Simplifying (x+1)/(x^2-4) + (2x-3)/(x^2+x-2) A Step-by-Step Guide

Introduction to Simplifying Rational Expressions

In mathematics, particularly in algebra, rational expressions are fundamental. These expressions, which are essentially fractions with polynomials in the numerator and denominator, appear frequently in various mathematical contexts, including calculus, equation solving, and real-world applications. This article delves into the simplification of a specific rational expression: ${\frac{x+1}{x^2-4} + \frac{2x-3}{x^2+x-2}}$. We will break down each step involved in the simplification process, ensuring a comprehensive understanding for readers of all levels. Mastering the simplification of rational expressions is crucial for anyone looking to excel in algebra and related fields. Before we dive into the specifics, it's important to grasp the underlying principles that govern these expressions. A rational expression is considered simplified when its numerator and denominator have no common factors other than 1. This often involves factoring polynomials, identifying common factors, and then canceling them out. The process mirrors the simplification of numerical fractions, where we reduce the fraction to its lowest terms. In our case, we're dealing with algebraic fractions, but the core concept remains the same. We aim to present the expression in its most concise and manageable form. This not only makes the expression easier to work with but also reveals its underlying structure and properties. The journey to simplifying rational expressions can be both challenging and rewarding, and this article serves as your guide through the process.

Understanding the Components of the Expression

To effectively simplify the given rational expression, ${\frac{x+1}{x^2-4} + \frac{2x-3}{x^2+x-2}}$, we must first meticulously examine each component. This involves breaking down the expression into its individual fractions and analyzing their numerators and denominators. The first fraction, ${\frac{x+1}{x^2-4}}$, has a numerator of ${x+1}$, which is a simple linear expression. The denominator, ${x^2-4}$, is a quadratic expression that can be recognized as a difference of squares. This is a crucial observation because the difference of squares is a standard factoring pattern that we can exploit. Specifically, ${x^2-4}$ can be factored into ${(x-2)(x+2)}$. Recognizing these patterns is a cornerstone of algebraic manipulation. The second fraction, ${\frac{2x-3}{x^2+x-2}}$, presents a slightly different challenge. The numerator, ${2x-3}$, is another linear expression. However, the denominator, ${x^2+x-2}$, is a quadratic expression that requires a bit more effort to factor. We need to find two numbers that multiply to -2 and add up to 1 (the coefficient of the ${x}$ term). These numbers are 2 and -1, which allows us to factor the denominator into ${(x+2)(x-1)}$. Understanding how to factor quadratic expressions is essential for simplifying rational expressions. It allows us to identify common factors between the numerator and denominator, which is the key to simplification. By carefully examining the components of the expression, we lay the groundwork for the subsequent steps in the simplification process. This initial analysis provides us with the necessary insights to apply factoring techniques and ultimately simplify the expression.

Factoring the Denominators: A Step-by-Step Approach

The core of simplifying rational expressions lies in the ability to factor polynomials, particularly the denominators. In our expression, ${\frac{x+1}{x^2-4} + \frac{2x-3}{x^2+x-2}}$, we have two denominators to factor: ${x^2-4}$ and ${x^2+x-2}$. Let's start with the first denominator, ${x^2-4}$. As mentioned earlier, this expression is a classic example of a difference of squares. The difference of squares pattern states that ${a^2 - b^2}$ can be factored into ${(a-b)(a+b)}$. Applying this pattern to ${x^2-4}$, we can see that ${a=x}$ and ${b=2}$, since ${4=2^2}$. Therefore, ${x^2-4}$ factors into ${(x-2)(x+2)}$. This factorization is crucial because it reveals the factors that may potentially cancel out with terms in the numerator or other denominators. Next, we turn our attention to the second denominator, ${x^2+x-2}$. This is a quadratic trinomial, and we need to find two numbers that multiply to the constant term (-2) and add up to the coefficient of the ${x}$ term (1). After some thought, we can identify these numbers as 2 and -1. This means we can rewrite the middle term ${x}$ as ${2x - x}$. So, ${x^2+x-2}$ becomes ${x^2 + 2x - x - 2}$. Now, we can factor by grouping. We group the first two terms and the last two terms: ${(x^2 + 2x) + (-x - 2)}$. We factor out the greatest common factor (GCF) from each group. From the first group, we factor out an ${x}$, leaving us with ${x(x+2)}$. From the second group, we factor out a -1, leaving us with ${-1(x+2)}$. Now we have ${x(x+2) - 1(x+2)}$. Notice that both terms have a common factor of ${(x+2)}$. We factor out ${(x+2)}$, which gives us ${(x+2)(x-1)}$. Thus, ${x^2+x-2}$ factors into ${(x+2)(x-1)}$. By systematically factoring each denominator, we've unveiled the underlying structure of the rational expressions. This step is essential for identifying common factors and proceeding with the simplification process.

