Finding Zeros Of Rational Functions A Step By Step Guide

$F(x)=\frac{(x+3)(x-1)}{(x-2)(x+2)}$

A. -2 B. 1 C. -5 D. -3 E. 2 F. 3

In mathematics, determining the zeros of a function is a fundamental concept, especially when dealing with rational functions. Rational functions, which are expressed as the ratio of two polynomials, present unique characteristics when finding their zeros. This article aims to provide a comprehensive understanding of how to identify the zeros of a given rational function, specifically focusing on the function $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$. We will explore the underlying principles, step-by-step methods, and common pitfalls to avoid, ensuring a clear grasp of this essential mathematical skill. By the end of this guide, you will be well-equipped to tackle similar problems and understand the broader implications of zeros in the context of functions.

Identifying Zeros of a Rational Function

To effectively identify zeros of a function, we must first understand what a zero actually represents. A zero of a function, also known as a root or x-intercept, is a value of $x$ for which the function $F(x)$ equals zero. In the context of a rational function, this occurs when the numerator of the fraction equals zero, while the denominator does not. The zeros of a function provide crucial information about the behavior of the function, such as where it intersects the x-axis and where the function's value changes sign. Specifically, for the given function $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$, we are interested in finding the values of $x$ that make the function equal to zero. These values can then be cross-referenced with the provided options (A through F) to select the correct answers. Understanding this foundational concept is paramount for correctly solving the problem and for applying this knowledge to more complex mathematical scenarios. The process involves setting the numerator equal to zero and solving for $x$, a technique that will be detailed in the subsequent sections.

Step-by-Step Method to Find Zeros

Finding the zeros of a function such as $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$ involves a systematic approach. Here’s a detailed, step-by-step method to guide you through the process:

1. Set the Function Equal to Zero: The first step is to set the entire function $F(x)$ equal to zero. This is because we are looking for the values of $x$ that make the function's output zero. Mathematically, this is represented as:

   $\frac{(x+3)(x-1)}{(x-2)(x+2)} = 0$

2. Focus on the Numerator: A rational function equals zero only when its numerator equals zero. The denominator cannot be zero because division by zero is undefined. Therefore, we focus solely on the numerator:

   $(x+3)(x-1) = 0$

3. Solve for x: To find the zeros of a function, we set each factor in the numerator equal to zero and solve for $x$. This is based on the principle that if the product of factors is zero, then at least one of the factors must be zero. Thus, we have two equations:

   • $x + 3 = 0$
   • $x - 1 = 0$

4. Isolate x: Solve each equation for $x$:

   • For $x + 3 = 0$, subtract 3 from both sides to get $x = -3$.
   • For $x - 1 = 0$, add 1 to both sides to get $x = 1$.

5. Verify Solutions: It is crucial to verify that these values do not make the denominator zero. If a value makes the denominator zero, it is not a zero of the function but rather a point of discontinuity (a vertical asymptote). In this case, the denominator is $(x-2)(x+2)$.

   • For $x = -3$, the denominator is $(-3-2)(-3+2) = (-5)(-1) = 5$, which is not zero.
   • For $x = 1$, the denominator is $(1-2)(1+2) = (-1)(3) = -3$, which is also not zero.

6. Identify the Zeros: The zeros of the function are the values of $x$ that satisfy the equation and do not make the denominator zero. In this case, the zeros are $x = -3$ and $x = 1$.

By following these steps, you can systematically find the zeros of any rational function. This process ensures that you not only identify the values that make the function zero but also verify that these values are valid within the function's domain. The next section will apply this method to the specific problem at hand, checking the provided options to confirm the correct answers.

Applying the Method to the Given Function

Now, let’s apply the step-by-step method described above to the given function, $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$, and determine the zeros of a function by checking the provided options:

1. Function Setup: The function is already set up as a rational function: $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$.

2. Numerator Focus: As established, we focus on the numerator to find the zeros:

   $(x+3)(x-1) = 0$

3. Solve for x: We set each factor in the numerator equal to zero:

   • $x + 3 = 0$
   • $x - 1 = 0$

4. Isolate x: Solving these equations gives us:

   • $x = -3$
   • $x = 1$

5. Verify Solutions: We need to ensure that these values do not make the denominator zero. The denominator is $(x-2)(x+2)$.

