Finding Zeros Of Polynomial Function F(x) = (x+1)(x-8)(5x+2)

In mathematics, finding the zeros of a function is a fundamental task. Zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. These points are crucial for understanding the behavior of the function and its graph. This article will guide you through the process of identifying the zeros of the polynomial function f(x) = (x+1)(x-8)(5x+2). Understanding the zeros of a function helps in analyzing its graph, determining its domain and range, and solving related equations and inequalities. In the context of polynomials, the zeros provide essential information about the function's behavior, such as where it crosses the x-axis and the intervals where it is positive or negative. The zeros are also critical in various applications, including physics, engineering, and economics, where polynomial functions are used to model real-world phenomena. For example, in physics, the zeros of a polynomial function can represent the points where an object's trajectory intersects the ground. In engineering, they can indicate the stability of a system. In economics, they might represent equilibrium points in a market model. Therefore, mastering the technique of finding zeros is not only a key skill in mathematics but also a valuable tool in many other disciplines. In the following sections, we will systematically break down the process of finding the zeros of the given polynomial function, ensuring a clear and comprehensive understanding of each step. This article aims to provide not just the solution, but also the reasoning and methodology behind it, making it easier for readers to apply these techniques to other similar problems. By the end of this article, you will have a solid understanding of how to identify zeros and their significance in the broader context of mathematical analysis and applications. This understanding will empower you to tackle more complex problems and gain a deeper appreciation of the role of polynomial functions in various fields.

Understanding Zeros of a Function

The zeros of a function f(x) are the values of x that make f(x) = 0. In simpler terms, these are the points where the graph of the function intersects the x-axis. For a polynomial function in factored form, such as the one we have, finding the zeros is relatively straightforward. Each factor of the form (x - a) contributes a zero at x = a. This is because when x = a, the factor becomes zero, and thus the entire function becomes zero. Conversely, if the factor is of the form (x + a), the zero occurs at x = -a. This fundamental concept is crucial for quickly identifying the zeros of polynomial functions, especially when they are presented in factored form. Understanding zeros is essential not only for solving equations but also for sketching the graph of a function. The zeros provide critical anchor points that help in visualizing the function's behavior. For instance, between any two consecutive zeros, the function will either be entirely positive or entirely negative. This property allows us to determine the intervals where the function lies above or below the x-axis. Moreover, the zeros play a significant role in the application of polynomial functions in various fields. In physics, for example, the zeros can represent the points where a projectile hits the ground. In engineering, they can indicate the points of equilibrium in a system. In economics, they might represent the break-even points in a business model. Therefore, the ability to find and interpret zeros is a valuable skill that extends beyond the realm of pure mathematics. In the subsequent sections, we will apply this understanding to the given function, f(x) = (x+1)(x-8)(5x+2), to systematically identify its zeros. By breaking down the function into its factors and setting each factor to zero, we will demonstrate how to accurately determine the values of x that make the function equal to zero. This step-by-step approach will not only provide the solution but also enhance your comprehension of the underlying principles.

Analyzing the Given Function: f(x) = (x+1)(x-8)(5x+2)

Our given function is f(x) = (x+1)(x-8)(5x+2). This function is a polynomial expressed in factored form, which makes it easier to identify its zeros. Each factor corresponds to a potential zero of the function. The factors are (x+1), (x-8), and (5x+2). To find the zeros, we set each factor equal to zero and solve for x. This approach leverages 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. This property is the cornerstone of solving polynomial equations in factored form. By applying the zero-product property, we can systematically determine the values of x that make the entire function equal to zero. Analyzing the factors individually allows us to break down the problem into smaller, more manageable steps. For each factor, we will perform a simple algebraic manipulation to isolate x and find its value. This process not only yields the zeros but also reinforces the understanding of how each factor contributes to the overall behavior of the function. For example, the factor (x+1) implies that the function will be zero when x is equal to -1. Similarly, the factor (x-8) indicates a zero at x = 8. The factor (5x+2) requires a bit more algebraic manipulation, but the principle remains the same: we set the factor to zero and solve for x. This methodical approach ensures that we do not overlook any potential zeros and that we accurately determine all the values of x that satisfy the equation f(x) = 0. In the following sections, we will delve into the specific steps required to solve for x in each factor, providing a clear and concise explanation of the algebraic manipulations involved. This will not only give you the solutions but also equip you with the skills to tackle similar problems with confidence. Understanding the factored form of a polynomial function is a powerful tool for analyzing its behavior and finding its zeros, making this a crucial concept in algebra and calculus.

