, and $a_n$ is called the leading coefficient. Understanding the leading coefficient is crucial as it significantly impacts the end behavior of the polynomial function. A polynomial's roots are the values of x for which f(x) = 0. These roots are also known as zeros of the polynomial. The behavior of a polynomial near its roots is dictated by the multiplicity of each root. For example, a root with multiplicity 1 means the polynomial crosses the x-axis at that point, while a root with multiplicity 2 means the polynomial touches the x-axis and turns around. These fundamental concepts are the building blocks for constructing and analyzing polynomial functions. The relationship between coefficients, roots, and their multiplicities offers deep insights into the nature and behavior of polynomials, making them indispensable tools in mathematical modeling and analysis. By understanding these core principles, we can effectively construct polynomials that fit specific criteria, such as those described in our problem.

Key Components of Polynomial Functions

1. Roots: The roots of a polynomial function, also known as zeros, are the values of x for which the function equals zero, i.e., f(x) = 0. Roots are critical in determining the factors of the polynomial. For instance, if a is a root, then (x - a) is a factor of the polynomial. The roots provide the x-intercepts of the graph of the polynomial function.
2. Multiplicity: The multiplicity of a root refers to the number of times a particular root appears as a solution of the polynomial equation. If a root 'a' has a multiplicity of m, it means the factor (x - a) appears m times in the factored form of the polynomial. The multiplicity of a root affects the behavior of the graph at the x-intercept. For example, a root with an odd multiplicity will cause the graph to cross the x-axis, while a root with an even multiplicity will cause the graph to touch the x-axis and turn around.
3. Leading Coefficient: The leading coefficient is the coefficient of the term with the highest power of x in the polynomial. It plays a crucial role in determining the end behavior of the polynomial function. The sign of the leading coefficient dictates whether the polynomial rises or falls as x approaches positive or negative infinity. For example, if the leading coefficient is positive and the degree of the polynomial is even, the function will rise on both ends. If it's positive and the degree is odd, it will fall to the left and rise to the right. The magnitude of the leading coefficient also affects the steepness of the graph.

We are tasked with finding the polynomial function that satisfies the following conditions:

• Leading coefficient: 2
• Root: -4 with multiplicity 3
• Root: 10 with multiplicity 1

To solve this, we will utilize the relationship between the roots, their multiplicities, and the leading coefficient to construct the polynomial function. Each root contributes a factor to the polynomial, with the multiplicity determining the power of that factor. The leading coefficient then scales the entire polynomial. The goal is to assemble these components in the correct manner to arrive at the function that meets all the specified criteria. This process demonstrates the practical application of polynomial theory, illustrating how the algebraic form of a polynomial is intrinsically linked to its roots and overall behavior. By carefully considering each condition, we can systematically build the desired polynomial function, ensuring it matches the given characteristics precisely. This method is a cornerstone of polynomial analysis and function construction.

Given the roots and their multiplicities, we can construct the polynomial function in factored form. A root of -4 with multiplicity 3 corresponds to the factor $(x - (-4))^3$, which simplifies to $(x + 4)^3$. A root of 10 with multiplicity 1 corresponds to the factor $(x - 10)^1$, which is simply $(x - 10)$. The leading coefficient is given as 2, so we multiply the entire expression by 2. Therefore, the polynomial function can be written as:

$f(x) = 2(x + 4)^3 (x - 10)$

This factored form clearly shows the roots and their respective multiplicities. The factor $(x + 4)^3$ indicates the root -4 with multiplicity 3, and the factor $(x - 10)$ indicates the root 10 with multiplicity 1. The leading coefficient 2 ensures that the polynomial has the desired scaling. Expanding this expression would yield the polynomial in its standard form, but the factored form is often more useful for identifying roots and understanding the function's behavior. This construction highlights the power of factored form in polynomial representation, allowing for a direct link between the roots and the function's expression. By assembling these factors and the leading coefficient, we have successfully created a polynomial function that meets the given specifications, demonstrating the core principles of polynomial construction.

Now, let's evaluate the given options based on our constructed polynomial function:

A. $f(x) = 2(x - 4)(x - 4)(x - 4)(x + 10)$ B. $f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10)$ C. $f(x) = 3(x - 4)(x - 4)(x + 10)$ D. Discussion category: mathematics

Comparing these options with our constructed function $f(x) = 2(x + 4)^3 (x - 10)$, we can see that option B matches our result perfectly. Option A has incorrect signs for the root -4, and option C has an incorrect leading coefficient and misses a factor. The Discussion category is not a valid polynomial function.

Option B, $f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10)$, correctly represents the polynomial function with a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. The factors $(x + 4)(x + 4)(x + 4)$ correspond to the root -4 with multiplicity 3, and the factor $(x - 10)$ corresponds to the root 10 with multiplicity 1. This methodical comparison demonstrates the importance of understanding the relationship between roots, multiplicities, and factored form in identifying the correct polynomial function. By carefully analyzing each option against our derived function, we can confidently select the accurate representation that satisfies all given conditions, reinforcing the core principles of polynomial construction.

The correct option is:

B. $f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10)$

This polynomial function satisfies all the given conditions: a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. The factored form clearly shows the roots and their multiplicities, making it easy to verify that the function meets the specified criteria. This solution underscores the importance of accurately translating roots and multiplicities into factors and combining them with the correct leading coefficient. The ability to construct and identify polynomial functions based on their roots and leading coefficients is a fundamental skill in algebra and calculus, with applications spanning various mathematical and scientific domains. This exercise exemplifies the practical application of polynomial theory and the critical role of factored form in understanding and manipulating polynomial expressions.

In conclusion, we successfully identified the polynomial function with a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. The process involved understanding the relationship between roots, multiplicities, and the leading coefficient, and then constructing the polynomial in factored form. The correct function, $f(x) = 2(x + 4)^3 (x - 10)$, was found to be option B. This exercise highlights the fundamental principles of polynomial construction and analysis. The ability to translate roots and multiplicities into factored form, combined with the leading coefficient, is crucial for solving a wide range of problems in mathematics and related fields. Understanding these concepts allows for a deeper comprehension of polynomial behavior and their applications in various contexts. Mastering polynomial construction techniques provides a strong foundation for more advanced mathematical studies and practical problem-solving scenarios.