Zeroes Of The Graph Of F(x) = -x⁵ + 9x⁴ - 18x³ A Comprehensive Analysis

by ADMIN 72 views

Introduction: Understanding Polynomial Zeroes

In the realm of mathematics, particularly in the study of polynomial functions, zeroes play a pivotal role. The zeroes of a polynomial function, also known as roots or x-intercepts, are the values of x for which the function equals zero. These values provide critical insights into the behavior and characteristics of the polynomial graph. This article delves into the process of identifying and characterizing the zeroes of a specific polynomial function, f(x) = -x⁵ + 9x⁴ - 18x³. We will explore the concept of multiplicity, which describes the number of times a particular zero appears as a root of the polynomial. By understanding the zeroes and their multiplicities, we can gain a comprehensive understanding of the function's graph and its properties.

Finding the zeroes of a polynomial function involves determining the values of x that make the function equal to zero. These values are crucial for understanding the behavior of the polynomial graph. Each zero corresponds to a point where the graph intersects or touches the x-axis. The multiplicity of a zero indicates how many times that zero appears as a root of the polynomial equation. This information is essential for accurately sketching the graph and understanding its local behavior around each zero. Understanding these concepts allows us to analyze and interpret polynomial functions more effectively.

The given function, f(x) = -x⁵ + 9x⁴ - 18x³, is a quintic polynomial (a polynomial of degree 5). To find its zeroes, we need to solve the equation -x⁵ + 9x⁴ - 18x³ = 0. This involves factoring the polynomial and setting each factor equal to zero. Factoring is a key technique in polynomial algebra, allowing us to break down complex expressions into simpler, more manageable forms. By identifying common factors and applying factoring methods, we can rewrite the polynomial in a way that reveals its roots. This process not only helps in finding the zeroes but also in simplifying the polynomial for further analysis, such as determining its end behavior and turning points. The degree of the polynomial is significant as it indicates the maximum number of zeroes the polynomial can have, including both real and complex roots. In this case, the quintic polynomial can have up to five zeroes, which may include real numbers and complex numbers. Complex zeroes always come in conjugate pairs if the polynomial has real coefficients, but in this context, we are primarily focused on the real zeroes and their multiplicities.

Step-by-Step Solution: Finding the Zeroes

The first step in finding the zeroes of the polynomial function f(x) = -x⁵ + 9x⁴ - 18x³ is to factor out the greatest common factor (GCF). The GCF of the terms in this polynomial is -x³. Factoring out -x³, we get:

f(x) = -x³(x² - 9x + 18)

This simplifies the polynomial and makes it easier to find the roots. Factoring is a fundamental technique in algebra that helps to break down complex expressions into simpler components. By identifying and extracting the GCF, we reduce the complexity of the polynomial, making it easier to analyze and solve. This step is crucial because it allows us to isolate the roots more effectively. Without factoring, solving the quintic polynomial directly would be significantly more challenging. The factored form not only reveals the roots but also their multiplicities, which are essential for understanding the behavior of the graph near those points.

Next, we need to factor the quadratic expression (x² - 9x + 18). This quadratic can be factored into two binomials by finding two numbers that multiply to 18 and add to -9. Those numbers are -3 and -6. Therefore, the factored form of the quadratic is:

(x - 3)(x - 6)

Factoring the quadratic expression is a crucial step in determining the remaining zeroes of the polynomial. This process involves identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic. In this case, we needed two numbers that multiply to 18 (the constant term) and add up to -9 (the coefficient of the linear term). The correct factors, -3 and -6, allow us to rewrite the quadratic in its factored form. This factorization is essential because it transforms the quadratic expression into a product of two simpler binomials, each of which can be easily set to zero to find the roots. The factored form provides a clear pathway to the zeroes, making the polynomial equation much easier to solve. This step leverages the basic principles of quadratic factorization, a fundamental skill in algebra that is widely applicable in various mathematical contexts.