Step-by-Step Solution: Finding the Zeros

1. Setting the First Factor to Zero: (x + 1) = 0

To find the first zero, we set the factor (x + 1) equal to zero:

x + 1 = 0

Subtracting 1 from both sides, we get:

x = -1

So, the first zero is x = -1. This means that when x is -1, the function f(x) will be equal to zero. Graphically, this corresponds to the point where the function's graph crosses the x-axis at x = -1. Understanding this relationship between the zero and the graph is crucial for visualizing the function's behavior. The zero x = -1 is a critical point on the graph, providing a reference for understanding the function's values in the vicinity of this point. To the left of x = -1, the function may be positive or negative, and to the right, it may behave differently. This is why identifying zeros is a fundamental step in analyzing and sketching the graph of a polynomial function. In this case, the factor (x + 1) is a linear term, and its zero is a straightforward solution. However, for higher-degree polynomials, the process may involve more complex algebraic techniques, such as factoring or using the quadratic formula. Nonetheless, the underlying principle remains the same: we set each factor to zero and solve for x. This systematic approach ensures that we do not miss any potential zeros and that we accurately determine the values of x that make the function equal to zero. In the following steps, we will apply the same principle to the remaining factors of the function, building a complete set of zeros that will allow us to fully understand the function's behavior and graph. Each zero contributes to the overall shape and characteristics of the polynomial function, making their identification a crucial task in mathematical analysis.

2. Setting the Second Factor to Zero: (x - 8) = 0

Next, we set the second factor (x - 8) equal to zero:

x - 8 = 0

Adding 8 to both sides, we find:

x = 8

Thus, the second zero is x = 8. This indicates that the function f(x) becomes zero when x is 8. Similar to the first zero, this corresponds to a point where the graph of the function intersects the x-axis, but this time at x = 8. The zero x = 8 provides another critical reference point for analyzing the function's graph. The behavior of the function in the vicinity of x = 8 will be influenced by this zero, and understanding this behavior is crucial for sketching an accurate graph. Between the zeros x = -1 and x = 8, the function will either be entirely positive or entirely negative. This property allows us to determine the intervals where the function lies above or below the x-axis. The factor (x - 8), like the factor (x + 1), is a linear term, making the process of finding its zero relatively simple. However, the significance of this zero in the context of the entire polynomial function is substantial. It contributes to the overall shape and characteristics of the function, and its location on the x-axis is a key piece of information for understanding the function's behavior. In the subsequent step, we will address the final factor, (5x + 2), and find its corresponding zero. This will complete the set of zeros for the function f(x) = (x+1)(x-8)(5x+2), providing us with a comprehensive understanding of its behavior and graph. Each zero plays a vital role in shaping the function, and their accurate identification is essential for mathematical analysis and applications.

3. Setting the Third Factor to Zero: (5x + 2) = 0

Finally, we set the third factor (5x + 2) equal to zero:

5x + 2 = 0

Subtracting 2 from both sides gives:

5x = -2

Dividing both sides by 5, we get:

x = -2/5

Therefore, the third zero is x = -2/5. This zero, like the previous two, indicates a point where the function f(x) equals zero, and its graph intersects the x-axis. The zero x = -2/5 is particularly interesting because it is a fraction, demonstrating that zeros can be rational numbers. This further enriches our understanding of the function's behavior and graph. The factor (5x + 2) requires a bit more algebraic manipulation compared to the previous two factors, but the underlying principle remains the same: we set the factor to zero and solve for x. This methodical approach ensures accuracy and reinforces the importance of algebraic techniques in finding zeros. With the identification of x = -2/5 as a zero, we have now found all three zeros of the function f(x) = (x+1)(x-8)(5x+2). These zeros are x = -1, x = 8, and x = -2/5. These values are critical for understanding the function's behavior, sketching its graph, and solving related equations and inequalities. In the concluding section, we will summarize our findings and discuss the significance of these zeros in the broader context of polynomial functions and their applications. Each zero contributes to the unique shape and characteristics of the function, and their accurate identification is essential for mathematical analysis and real-world problem-solving. The process of finding these zeros has not only provided us with the solutions but also enhanced our understanding of the underlying principles of polynomial functions.

Conclusion: Identifying the Zeros

In conclusion, the zeros of the function f(x) = (x+1)(x-8)(5x+2) are x = -1, x = -2/5, and x = 8. Therefore, the correct answer is D. -1, -2/5, 8. These zeros are crucial points on the graph of the function, representing the x-intercepts where the graph crosses the x-axis. Understanding and identifying zeros is a fundamental skill in algebra and calculus, with applications in various fields such as physics, engineering, and economics. Zeros provide critical information about the behavior of a function, including its intervals of positivity and negativity, its turning points, and its overall shape. The process of finding zeros often involves setting each factor of a polynomial function to zero and solving for x, as we demonstrated in this article. This approach leverages the zero-product property, which is a cornerstone of solving polynomial equations. The ability to accurately identify zeros is not only essential for sketching graphs but also for solving equations, inequalities, and optimization problems. In the broader context of mathematics, zeros play a vital role in understanding the relationships between algebraic expressions and their graphical representations. They serve as anchor points that help in visualizing the function's behavior and predicting its values at different points. Moreover, zeros are often used in real-world applications to model phenomena and solve practical problems. For example, in physics, zeros can represent the points where a projectile hits the ground. In engineering, they can indicate the points of equilibrium in a system. In economics, they might represent the break-even points in a business model. Therefore, mastering the technique of finding zeros is not only a key skill in mathematics but also a valuable tool in many other disciplines. This article has provided a step-by-step guide to identifying the zeros of a polynomial function, ensuring a clear and comprehensive understanding of the process. By following this approach, you can confidently tackle similar problems and gain a deeper appreciation of the role of zeros in mathematical analysis and applications.