Function With Zeros At -2, -1, And 4 A Step By Step Solution

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In the realm of mathematics, particularly when dealing with polynomial functions, identifying the function that possesses specific zeros is a fundamental skill. Zeros, also known as roots, are the x-values where the function intersects the x-axis, making the function's value equal to zero. This article delves into the process of determining the correct function given a set of zeros, providing a step-by-step approach and illustrative examples. We will explore how to construct a polynomial function from its zeros, understand the significance of multiplicity, and apply these concepts to solve a practical problem. Whether you are a student grappling with polynomial functions or a math enthusiast seeking to deepen your understanding, this guide will equip you with the knowledge and techniques to confidently tackle such problems.

Understanding Zeros of a Function

Before we dive into the process of finding a function with specific zeros, it's crucial to have a solid understanding of what zeros are and how they relate to the function's equation. In mathematical terms, a zero of a function f(x) is a value x for which f(x) = 0. Graphically, these zeros represent the points where the function's graph intersects the x-axis. The zeros of a function provide valuable information about its behavior and shape.

The Connection Between Zeros and Factors

The cornerstone of finding a function with specific zeros lies in the relationship between zeros and factors. If x = a is a zero of a function, then (x - a) is a factor of that function. This principle stems from the Factor Theorem, which states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. Understanding this connection is paramount to constructing a function from its given zeros.

For instance, if we know that a function has a zero at x = 2, then we can immediately deduce that (x - 2) is a factor of the function. Similarly, if x = -3 is a zero, then (x + 3) is a factor. This simple yet powerful relationship forms the basis for building polynomial functions with predetermined zeros.

The Role of Multiplicity

In some cases, a zero may appear more than once. This is known as the multiplicity of the zero. The multiplicity of a zero affects the behavior of the graph at that point. If a zero has a multiplicity of 1, the graph crosses the x-axis at that point. If the multiplicity is an even number (e.g., 2, 4, 6), the graph touches the x-axis but does not cross it, creating a turning point. If the multiplicity is an odd number greater than 1 (e.g., 3, 5, 7), the graph flattens out as it crosses the x-axis.

For example, consider the function f(x) = (x - 1)². This function has a zero at x = 1 with a multiplicity of 2. The graph of this function will touch the x-axis at x = 1 but will not cross it, forming a parabola that opens upwards and touches the x-axis at its vertex.

Understanding multiplicity is crucial for accurately constructing a function from its zeros and for interpreting the behavior of its graph. When given a set of zeros, it's important to note whether any zeros have a multiplicity greater than 1, as this will impact the form of the function.

Constructing a Function from Zeros

Now that we have a firm grasp of the relationship between zeros and factors, and the concept of multiplicity, let's delve into the process of constructing a function from its zeros. This involves reversing the process of finding zeros from a function, allowing us to build a function with specific characteristics.

Step-by-Step Approach

Here's a step-by-step approach to constructing a function from its zeros:

  1. Identify the Zeros: Begin by clearly identifying all the zeros of the function. This is the starting point for the entire process. For example, if you are given that the zeros are -2, -1, and 4, write them down explicitly.
  2. Determine the Factors: For each zero, create a corresponding factor. Remember, if x = a is a zero, then (x - a) is a factor. So, if the zeros are -2, -1, and 4, the corresponding factors are (x + 2), (x + 1), and (x - 4).
  3. Account for Multiplicity: If any zeros have a multiplicity greater than 1, raise the corresponding factor to the power of its multiplicity. For instance, if the zero 4 has a multiplicity of 2, the factor (x - 4) would be squared, resulting in (x - 4)².
  4. Multiply the Factors: Multiply all the factors together to obtain the function. This will give you a polynomial function that has the specified zeros with their respective multiplicities. For example, if the factors are (x + 2), (x + 1), and (x - 4), the function would be f(x) = (x + 2)(x + 1)(x - 4).
  5. Consider the Leading Coefficient: The leading coefficient can affect the vertical stretch or compression of the graph. If no specific leading coefficient is given, you can assume it to be 1. However, if additional information is provided, such as a point the function passes through, you can solve for the leading coefficient by substituting the point's coordinates into the function and solving for the coefficient.

