Partial Fraction Decomposition Of (2 + X^2) / ((x + 1)(x^2 + 4))

Partial fraction decomposition is a crucial technique in algebra and calculus, allowing us to break down complex rational expressions into simpler fractions. This method is invaluable for integrating rational functions, solving differential equations, and simplifying algebraic expressions. In this comprehensive guide, we will delve into the partial fraction decomposition of the rational expression (2 + x^2) / ((x + 1)(x^2 + 4)), exploring the underlying principles and step-by-step procedures. Our focus will be on identifying the correct form of the decomposition, a foundational step in the broader process.

When dealing with rational functions, which are essentially fractions where the numerator and denominator are polynomials, integration can often be a daunting task. Partial fraction decomposition comes to the rescue by transforming these complex fractions into a sum of simpler fractions that are much easier to integrate. The technique is particularly useful when the denominator of the rational function can be factored into linear and irreducible quadratic factors. The core idea is to express the given rational function as a sum of fractions, each having one of these factors (or a power of these factors) as its denominator. This decomposition simplifies the integration process significantly, making it a cornerstone of calculus and related fields.

The key to successful partial fraction decomposition lies in correctly setting up the form of the decomposition. This depends entirely on the structure of the denominator of the original rational function. Specifically, we need to consider the types of factors present in the denominator—whether they are linear, repeated linear, irreducible quadratic, or repeated irreducible quadratic. Each type of factor necessitates a different approach in setting up the partial fractions. For instance, a linear factor like (x + 1) will give rise to a term of the form A / (x + 1), where A is a constant. On the other hand, an irreducible quadratic factor like (x^2 + 4) will require a term of the form (Bx + C) / (x^2 + 4), where B and C are constants. Understanding these distinctions is crucial for choosing the appropriate form of the decomposition and subsequently finding the values of the unknown coefficients.

This guide will specifically address the rational expression (2 + x^2) / ((x + 1)(x^2 + 4)). We will walk through the process of identifying the correct form of its partial fraction decomposition, explaining the reasoning behind each step. This will involve recognizing the factors in the denominator and applying the appropriate rules for setting up the partial fractions. By the end of this discussion, you will have a solid understanding of how to approach similar problems and confidently decompose rational expressions into simpler, manageable forms. This skill is not just valuable for academic pursuits in mathematics but also for practical applications in engineering, physics, and computer science, where rational functions frequently appear in various models and analyses.

The initial and crucial step in partial fraction decomposition is to meticulously analyze the denominator of the given rational expression. The form of the decomposition hinges entirely on the nature of the factors present in the denominator. In our case, the denominator is (x + 1)(x^2 + 4). Let's break down these factors to understand their implications for the decomposition.

The first factor, (x + 1), is a linear factor. A linear factor is a polynomial of degree one, meaning the highest power of the variable x is 1. Linear factors are the simplest type of factors we encounter in partial fraction decomposition, and they lead to straightforward terms in the decomposition. Specifically, for each distinct linear factor (ax + b) in the denominator, we include a term of the form A / (ax + b) in the partial fraction decomposition, where A is a constant that needs to be determined. In our case, the linear factor (x + 1) will contribute a term of the form A / (x + 1) to the decomposition.

The second factor, (x^2 + 4), is an irreducible quadratic factor. An irreducible quadratic factor is a polynomial of degree two (quadratic) that cannot be factored further into linear factors over the real numbers. This irreducibility is key because it dictates the form of the corresponding term in the partial fraction decomposition. Unlike linear factors, irreducible quadratic factors require a slightly more complex term in the decomposition. For each distinct irreducible quadratic factor (ax^2 + bx + c) in the denominator, we include a term of the form (Bx + C) / (ax^2 + bx + c) in the partial fraction decomposition, where B and C are constants that need to be determined. The presence of the 'Bx' term in the numerator is what distinguishes the term for an irreducible quadratic factor from that of a linear factor.