   • For $x = -3$, the denominator is $(-3-2)(-3+2) = (-5)(-1) = 5$, which is not zero.
   • For $x = 1$, the denominator is $(1-2)(1+2) = (-1)(3) = -3$, which is also not zero.

6. Identify the Zeros: The zeros of the function are $x = -3$ and $x = 1$.

7. Check the Options: Now, let's compare our results with the provided options:

   • A. -2: This is not a zero because $(-2+3)(-2-1) = (1)(-3) = -3 \neq 0$.
   • B. 1: This is a zero, as we found $x = 1$.
   • C. -5: This is not a zero because $(-5+3)(-5-1) = (-2)(-6) = 12 \neq 0$.
   • D. -3: This is a zero, as we found $x = -3$.
   • E. 2: This value makes the denominator zero, so it is not a zero but a vertical asymptote.
   • F. 3: This is not a zero because $(3+3)(3-1) = (6)(2) = 12 \neq 0$.

Therefore, the correct options are B. 1 and D. -3. This step-by-step application not only solves the problem but also reinforces the methodology, ensuring a solid understanding of how to find the zeros of a function.

Common Mistakes and How to Avoid Them

When finding the zeros of a function, particularly rational functions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls and knowing how to avoid them is crucial for accuracy. Here are some frequent errors and strategies to prevent them:

1. Forgetting to Check the Denominator: One of the most common mistakes is finding the values that make the numerator zero but failing to check if these values also make the denominator zero. If a value makes both the numerator and the denominator zero, it is not a zero of the function but may indicate a hole (a removable discontinuity). Zeros are only valid if they do not make the denominator zero. To avoid this, always substitute the potential zeros into the denominator and verify that the result is not zero.

2. Incorrectly Factoring: Errors in factoring the numerator or denominator can lead to incorrect zeros. Ensure you factor correctly using methods such as the quadratic formula, completing the square, or recognizing common factor patterns. Double-check your factoring by expanding the factored expression to see if it matches the original polynomial.

3. Algebraic Errors: Simple algebraic mistakes, such as sign errors or incorrect distribution, can derail the entire process. Take your time and carefully review each step. Use tools like calculators or online solvers to verify intermediate steps if necessary.

4. Ignoring Extraneous Solutions: In some cases, especially when dealing with more complex equations, solving the numerator may yield extraneous solutions that do not satisfy the original equation. Always plug the potential solutions back into the original function to ensure they are valid.

5. Misunderstanding the Concept of Zeros: A fundamental misunderstanding of what a zero represents can lead to confusion. Remember, a zero is a value of $x$ that makes the function $F(x)$ equal to zero. It is where the graph of the function intersects the x-axis. Keeping this concept clear will help you approach problems more logically.

By being mindful of these common mistakes and consistently applying the strategies to avoid them, you can improve your accuracy and confidence in finding the zeros of a function. The next section will summarize the key takeaways from this guide, reinforcing the essential steps and concepts.

Conclusion: Key Takeaways

In conclusion, finding the zeros of a function, especially rational functions, is a critical skill in mathematics. This guide has provided a comprehensive approach to identifying these zeros, focusing on the function $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$ as an example. The key takeaways include:

• Understanding the Definition: A zero of a function is a value of $x$ that makes the function equal to zero. For rational functions, this occurs when the numerator is zero, and the denominator is not.
• Step-by-Step Method:
  1. Set the function equal to zero.
  2. Focus on the numerator.
  3. Solve for $x$ by setting each factor in the numerator equal to zero.
  4. Verify that the solutions do not make the denominator zero.
  5. Identify the zeros of the function.
• Application to the Given Function: Applying this method to $F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}$ reveals that the zeros are $x = -3$ and $x = 1$.
• Common Mistakes to Avoid:
  • Forgetting to check the denominator.
  • Incorrect factoring.
  • Algebraic errors.
  • Ignoring extraneous solutions.
  • Misunderstanding the concept of zeros.

By mastering these concepts and techniques, you can confidently tackle problems involving the zeros of a function. Remember to practice regularly and apply these methods to a variety of functions to solidify your understanding. The ability to find zeros is not only essential for solving mathematical problems but also for understanding the behavior and properties of functions in various fields of study.