Simplifying (x+2)^2 / (x^2+x-2) A Step-by-Step Guide
In the realm of algebra, simplifying rational expressions is a fundamental skill. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, by mastering a few key techniques, these expressions can be simplified, making them easier to understand and manipulate. This article delves into the process of simplifying a specific rational expression, providing a step-by-step guide and highlighting the underlying principles. We will explore the expression (x+2)^2 / (x^2+x-2), breaking down each step involved in its simplification. Understanding these techniques is crucial for success in algebra and calculus, as it allows for efficient problem-solving and a deeper grasp of mathematical concepts.
The heart of simplifying rational expressions lies in factorization. Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that multiply together to give the original polynomial. This is a critical step because it allows us to identify common factors in the numerator and denominator, which can then be canceled out. To effectively factor, one needs to be familiar with various factoring techniques, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions. In the expression (x+2)^2 / (x^2+x-2), the denominator, x^2 + x - 2, is a quadratic expression that can be factored into two binomials. The numerator, (x+2)^2, is already in a partially factored form, as it represents the square of the binomial (x+2). The ability to recognize these patterns and apply appropriate factoring techniques is paramount in simplifying rational expressions.
Once the polynomials in the numerator and denominator are factored, the next step is to identify and cancel out any common factors. A common factor is an expression that appears in both the numerator and the denominator. Canceling out common factors is essentially dividing both the numerator and the denominator by the same expression, which simplifies the overall rational expression without changing its value. In our example, after factoring the denominator, we will look for factors that match the (x+2) term in the numerator. This cancellation process is based on the fundamental principle of fractions, which states that dividing both the numerator and denominator by the same non-zero quantity does not change the fraction's value. Therefore, the act of canceling common factors is a legitimate and powerful technique in simplifying rational expressions. However, it is crucial to remember that only factors, not terms, can be canceled. A term is a part of an expression that is separated by a plus or minus sign, while a factor is a part of an expression that is multiplied. This distinction is essential to avoid common errors in simplification.
Finally, after canceling all common factors, we are left with the simplified form of the rational expression. This simplified form is equivalent to the original expression, but it is expressed in its most concise and manageable form. In the case of (x+2)^2 / (x^2+x-2), the simplification process will lead us to a new expression that is easier to work with in subsequent algebraic manipulations or calculus operations. The simplified expression will reveal the essential mathematical relationship represented by the original expression, without the clutter of unnecessary terms. This simplification process is not just about obtaining a shorter expression; it is about gaining a deeper understanding of the underlying mathematical structure. Furthermore, simplified expressions are less prone to errors in calculations and interpretations, making them invaluable in various mathematical applications. Therefore, mastering the art of simplifying rational expressions is a vital step towards mathematical proficiency.
Step-by-Step Simplification of (x+2)^2 / (x^2+x-2)
Let's apply these principles to the expression (x+2)^2 / (x^2+x-2). This section will provide a detailed walkthrough of the simplification process, illustrating each step with clarity and precision. By following this example, readers can gain a practical understanding of how to simplify rational expressions and develop their problem-solving skills. The goal is to transform the given expression into its simplest form, which will not only make it easier to work with but also reveal its underlying mathematical structure. This step-by-step approach will emphasize the importance of careful factoring, identifying common factors, and performing cancellations accurately. It will also highlight common pitfalls and how to avoid them, ensuring a solid grasp of the simplification process.
1. Factoring the Denominator
The first key step in simplifying the rational expression (x+2)^2 / (x^2+x-2) is to factor the denominator. The denominator, in this case, is the quadratic expression x^2 + x - 2. Factoring this quadratic involves finding two binomials that multiply together to give x^2 + x - 2. We need to identify two numbers that multiply to -2 (the constant term) and add up to 1 (the coefficient of the x term). These numbers are 2 and -1. Therefore, we can factor the denominator as follows:
x^2 + x - 2 = (x + 2)(x - 1)
This factorization is crucial because it reveals the factors that make up the denominator, which will be essential for identifying common factors with the numerator. The ability to accurately factor quadratic expressions is a cornerstone of simplifying rational expressions. Incorrect factorization will lead to incorrect simplification and a flawed understanding of the expression. Thus, it is vital to practice and master factoring techniques to ensure proficiency in this fundamental step. The factored form of the denominator, (x + 2)(x - 1), now allows us to see the potential for common factors with the numerator, which is our next focus.