To further clarify why (x^2 + 4) is irreducible, consider attempting to factor it. We might try to express it as (x + a)(x + b) for some real numbers a and b. However, expanding this product gives x^2 + (a + b)x + ab. For this to be equal to x^2 + 4, we would need a + b = 0 and ab = 4. The condition a + b = 0 implies a = -b, and substituting this into ab = 4 gives -b^2 = 4, or b^2 = -4. Since there is no real number whose square is -4, we conclude that (x^2 + 4) cannot be factored over the real numbers and is indeed irreducible. This irreducibility is what necessitates the numerator to be of the form Bx + C in the partial fraction decomposition.

Understanding the distinction between linear and irreducible quadratic factors is paramount in setting up the correct form of the partial fraction decomposition. Misidentifying these factors can lead to an incorrect setup and, consequently, an inability to solve for the constants. In the context of our given expression, (2 + x^2) / ((x + 1)(x^2 + 4)), we now recognize that the denominator contains one linear factor (x + 1) and one irreducible quadratic factor (x^2 + 4). This understanding will guide us in constructing the appropriate form of the decomposition in the next section.

Based on our analysis of the denominator factors, we can now construct the correct form for the partial fraction decomposition of the expression (2 + x^2) / ((x + 1)(x^2 + 4)). We identified that the denominator has a linear factor, (x + 1), and an irreducible quadratic factor, (x^2 + 4). Each of these factors will contribute a specific term to the decomposition, and understanding the rules for each type of factor is crucial for setting up the correct form.

For the linear factor (x + 1), the corresponding term in the partial fraction decomposition will be of the form A / (x + 1), where A is a constant that we will need to determine. This is a standard rule for linear factors: each distinct linear factor in the denominator contributes a term with a constant in the numerator and the linear factor itself in the denominator. The constant A represents the weight or contribution of this particular fraction to the overall rational expression. It's important to note that if the linear factor were raised to a power, such as (x + 1)^2, we would need to include additional terms in the decomposition to account for the repeated factor. However, in our case, the linear factor appears only once, so we only need the single term A / (x + 1).

For the irreducible quadratic factor (x^2 + 4), the corresponding term in the partial fraction decomposition will be of the form (Bx + C) / (x^2 + 4), where B and C are constants that we will need to determine. This is the standard form for irreducible quadratic factors: the numerator is a linear expression (Bx + C), and the denominator is the irreducible quadratic factor itself. The linear expression in the numerator is necessary because when we combine the fractions back together, we need to be able to obtain a numerator that could potentially be of degree 2 (since the original numerator is of degree 2). The constants B and C allow for the flexibility to match the coefficients in the original numerator when the fractions are combined.

Combining the terms for the linear and irreducible quadratic factors, we arrive at the complete form of the partial fraction decomposition for (2 + x^2) / ((x + 1)(x^2 + 4)). The decomposition will look like this:

(2 + x^2) / ((x + 1)(x^2 + 4)) = A / (x + 1) + (Bx + C) / (x^2 + 4)

This equation represents the core of the partial fraction decomposition. It states that the original rational expression can be expressed as the sum of two simpler fractions: one with the linear factor in the denominator and a constant in the numerator, and the other with the irreducible quadratic factor in the denominator and a linear expression in the numerator. The next step in the process would be to solve for the constants A, B, and C. This typically involves multiplying both sides of the equation by the original denominator, equating coefficients of like terms, and solving the resulting system of equations. However, for the purpose of this discussion, we are primarily focused on identifying the correct form of the decomposition, which we have now successfully achieved.

The importance of setting up the correct form cannot be overstated. It is the foundation upon which the entire decomposition rests. An incorrect form will lead to incorrect values for the constants and an ultimately incorrect decomposition. Therefore, a thorough understanding of the rules for linear and irreducible quadratic factors, as well as careful analysis of the denominator, is essential for mastering partial fraction decomposition.

Now that we have established the correct form for the partial fraction decomposition of (2 + x^2) / ((x + 1)(x^2 + 4)), we can evaluate the given options to determine which one matches our derived form. The correct form, as we determined, is:

(2 + x^2) / ((x + 1)(x^2 + 4)) = A / (x + 1) + (Bx + C) / (x^2 + 4)

Let's examine each option in light of this form:

Option A: A / (x + 1) + B / (x^2 + 4)

This option includes a term for the linear factor (x + 1) in the denominator, which is A / (x + 1), and a term for the irreducible quadratic factor (x^2 + 4). However, the term for the irreducible quadratic factor is simply B / (x^2 + 4), which is a constant divided by the quadratic. This is incorrect because, as we discussed, an irreducible quadratic factor requires a linear expression in the numerator, not just a constant. Therefore, Option A is not the correct form.