Combining the factored GCF and the factored quadratic, we get the fully factored form of the polynomial:

f(x) = -x³(x - 3)(x - 6)

This fully factored form is crucial for identifying the zeroes and their multiplicities. The factored form of a polynomial provides a clear view of its roots and how many times each root appears. Each factor corresponds to a zero of the polynomial, and the exponent of the factor indicates the multiplicity of that zero. In this case, the fully factored form reveals three distinct zeroes: 0, 3, and 6. The multiplicity of each zero is directly related to the exponent of its corresponding factor. The factor -x³ indicates that 0 is a zero with multiplicity 3, while the factors (x - 3) and (x - 6) each have an exponent of 1, indicating that 3 and 6 are zeroes with multiplicity 1. This fully factored form is essential for accurately sketching the graph of the polynomial and understanding its behavior near the x-intercepts. It provides a complete and concise representation of the polynomial's roots, which is fundamental for further analysis and applications.

Identifying the Zeroes and Their Multiplicities

From the factored form f(x) = -x³(x - 3)(x - 6), we can identify the zeroes and their multiplicities:

  • -x³ gives us a zero of 0 with multiplicity 3.
  • (x - 3) gives us a zero of 3 with multiplicity 1.
  • (x - 6) gives us a zero of 6 with multiplicity 1.

Multiplicity plays a crucial role in understanding the behavior of the graph at each zero. A zero with an odd multiplicity (like 1 or 3) will cause the graph to cross the x-axis at that point. A zero with an even multiplicity (like 2) will cause the graph to touch the x-axis and turn around. In our case, the zero at 0 has a multiplicity of 3, so the graph crosses the x-axis at x = 0. The zeroes at 3 and 6 have multiplicities of 1, so the graph also crosses the x-axis at these points. This information is essential for accurately sketching the polynomial function's graph and understanding its overall shape. The multiplicities help determine the local behavior of the graph around each zero, which is a key aspect of polynomial analysis.

In summary, understanding the zeroes and their multiplicities is essential for sketching the graph of a polynomial function. The zeroes tell us where the graph intersects or touches the x-axis, and the multiplicities tell us how the graph behaves at these points. Multiplicity helps us distinguish between points where the graph crosses the x-axis and points where it merely touches and turns around. This detailed understanding of the zeroes and their multiplicities provides valuable insights into the overall behavior and characteristics of the polynomial function. The information derived from the zeroes and their multiplicities is also crucial for solving polynomial equations and analyzing various real-world applications involving polynomial models.

Conclusion: The Zeroes of f(x) = -x⁵ + 9x⁴ - 18x³

In conclusion, the zeroes of the graph of f(x) = -x⁵ + 9x⁴ - 18x³ are:

  • 0 with multiplicity 3
  • 3 with multiplicity 1
  • 6 with multiplicity 1

Therefore, the correct answer is B. This comprehensive analysis highlights the importance of factoring and understanding multiplicities when working with polynomial functions. The process of finding zeroes and their multiplicities involves several key steps, including factoring out the greatest common factor, factoring quadratic expressions, and interpreting the results. Each step contributes to a deeper understanding of the polynomial's behavior and graphical representation. The concept of multiplicity is particularly significant as it determines how the graph interacts with the x-axis at each zero. A high multiplicity indicates a more pronounced effect on the graph's shape near that zero. This knowledge is not only valuable in mathematical contexts but also in various scientific and engineering applications where polynomial functions are used to model real-world phenomena. By mastering these techniques, students and professionals alike can effectively analyze and manipulate polynomial functions to solve a wide range of problems.

Understanding the zeroes and their multiplicities is fundamental to polynomial analysis. This knowledge allows us to accurately sketch the graph, predict the function's behavior, and solve related equations. The ability to identify and interpret the zeroes of a polynomial is a crucial skill in algebra and calculus. It forms the basis for more advanced topics such as polynomial division, synthetic division, and the Remainder Theorem. Furthermore, understanding zeroes and their multiplicities is essential for applications in various fields, including physics, engineering, and computer science. Polynomial functions are used to model a wide range of phenomena, and the ability to analyze these functions is a valuable asset in many disciplines. The concepts discussed in this article provide a solid foundation for further exploration of polynomial functions and their applications.