Illustrative Examples

Let's solidify our understanding with a couple of examples:

Example 1: Construct a function with zeros at x = 1, x = -2, and x = 3, each with a multiplicity of 1.

  • Zeros: 1, -2, 3
  • Factors: (x - 1), (x + 2), (x - 3)
  • Function: f(x) = (x - 1)(x + 2)(x - 3)

Example 2: Construct a function with a zero at x = -1 with a multiplicity of 2 and a zero at x = 4 with a multiplicity of 1.

  • Zeros: -1 (multiplicity 2), 4
  • Factors: (x + 1)², (x - 4)
  • Function: f(x) = (x + 1)²(x - 4)

By following these steps, you can confidently construct a polynomial function from any given set of zeros and multiplicities. This skill is essential for solving a wide range of mathematical problems and for understanding the behavior of polynomial functions.

Solving the Problem: Identifying the Correct Function

Now, let's apply our knowledge to solve the problem presented. We are tasked with identifying the function that has zeros at -2, -1, and 4. To do this, we will use the principles we've discussed to construct the function and then compare it to the given options.

Step 1: Determine the Factors

Given the zeros -2, -1, and 4, we can determine the corresponding factors:

  • Zero: -2, Factor: (x + 2)
  • Zero: -1, Factor: (x + 1)
  • Zero: 4, Factor: (x - 4)

Step 2: Construct the Function

Multiplying these factors together, we get the function:

f(x) = (x + 2)(x + 1)(x - 4)

Step 3: Compare with the Given Options

Now, let's compare our constructed function with the options provided:

  • f(x) = (x - 2)(x - 1)(x + 4)²
  • f(x) = (x + 2)(x + 1)(x - 4)
  • f(x) = (x + 2)²(x + 1)(x - 4)
  • f(x) = x(x - 2)(x - 1)(x - 4)

By direct comparison, we can see that the second option, f(x) = (x + 2)(x + 1)(x - 4), matches the function we constructed. Therefore, this is the correct answer.

Why the Other Options are Incorrect

It's also beneficial to understand why the other options are incorrect. This reinforces our understanding of the relationship between zeros and factors.

  • f(x) = (x - 2)(x - 1)(x + 4)²: This function has zeros at 2, 1, and -4, which does not match the required zeros.
  • f(x) = (x + 2)²(x + 1)(x - 4): This function has a zero at -2 with a multiplicity of 2, which means the graph would touch the x-axis at x = -2 but not cross it. This is different from having a simple zero at -2.
  • f(x) = x(x - 2)(x - 1)(x - 4): This function has zeros at 0, 2, 1, and 4, which includes zeros not specified in the problem.

By analyzing why the incorrect options don't fit the criteria, we gain a deeper appreciation for the importance of matching factors with their corresponding zeros.

Conclusion: Mastering Zeros and Factors

In this article, we have explored the crucial concept of zeros of a function and their relationship to factors. We've learned how to construct a polynomial function from its zeros, taking into account the multiplicity of each zero. By understanding the Factor Theorem and the effect of multiplicity on the graph of a function, we can confidently identify the function that corresponds to a given set of zeros.

The step-by-step approach we've outlined provides a systematic way to tackle such problems. By identifying the zeros, determining the factors, accounting for multiplicity, and multiplying the factors, we can construct the desired function. Comparing the constructed function with the given options allows us to pinpoint the correct answer.

This skill is not only essential for solving mathematical problems but also for understanding the behavior of polynomial functions and their graphical representations. By mastering the concepts of zeros and factors, you'll be well-equipped to tackle more advanced topics in algebra and calculus. Remember to practice these techniques with various examples to solidify your understanding and build your confidence in working with polynomial functions.

Whether you're a student preparing for an exam or simply a math enthusiast eager to expand your knowledge, the principles discussed in this guide will serve as a valuable tool in your mathematical journey. Keep exploring, keep practicing, and keep building your understanding of the fascinating world of mathematics.