Polynomial Function Zeros Finding Zeros Of F(x) = -1/2x³ + X² + 5/2x - 3

Jul 13, 2025 by ADMIN 73 views

Exploring Polynomial Functions Finding Zeros of f(x) = -1/2x³ + x² + 5/2x - 3 Given c = -2

In the fascinating realm of mathematics, polynomial functions hold a pivotal role, weaving their way through various applications and theoretical explorations. Understanding the behavior and properties of these functions is crucial for students, educators, and professionals alike. This article delves into the intricacies of a specific polynomial function, $f(x) = -\frac{1}{2}x^3 + x^2 + \frac{5}{2}x - 3$ , with a known zero at $c = -2$ . Our journey will encompass identifying all the remaining zeros, a critical step in fully grasping the function's characteristics. We will methodically dissect the function, employing techniques such as synthetic division and quadratic formula application, ensuring clarity and comprehension at each stage. The goal is to not only find the zeros but also to elucidate the underlying mathematical principles that govern these solutions. As we proceed, we will emphasize the importance of each step, highlighting how it contributes to the broader understanding of polynomial functions and their zeros. This detailed exploration aims to provide a comprehensive guide, making the process accessible and informative for all readers, regardless of their mathematical background. The adventure into this polynomial's world promises to be enlightening, equipping you with the skills to tackle similar challenges with confidence.

a. Identifying All the Remaining Zeros

To pinpoint the remaining zeros of the polynomial function $f(x) = -\frac{1}{2}x^3 + x^2 + \frac{5}{2}x - 3$ , given that $c = -2$ is a zero, we embark on a structured mathematical expedition. The journey begins with the strategic application of synthetic division, a method that efficiently reduces the polynomial's degree. Knowing that -2 is a zero means $(x + 2)$ is a factor of $f(x)$ . Synthetic division allows us to divide $f(x)$ by $(x + 2)$ , revealing the quotient, which is a polynomial of a lesser degree. This quotient holds the key to finding the remaining zeros. The process involves setting up the synthetic division table with the coefficients of $f(x)$ and the known zero, -2. Performing the synthetic division, we methodically bring down the coefficients, multiply, and add, ultimately obtaining the coefficients of the quotient polynomial. This step is crucial as it transforms our cubic polynomial into a more manageable quadratic form. The quadratic equation, derived from the quotient, can then be solved using various methods, such as factoring, completing the square, or the quadratic formula. Each method offers a unique pathway to the solution, and the choice often depends on the specific characteristics of the quadratic. In this exploration, we'll likely employ the quadratic formula, a universally applicable tool that guarantees the identification of all real and complex roots. As we unveil these zeros, we gain a deeper understanding of the function's behavior, particularly its points of intersection with the x-axis. This comprehensive approach not only answers the immediate question but also enriches our mathematical toolkit, preparing us for future polynomial explorations.

Step-by-Step Process of Finding Zeros

Applying Synthetic Division: Use the given zero, $c = -2$ , to perform synthetic division on the polynomial $f(x) = -\frac{1}{2}x^3 + x^2 + \frac{5}{2}x - 3$ . This process will reduce the cubic polynomial to a quadratic polynomial. Synthetic division is an efficient method for dividing a polynomial by a linear factor $(x - c)$ , where $c$ is a known root. It simplifies the process of finding the quotient and the remainder, which are crucial for further analysis. The setup involves writing the coefficients of the polynomial and the value of $c$ in a specific format, then performing a series of arithmetic operations to obtain the coefficients of the quotient. The last number in the result is the remainder, which should be zero if $c$ is indeed a root of the polynomial. This step is not just a computational procedure; it's a bridge that connects the known root to the remaining factors of the polynomial.
Deriving the Quadratic Quotient: From the result of the synthetic division, identify the coefficients of the quadratic quotient. This quadratic equation represents the reduced form of the original polynomial after dividing out the factor corresponding to the known zero. The coefficients obtained from synthetic division directly translate into the coefficients of the quadratic equation. For instance, if the synthetic division results in the numbers $a$ , $b$ , and $c$ , the quadratic quotient will be $ax^2 + bx + c$ . This quadratic equation is the key to finding the remaining zeros of the original polynomial. It encapsulates the remaining roots in a more manageable form, allowing us to apply standard techniques for solving quadratic equations. The transition from a cubic to a quadratic polynomial is a significant simplification, making the problem more accessible and solvable.
Solving the Quadratic Equation: Solve the resulting quadratic equation to find the remaining zeros of the polynomial. This can be done by factoring, completing the square, or using the quadratic formula. The choice of method often depends on the specific characteristics of the quadratic equation. Factoring is a straightforward approach when the quadratic can be easily expressed as a product of two linear factors. Completing the square is a more versatile method that can be used for any quadratic equation, but it involves algebraic manipulations that can be cumbersome. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , is a universal solution that works for all quadratic equations, regardless of their complexity. It provides a direct and reliable way to find the roots, ensuring that no solution is missed. The solutions obtained from the quadratic equation are the remaining zeros of the original polynomial, completing the picture of the function's roots.