Rewriting the Expression with Factored Denominators

After successfully factoring the denominators, the next crucial step is to rewrite the original expression, ${\frac{x+1}{x^2-4} + \frac{2x-3}{x^2+x-2}}$, using these factored forms. This transformation sets the stage for identifying common denominators and combining the fractions. Recall that we factored ${x^2-4}$ into ${(x-2)(x+2)}$ and ${x^2+x-2}$ into ${(x+2)(x-1)}$. Substituting these factored forms into the original expression, we get: ${\frac{x+1}{(x-2)(x+2)} + \frac{2x-3}{(x+2)(x-1)}}$. This rewritten expression provides a clearer view of the structure of the fractions and highlights the common factors in the denominators. Specifically, we can see that both denominators share a factor of ${(x+2)}$. This common factor is key to finding a common denominator, which is necessary to add the two fractions. Rewriting the expression with factored denominators is not just a cosmetic change; it's a strategic move that simplifies the subsequent steps. It allows us to see the expression in a new light, making it easier to manipulate and simplify. The process of factoring and rewriting is a fundamental technique in algebra, and it's particularly useful when dealing with rational expressions. By mastering this technique, you can tackle more complex algebraic problems with confidence. This step is a bridge between factoring and combining fractions, and it's essential for achieving the final simplified form of the expression.

Finding the Least Common Denominator (LCD)

To combine the two fractions in our expression, ${\frac{x+1}{(x-2)(x+2)} + \frac{2x-3}{(x+2)(x-1)}}$, we need to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In other words, it's the least common multiple of the denominators. To find the LCD, we look at the factored forms of the denominators: ${(x-2)(x+2)}$ and ${(x+2)(x-1)}$. The LCD must include all the unique factors present in both denominators, raised to the highest power they appear in either denominator. In this case, the unique factors are ${(x-2)}$, ${(x+2)}$, and ${(x-1)}$. Each of these factors appears only once in either denominator, so the LCD is simply the product of these factors: ${(x-2)(x+2)(x-1)}$. Understanding how to find the LCD is crucial for adding or subtracting fractions, whether they are numerical or algebraic. The LCD ensures that we are working with equivalent fractions that have the same denominator, which allows us to combine the numerators. Finding the LCD is a systematic process that involves identifying all the unique factors in the denominators and including them in the LCD. This process may seem complex at first, but with practice, it becomes a routine part of simplifying rational expressions. The LCD is the foundation upon which we build the combined fraction, and it's essential for achieving the final simplified form. By correctly identifying the LCD, we can proceed with confidence, knowing that we are on the right track.

Combining the Fractions Using the LCD

Now that we've determined the least common denominator (LCD) to be ${(x-2)(x+2)(x-1)}$, we can proceed to combine the fractions in our expression: ${\frac{x+1}{(x-2)(x+2)} + \frac{2x-3}{(x+2)(x-1)}}$. To do this, we need to rewrite each fraction with the LCD as its denominator. This involves multiplying the numerator and denominator of each fraction by the factors that are missing from its original denominator. For the first fraction, ${\frac{x+1}{(x-2)(x+2)}}$, the denominator is missing the factor ${(x-1)}$. So, we multiply both the numerator and denominator by ${(x-1)}$: ${\frac{(x+1)(x-1)}{(x-2)(x+2)(x-1)}}$. For the second fraction, ${\frac{2x-3}{(x+2)(x-1)}}$, the denominator is missing the factor ${(x-2)}$. So, we multiply both the numerator and denominator by ${(x-2)}$: ${\frac{(2x-3)(x-2)}{(x-2)(x+2)(x-1)}}$. Now that both fractions have the same denominator, we can add them by adding their numerators: ${\frac{(x+1)(x-1) + (2x-3)(x-2)}{(x-2)(x+2)(x-1)}}$. Next, we need to expand the products in the numerator. ${(x+1)(x-1)}$ is a difference of squares, which expands to ${x^2 - 1}$. ${(2x-3)(x-2)}$ expands to ${2x^2 - 4x - 3x + 6}$, which simplifies to ${2x^2 - 7x + 6}$. Substituting these expansions into the numerator, we get: ${\frac{x^2 - 1 + 2x^2 - 7x + 6}{(x-2)(x+2)(x-1)}}$. Now, we combine like terms in the numerator: ${x^2 + 2x^2 = 3x^2}$, ${-7x}$ remains as is, and ${-1 + 6 = 5}$. So, the numerator simplifies to ${3x^2 - 7x + 5}$. Our expression now looks like this: ${\frac{3x^2 - 7x + 5}{(x-2)(x+2)(x-1)}}$. Combining the fractions using the LCD is a multi-step process that involves rewriting the fractions with a common denominator, expanding the products in the numerator, and then combining like terms. This process is a cornerstone of algebraic manipulation, and it's essential for simplifying rational expressions. By mastering this technique, you can confidently tackle more complex algebraic problems.