2. Factoring the Numerator
Now, let's consider the numerator of our rational expression, (x+2)^2. While the numerator is already in a partially factored form, it's helpful to expand it to clearly see its factors. The expression (x+2)^2 means (x+2) multiplied by itself. Therefore, we can write it as:
(x+2)^2 = (x+2)(x+2)
This expansion makes it explicitly clear that the numerator has two factors of (x+2). This step might seem trivial, but it is essential for a clear understanding of the factors involved and for accurate cancellation in the next step. By writing the numerator in this expanded form, we can easily identify common factors with the factored denominator. It is important to remember that even when an expression appears to be already factored, expanding it can sometimes reveal hidden common factors or simplify the overall process. In this case, the expansion clarifies the presence of the (x+2) factor, which we know is also present in the factored denominator.
3. Identifying and Canceling Common Factors
With both the numerator and the denominator factored, we can now identify and cancel common factors in the rational expression. Our expression currently looks like this:
(x+2)(x+2) / (x+2)(x-1)
We can see that the factor (x+2) appears in both the numerator and the denominator. This means we can cancel one (x+2) factor from both the numerator and the denominator. This cancellation is based on the principle that dividing both the numerator and the denominator by the same non-zero quantity does not change the value of the fraction. After canceling the common factor, we are left with:
(x+2) / (x-1)
This step is the heart of the simplification process, as it reduces the complexity of the expression. It is crucial to perform this cancellation accurately, ensuring that only common factors are canceled and not terms. The ability to correctly identify and cancel common factors is a key skill in simplifying rational expressions. This step transforms the original expression into a more manageable form, which is now easier to understand and use in further calculations.
The Simplified Expression and Its Implications
After canceling the common factors, we arrive at the simplified expression: (x+2) / (x-1). This is the final simplified form of the original rational expression (x+2)^2 / (x^2+x-2). This simplified expression is mathematically equivalent to the original, but it is now expressed in its most concise form. This simplification process has several important implications, both in terms of mathematical understanding and practical application.
Understanding the Simplified Form
The simplified form, (x+2) / (x-1), provides a clearer representation of the relationship between the numerator and the denominator. By removing the common factor, we have eliminated a redundancy in the expression, revealing its essential structure. This simplified form is easier to analyze and interpret. For example, it is now more straightforward to identify the values of x for which the expression is undefined (i.e., when the denominator is zero). In this case, the expression is undefined when x = 1. The simplified form also makes it easier to visualize the behavior of the function represented by the expression, such as its asymptotes and intercepts. Furthermore, the simplified expression is less prone to errors in calculations and manipulations, making it a valuable tool in various mathematical contexts.
Practical Applications
The ability to simplify rational expressions has numerous practical applications in algebra, calculus, and other areas of mathematics. Simplified expressions are easier to work with in solving equations, graphing functions, and performing other mathematical operations. For instance, when solving an equation involving rational expressions, simplifying the expressions first can significantly reduce the complexity of the problem. In calculus, simplifying rational expressions is often a necessary step before performing operations such as differentiation and integration. Simplified expressions also play a crucial role in real-world applications, such as modeling physical phenomena, analyzing data, and solving engineering problems. In these contexts, using simplified expressions can lead to more accurate results and a deeper understanding of the underlying processes. Therefore, mastering the skill of simplifying rational expressions is not just an academic exercise; it is a practical tool that enhances problem-solving abilities in a wide range of fields.
Importance of Checking for Extraneous Solutions
While simplifying rational expressions is a powerful technique, it is essential to be aware of the potential for introducing extraneous solutions. An extraneous solution is a value that satisfies the simplified equation but does not satisfy the original equation. This can occur because the simplification process may eliminate restrictions on the variable that were present in the original expression. In the case of (x+2)^2 / (x^2+x-2), the original expression is undefined when the denominator, x^2 + x - 2, is equal to zero. This occurs when x = -2 and x = 1. Therefore, these values are not in the domain of the original expression. However, after simplifying to (x+2) / (x-1), the restriction x = -2 is no longer apparent. If we were to solve an equation involving this simplified expression, we might obtain x = -2 as a solution, which would be extraneous. To avoid this, it is crucial to always check the solutions obtained from the simplified expression in the original expression to ensure they are valid. This step is a critical part of the problem-solving process and ensures the accuracy of the final result.
In conclusion, the simplification of the rational expression (x+2)^2 / (x^2+x-2) to (x+2) / (x-1) demonstrates the power and importance of factoring and canceling common factors. This process not only makes the expression more manageable but also reveals its underlying structure and simplifies further mathematical manipulations. However, it is crucial to remember the potential for extraneous solutions and to always check the validity of solutions in the original expression. By mastering these techniques, one can gain a deeper understanding of rational expressions and enhance their problem-solving skills in algebra and beyond.
Answer
The correct answer is A) (x+2) / (x-1).