By meticulously following these steps, we not only find the remaining zeros but also reinforce our understanding of polynomial division and the relationship between zeros and factors. This process is a cornerstone of polynomial analysis, equipping us with the tools to dissect and comprehend complex functions.

b. Factoring f(x) Completely

Factoring a polynomial completely is akin to disassembling a complex machine into its fundamental components. For our polynomial $f(x) = -\frac{1}{2}x^3 + x^2 + \frac{5}{2}x - 3$ , knowing all its zeros is the key to this disassembly. Each zero corresponds to a factor of the polynomial, and expressing $f(x)$ as a product of these factors provides a comprehensive understanding of its structure. The process begins by recognizing the relationship between zeros and factors: if $c$ is a zero of $f(x)$ , then $(x - c)$ is a factor. Given that $c = -2$ is a zero, $(x + 2)$ is one factor. The remaining zeros, which we identified in part (a), will yield the other factors. These factors will likely be linear or quadratic, depending on the nature of the zeros (real or complex). The leading coefficient of the polynomial, $-\frac{1}{2}$ in this case, also plays a crucial role in the complete factorization. It must be included as a factor to ensure the factored form is equivalent to the original polynomial. The complete factorization not only reveals the zeros but also provides insights into the polynomial's behavior, such as its end behavior and turning points. This factored form is a powerful tool for solving equations, graphing, and further analysis of the function. It's a concise representation that encapsulates the polynomial's essence, making it easier to work with and understand.

Steps to Factor Completely

Expressing f(x) as a Product of Linear Factors: Use the zeros found in part (a), including the given zero $c = -2$ , to express $f(x)$ as a product of linear factors. Each zero $r$ corresponds to a factor $(x - r)$ . If a zero is repeated, its corresponding factor will appear multiple times. This step is crucial as it translates the zeros, which are essentially solutions to the equation $f(x) = 0$ , into the building blocks of the polynomial's structure. For example, if the zeros are $-2$ , $r_1$ , and $r_2$ , the factors will be $(x + 2)$ , $(x - r_1)$ , and $(x - r_2)$ . The polynomial can then be written as a product of these factors, along with a constant factor that accounts for the leading coefficient. This representation provides a clear and direct link between the roots of the polynomial and its algebraic form, making it easier to visualize and analyze the function's behavior.
Including the Leading Coefficient: Remember to include the leading coefficient of $f(x)$ in the factored form. This is essential to ensure that the factored form is equivalent to the original polynomial. The leading coefficient, in this case $-\frac{1}{2}$ , scales the polynomial and affects its overall shape and orientation. Failing to include it would result in a factored form that, while having the correct zeros, does not accurately represent the original polynomial. The leading coefficient acts as a multiplier, ensuring that the factored form expands back to the original polynomial. This step is a critical detail that completes the factoring process, ensuring accuracy and consistency.

By adhering to these steps, we achieve a complete and accurate factorization of the polynomial, unlocking its structural secrets and paving the way for deeper analysis and applications.

c. Solving the Equation f(x) = 0

Solving the equation $f(x) = 0$ is the culmination of our exploration, where we directly apply our understanding of zeros and factorization to find the values of $x$ that make the function equal to zero. These values, the zeros of the function, are the points where the graph of $f(x)$ intersects the x-axis. Having factored $f(x)$ completely in part (b), we have essentially laid the groundwork for this final step. The factored form of the polynomial transforms the equation $f(x) = 0$ into a product of factors equal to zero. This is a pivotal transformation, as it allows us to apply the zero-product property: if a product of factors is zero, then at least one of the factors must be zero. This property elegantly simplifies the problem, breaking it down into a series of simpler equations, one for each factor. Each linear factor $(x - c)$ gives us a solution $x = c$ , which is a zero of the polynomial. By setting each factor equal to zero and solving, we systematically uncover all the solutions to the equation $f(x) = 0$ . These solutions are not just abstract numbers; they are key characteristics of the polynomial function, defining its behavior and graph. Solving $f(x) = 0$ is therefore not just a mathematical exercise; it's a crucial step in understanding the function's essence.

Process to Solve f(x) = 0

Applying the Zero-Product Property: Set each factor obtained in part (b) equal to zero and solve for $x$ . The zero-product property states that if the product of several factors is zero, then at least one of the factors must be zero. This principle is the cornerstone of solving polynomial equations once they are factored. By setting each factor to zero, we transform a single complex equation into a set of simpler, linear equations. Each linear equation corresponds to a zero of the polynomial, and solving it yields that zero. This step is a direct application of the fundamental theorem of algebra, which connects the roots of a polynomial to its factors. It's a powerful and efficient method for finding all the solutions to the equation $f(x) = 0$ .
Listing All Solutions: List all the solutions obtained from the factors. These solutions are the zeros of the polynomial function $f(x)$ . Each solution represents a point where the graph of the function intersects the x-axis. These zeros are not just isolated numbers; they are integral to understanding the function's behavior, its graph, and its applications. The set of all solutions provides a complete picture of where the function equals zero, which is a fundamental characteristic. This step is the culmination of our efforts, providing a definitive answer to the question of solving $f(x) = 0$ . It's a testament to the power of factorization and the zero-product property in unraveling the solutions of polynomial equations.

In summary, this exploration has taken us through the essential steps of analyzing a polynomial function: identifying zeros, factoring completely, and solving the equation $f(x) = 0$ . We began with a cubic polynomial and a known zero, and through synthetic division, we reduced the problem to solving a quadratic equation. The quadratic formula provided the remaining zeros, which then allowed us to factor the polynomial completely. Finally, by applying the zero-product property, we solved for all the values of $x$ that make the function equal to zero. This methodical approach underscores the interconnectedness of these concepts in polynomial analysis. Each step builds upon the previous one, leading to a comprehensive understanding of the function's behavior. The zeros, factors, and solutions are not just mathematical abstractions; they are key characteristics that define the polynomial and its graph. This journey through $f(x) = -\frac{1}{2}x^3 + x^2 + \frac{5}{2}x - 3$ serves as a paradigm for analyzing other polynomials, equipping you with the skills and knowledge to tackle similar challenges with confidence. The ability to dissect and understand polynomial functions is a valuable asset in mathematics and its applications, opening doors to deeper insights and problem-solving capabilities.