Option B: A / (x + 1) + (Bx + C) / (x^2 + 4)

This option perfectly matches the form we derived. It includes the term A / (x + 1) for the linear factor (x + 1) and the term (Bx + C) / (x^2 + 4) for the irreducible quadratic factor (x^2 + 4). The numerator (Bx + C) correctly represents a linear expression, which is necessary for the irreducible quadratic factor. Thus, Option B is the correct form for the partial fraction decomposition.

Option C: A / (x + 1) + B / (x + 2) + C / (x - 2)

This option attempts to decompose the rational expression into fractions with linear factors in the denominator. However, it incorrectly factors the term (x^2 + 4) into (x + 2) and (x - 2). As we established earlier, (x^2 + 4) is an irreducible quadratic factor and cannot be factored further over the real numbers. This means that the decomposition in Option C is fundamentally flawed because it misrepresents the structure of the denominator. Therefore, Option C is incorrect.

By carefully comparing each option to the correct form we derived, we can confidently identify Option B as the correct choice. This exercise highlights the importance of understanding the rules for setting up partial fraction decompositions based on the types of factors in the denominator. A thorough analysis of the denominator, coupled with a clear understanding of the required forms for linear and irreducible quadratic factors, is essential for correctly setting up the decomposition.

In conclusion, the partial fraction decomposition of the rational expression (2 + x^2) / ((x + 1)(x^2 + 4)) has the form given in Option B: A / (x + 1) + (Bx + C) / (x^2 + 4). This result underscores the importance of understanding the underlying principles of partial fraction decomposition and the critical role of accurately identifying the factors in the denominator.

We began by recognizing that the denominator (x + 1)(x^2 + 4) contains two distinct types of factors: a linear factor (x + 1) and an irreducible quadratic factor (x^2 + 4). This distinction is crucial because each type of factor necessitates a different approach in setting up the partial fractions. Linear factors, like (x + 1), contribute terms of the form A / (x + 1), where A is a constant. Irreducible quadratic factors, like (x^2 + 4), contribute terms of the form (Bx + C) / (x^2 + 4), where B and C are constants. The presence of the linear expression (Bx + C) in the numerator for the irreducible quadratic factor is a key characteristic that distinguishes it from the term for a linear factor.

By correctly identifying these factors and applying the appropriate rules, we were able to construct the correct form of the decomposition: A / (x + 1) + (Bx + C) / (x^2 + 4). This form ensures that when we combine the fractions back together, we have the flexibility to match the original numerator. The constants A, B, and C account for the different contributions of each fraction to the overall rational expression. Finding these constants is the next step in the process, but the correct setup is a prerequisite for successfully solving for them.

The incorrect options highlighted the pitfalls of misidentifying the factors or misapplying the rules. Option A, for example, used only a constant in the numerator for the irreducible quadratic factor, which is insufficient. Option C incorrectly factored the irreducible quadratic, leading to a fundamentally flawed decomposition. These errors demonstrate that a thorough understanding of the underlying principles is essential for avoiding common mistakes.

Mastering partial fraction decomposition is not just an academic exercise; it has practical applications in various fields, including calculus, differential equations, and engineering. The ability to break down complex rational expressions into simpler forms allows us to solve problems that would otherwise be intractable. Whether it's integrating rational functions, solving for the response of a system, or simplifying algebraic expressions, partial fraction decomposition is a valuable tool in the mathematician's and engineer's toolkit.

In conclusion, the correct form of the partial fraction decomposition is A / (x + 1) + (Bx + C) / (x^2 + 4). This result underscores the importance of understanding the nature of denominator factors and applying the appropriate rules for setting up the decomposition. By mastering these principles, you can confidently tackle a wide range of problems involving rational expressions and partial fraction decomposition.