Finding Zeros Of Polynomial Functions F(x) = (x-5)(x-4)(x-2)
In the realm of mathematics, polynomial functions hold a significant position, serving as fundamental building blocks for more complex mathematical models. These functions, characterized by their algebraic expressions involving variables raised to non-negative integer powers, exhibit a wide range of behaviors and properties that are crucial in various fields, including engineering, physics, and computer science. One of the most important aspects of understanding polynomial functions is identifying their zeros, also known as roots or solutions. Zeros are the values of the variable that make the function equal to zero. Finding the zeros of a polynomial function allows us to determine where the function intersects the x-axis on a graph, providing valuable insights into the function's behavior and characteristics. This article delves into the process of finding the zeros of a specific polynomial function, offering a step-by-step guide that will equip you with the skills to solve similar problems.
In this exploration, we will focus on the polynomial function f(x) = (x - 5)(x - 4)(x - 2). This function, expressed in its factored form, presents a unique opportunity to directly identify its zeros. The factored form of a polynomial provides a clear representation of the values that will make each factor equal to zero, subsequently making the entire function equal to zero. By carefully examining the factors, we can extract the zeros without the need for complex calculations or algebraic manipulations. This method not only simplifies the process of finding zeros but also enhances our understanding of the relationship between the factors of a polynomial and its roots. Understanding how to determine zeros from factored form is a crucial skill in algebra, laying the groundwork for tackling more complex polynomial equations and applications. It enables you to visualize the behavior of polynomial functions, predict their values, and solve real-world problems involving curves and relationships modeled by polynomials.
Our primary task is to determine the zeros of the polynomial function given by the equation f(x) = (x - 5)(x - 4)(x - 2). The zeros of a function are the values of x for which the function's output, f(x), is equal to zero. In other words, we need to find the values of x that satisfy the equation (x - 5)(x - 4)(x - 2) = 0. This particular polynomial is presented in a factored form, which significantly simplifies the process of finding the zeros. The factored form directly reveals the roots of the polynomial, as each factor corresponds to a potential zero. To find these zeros, we set each factor equal to zero and solve for x. This method is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero. By applying this property, we can systematically identify all the values of x that make the function equal to zero. Understanding this approach is crucial for solving polynomial equations and for analyzing the behavior of polynomial functions in various contexts. For instance, in graphical analysis, the zeros of a polynomial function represent the points where the graph of the function intersects the x-axis. These points are key features of the graph, providing insights into the function's behavior, such as its intervals of increase and decrease, and its overall shape. Moreover, in real-world applications, zeros can represent critical values or solutions in modeling physical phenomena, engineering problems, and economic scenarios. Mastering the technique of finding zeros from factored form not only enhances your algebraic skills but also equips you with a powerful tool for problem-solving in diverse fields.
To find the zeros of the polynomial function f(x) = (x - 5)(x - 4)(x - 2), we will apply the zero-product property. This property states that if the product of several factors is zero, then at least one of the factors must be zero. In our case, the function f(x) is expressed as a product of three factors: (x - 5), (x - 4), and (x - 2). Setting each of these factors equal to zero will allow us to determine the values of x that make the entire function equal to zero, which are the zeros of the function.
Step 1: Set each factor equal to zero
We begin by setting each factor of the polynomial function to zero. This gives us the following three equations:
- x - 5 = 0
- x - 4 = 0
- x - 2 = 0
Each equation represents a simple linear equation that can be easily solved for x.
Step 2: Solve each equation for x
Now, we solve each equation individually to find the values of x that satisfy them.
- Solving x - 5 = 0: To isolate x, we add 5 to both sides of the equation, which gives us x = 5.
- Solving x - 4 = 0: Similarly, we add 4 to both sides of the equation, resulting in x = 4.
- Solving x - 2 = 0: Adding 2 to both sides of the equation gives us x = 2.
Therefore, the solutions to these equations are x = 5, x = 4, and x = 2. These are the zeros of the polynomial function f(x).