Simplifying the Numerator: Combining Like Terms and Factoring

After combining the fractions, we arrived at the expression ${\frac{3x^2 - 7x + 5}{(x-2)(x+2)(x-1)}}$. The next step in simplifying this rational expression is to focus on the numerator, ${3x^2 - 7x + 5}$. We need to determine if the numerator can be factored further. Factoring the numerator, if possible, could reveal common factors with the denominator, allowing us to simplify the expression further. However, in this case, the quadratic expression ${3x^2 - 7x + 5}$ does not factor neatly using integer coefficients. We can verify this by attempting to find two numbers that multiply to ${3 imes 5 = 15}$ and add up to -7. There are no such integer pairs. When a quadratic expression does not factor easily, we can use the quadratic formula to find its roots. The quadratic formula is given by: ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$, where ${a}$, ${b}$, and ${c}$ are the coefficients of the quadratic expression ${ax^2 + bx + c}$. In our case, ${a = 3}$, ${b = -7}$, and ${c = 5}$. Plugging these values into the quadratic formula, we get: ${x = \frac{7 \pm \sqrt{(-7)^2 - 4(3)(5)}}{2(3)}}$

${x = \frac{7 \pm \sqrt{49 - 60}}{6}}$

${x = \frac{7 \pm \sqrt{-11}}{6}}$. Since the discriminant (the value inside the square root) is negative, the roots are complex numbers. This means that the quadratic expression ${3x^2 - 7x + 5}$ cannot be factored using real numbers. Since the numerator does not factor, we cannot simplify the expression further by canceling out common factors with the denominator. Therefore, the numerator remains as ${3x^2 - 7x + 5}$. Simplifying the numerator is a crucial step in the process of simplifying rational expressions. It involves combining like terms and attempting to factor the resulting expression. If factoring is possible, it can lead to further simplification by canceling out common factors with the denominator. However, if the numerator does not factor, it remains as is, and we move on to the next step in the process.

Checking for Common Factors and Final Simplification

After simplifying the numerator to ${3x^2 - 7x + 5}$, we have the expression ${\frac{3x^2 - 7x + 5}{(x-2)(x+2)(x-1)}}$. The next critical step is to check for any common factors between the numerator and the denominator. If there are common factors, we can cancel them out to further simplify the expression. In this case, we have already established that the numerator, ${3x^2 - 7x + 5}$, does not factor using real numbers. This means it is unlikely to have any common factors with the linear factors in the denominator, which are ${(x-2)}$, ${(x+2)}$, and ${(x-1)}$. To confirm this, we can check if any of the roots of the denominator are also roots of the numerator. The roots of the denominator are the values of ${x}$ that make the denominator equal to zero. These are ${x = 2}$, ${x = -2}$, and ${x = 1}$. We can substitute each of these values into the numerator and see if the result is zero. If it is, then that value is a root of the numerator, and there is a common factor. Substituting ${x = 2}$ into the numerator, we get: ${3(2)^2 - 7(2) + 5 = 12 - 14 + 5 = 3}$, which is not zero. Substituting ${x = -2}$ into the numerator, we get: ${3(-2)^2 - 7(-2) + 5 = 12 + 14 + 5 = 31}$, which is not zero. Substituting ${x = 1}$ into the numerator, we get: ${3(1)^2 - 7(1) + 5 = 3 - 7 + 5 = 1}$, which is not zero. Since none of the roots of the denominator are also roots of the numerator, there are no common factors between the numerator and the denominator. This means that the expression is already in its simplest form. Therefore, the simplified form of the given rational expression is: ${\frac{3x^2 - 7x + 5}{(x-2)(x+2)(x-1)}}$. Checking for common factors is a crucial step in the final stages of simplifying rational expressions. It ensures that the expression is indeed in its simplest form and that no further simplification is possible. By systematically checking for common factors, we can confidently present the final simplified expression.

Final Simplified Expression and Conclusion

After a thorough process of factoring, finding the least common denominator, combining fractions, and checking for common factors, we have arrived at the final simplified expression: ${\frac{3x^2 - 7x + 5}{(x-2)(x+2)(x-1)}}$. This expression represents the simplified form of the original expression, ${\frac{x+1}{x^2-4} + \frac{2x-3}{x^2+x-2}}$. In conclusion, simplifying rational expressions involves a series of steps that require a solid understanding of algebraic principles. These steps include factoring polynomials, identifying the least common denominator, combining fractions, simplifying the numerator, and checking for common factors. Each step is crucial in achieving the final simplified form. The process may seem complex at first, but with practice and a systematic approach, it becomes a manageable and rewarding task. Mastering the simplification of rational expressions is a valuable skill in algebra and calculus, and it lays the foundation for more advanced mathematical concepts. By understanding the underlying principles and following the steps outlined in this article, you can confidently simplify rational expressions and tackle more complex algebraic problems. The ability to simplify rational expressions not only enhances your mathematical skills but also provides a deeper understanding of the structure and properties of algebraic expressions. This knowledge is essential for success in various mathematical fields and real-world applications.