Step 3: Identify the zeros
The zeros of the polynomial function f(x) = (x - 5)(x - 4)(x - 2) are the values of x that make the function equal to zero. From our calculations, we have found three zeros: x = 5, x = 4, and x = 2. These values are the points where the graph of the function would intersect the x-axis. By identifying these zeros, we gain a crucial understanding of the function's behavior and characteristics. Zeros are fundamental in various applications of polynomial functions, including graphing, solving equations, and modeling real-world phenomena. Understanding how to find zeros is an essential skill in algebra and calculus, providing a foundation for more advanced mathematical concepts and problem-solving techniques. This step-by-step approach clearly demonstrates how to efficiently determine the zeros of a polynomial function when it is presented in factored form, enhancing your ability to analyze and interpret polynomial functions.
Having identified the zeros of the polynomial function f(x) = (x - 5)(x - 4)(x - 2) as 5, 4, and 2, we now need to select the correct option from the given choices. The options presented are in the form of sets of numbers, representing potential zeros of the function. To choose the correct answer, we must match our calculated zeros with the set that accurately represents them.
Let's review the options:
- A. 5, -4, 2
- B. 5, -4, -2
- C. 5, 4, 2
- D. 5, 4, -2
Comparing our calculated zeros (5, 4, and 2) with the given options, we can see that option C, which is 5, 4, 2, perfectly matches the zeros we found. The other options contain incorrect values or incorrect signs, making them incorrect choices. Option A includes -4 instead of 4, Option B includes both -4 and -2, and Option D includes -2 instead of 2. Therefore, only option C accurately represents the zeros of the given polynomial function. This process of elimination and comparison ensures that we select the correct answer with confidence. Choosing the correct option is a crucial step in problem-solving, as it confirms that our calculations and understanding of the problem are accurate. In this case, selecting option C demonstrates a clear understanding of how to find the zeros of a polynomial function in factored form and the ability to correctly interpret the results in the context of the given options. This skill is essential for success in mathematics and related fields, where accuracy and precision are paramount.
After carefully analyzing the problem and following the step-by-step solution, we have confidently determined that the zeros of the polynomial function f(x) = (x - 5)(x - 4)(x - 2) are 5, 4, and 2. This conclusion aligns perfectly with option C, which is presented as 5, 4, 2. This answer confirms our understanding of the zero-product property and its application in finding the roots of a polynomial function expressed in factored form. Option C accurately represents the values of x that make the function f(x) equal to zero, thus fulfilling the definition of zeros of a function. This correct identification demonstrates a strong grasp of algebraic concepts and problem-solving techniques essential for success in mathematics.
In conclusion, we have successfully identified the zeros of the polynomial function f(x) = (x - 5)(x - 4)(x - 2) as 5, 4, and 2 by applying the zero-product property. This exercise highlights the importance of understanding the relationship between the factored form of a polynomial and its zeros. The zeros of a polynomial function are fundamental in mathematics and have wide-ranging applications in various fields. They represent the points where the graph of the function intersects the x-axis, providing crucial information about the function's behavior, such as its intervals of positivity and negativity, and its turning points.
Furthermore, the zeros of a polynomial play a critical role in solving polynomial equations. Finding the zeros is equivalent to solving the equation f(x) = 0, which is a common task in algebra and calculus. These solutions are essential for modeling real-world phenomena, where polynomial functions are used to represent various relationships and processes. For instance, in physics, polynomial functions can describe the trajectory of a projectile, and the zeros of the function can represent the points where the projectile hits the ground. In engineering, polynomials can model the behavior of electrical circuits, and the zeros can indicate resonant frequencies. In economics, polynomial functions can be used to model cost and revenue functions, and the zeros can represent break-even points.
Understanding how to find the zeros of a polynomial function is a foundational skill that enables you to solve complex problems and make informed decisions in a variety of contexts. The ability to identify and interpret zeros enhances your mathematical proficiency and prepares you for more advanced topics in algebra, calculus, and beyond. This article has provided a clear and concise guide to finding the zeros of a polynomial in factored form, equipping you with the tools necessary to tackle similar problems and appreciate the significance of zeros in mathematical analysis and real-world applications.
