Partial Fraction Decomposition Of (6x+7)/(x+2)^2 Explained

by Omar Yusuf 59 views

Hey guys! Let's dive into the world of partial fraction decomposition, a super useful technique in calculus and algebra. Today, we're tackling a specific problem: figuring out the correct form for the partial fraction decomposition of the expression (6x + 7) / (x + 2)^2. It might seem a bit intimidating at first, but trust me, we'll break it down step by step so you'll be a pro in no time.

What is Partial Fraction Decomposition?

So, what exactly is partial fraction decomposition? Think of it like reverse engineering the process of adding fractions. When you add fractions, you combine them into a single fraction. Partial fraction decomposition does the opposite: it takes a single, complex fraction and breaks it down into simpler fractions. These simpler fractions are called partial fractions, and they're much easier to work with, especially when you're dealing with integration or other calculus operations.

Partial fraction decomposition is especially handy when you've got rational functions – that is, fractions where both the numerator and denominator are polynomials. The goal is to express a complicated rational function as a sum of simpler fractions that have linear or irreducible quadratic denominators. This makes integration, finding inverse Laplace transforms, and other operations significantly easier. The key idea behind partial fraction decomposition is to rewrite a rational function, like our (6x+7)/(x+2)^2 example, as a sum of simpler fractions whose denominators are factors of the original denominator. This is incredibly useful in calculus for integration, as simpler fractions are much easier to integrate than complex ones. To effectively use partial fraction decomposition, it's essential to understand the different forms that the partial fractions can take, depending on the factors in the denominator of the original rational function. For instance, repeated linear factors, like in our case with (x+2)^2, require a specific approach to ensure the decomposition is correct. We need to account for each power of the repeated factor, which is why we'll see terms like A/(x+2) and B/(x+2)^2 in the correct decomposition. Understanding this nuance is crucial for setting up the partial fractions correctly and solving for the unknown constants. Failing to account for repeated factors will lead to an incorrect decomposition, which will then propagate errors through any subsequent calculations, such as integration. So, paying close attention to the structure of the denominator is the first and most critical step in partial fraction decomposition.

Our Problem: (6x + 7) / (x + 2)^2

Okay, let's focus on our specific problem. We have the rational function (6x + 7) / (x + 2)^2. Notice anything special about the denominator? That's right, it's (x + 2) squared. This means we have a repeated linear factor. Repeated linear factors in the denominator require a specific approach when we're setting up our partial fraction decomposition. When we have a repeated linear factor like (x + a)^n, we need to include terms for each power of (x + a) up to n. So, for (x + 2)^2, we'll need terms with denominators of (x + 2) and (x + 2)^2. This is a crucial point to remember because it dictates the structure of our decomposition. If we were to ignore the fact that (x + 2) is repeated, we'd miss a term and the decomposition wouldn't be correct. Think of it like this: each power of the repeated factor contributes a unique piece to the overall fraction, and we need to account for each of those pieces. Failing to do so would be like trying to assemble a puzzle with missing pieces – the final picture just wouldn't be complete. This careful consideration of repeated factors is what sets partial fraction decomposition apart from simple fraction addition and is key to its power as a problem-solving tool in calculus and beyond. So, let's keep this in mind as we move forward and set up the correct form for our decomposition.

The Correct Form: Option A is the Winner!

Given the options, the correct form for the partial fraction decomposition of (6x + 7) / (x + 2)^2 is:

A. A / (x + 2) + B / (x + 2)^2

Why is this the right answer? Because, as we discussed, we have a repeated linear factor (x + 2)^2 in the denominator. This means we need a term for each power of (x + 2) up to the power of 2. That's why we have both A / (x + 2) and B / (x + 2)^2. Let's break down why the other options are incorrect to solidify our understanding. Option B includes a term (Bx + C) / (x + 2)^2. While it might seem plausible at first glance, this is not the correct form for a repeated linear factor. The numerator should be a constant, not a linear expression, when dealing with repeated linear factors. Option C introduces a term with 'x' in the denominator (A/x). This is completely unnecessary because our original denominator doesn't have 'x' as a factor by itself. Remember, we're trying to decompose the fraction based on the factors present in the original denominator. This highlights an important principle in partial fraction decomposition: the partial fractions should reflect the factors of the original denominator. Introducing extraneous factors, like 'x' in this case, would lead to an incorrect decomposition. The correct form ensures that we account for each factor and its multiplicity, leading to a valid representation of the original rational function. Option A correctly captures this by including terms for each power of the repeated linear factor (x + 2), making it the ideal choice for partial fraction decomposition in this scenario. Understanding these nuances is key to mastering this technique and applying it successfully in various mathematical contexts.

Why Other Options Are Incorrect

Let's quickly discuss why the other options aren't the right fit. This will help solidify your understanding of partial fraction decomposition.

  • B. A / (x + 2) + (Bx + C) / (x + 2)^2: This option is incorrect because the numerator of the fraction with the repeated factor squared should be a constant, not a linear expression like (Bx + C).
  • C. A / x + B / (x + 2) + C / (x + 2)^2: This one's wrong because we don't have a simple 'x' factor in the original denominator. We only have (x + 2) as a factor, and it's repeated.

Solving for A and B

Now that we know the correct form, how would we actually find the values of A and B? Here's the general approach:

  1. Multiply both sides of the equation [ (6x + 7) / (x + 2)^2 = A / (x + 2) + B / (x + 2)^2 ] by the original denominator, which is (x + 2)^2. This will clear all the fractions.
  2. Simplify and expand. After multiplying, you'll get: 6x + 7 = A(x + 2) + B. Expand the right side to get: 6x + 7 = Ax + 2A + B.
  3. Equate coefficients. This is a crucial step. We'll equate the coefficients of the terms with the same powers of x on both sides of the equation. This means we'll compare the coefficients of the 'x' terms and the constant terms. So, we get two equations: A = 6 (from the 'x' terms) and 2A + B = 7 (from the constant terms). This method, known as equating coefficients, is a powerful technique for solving systems of equations that arise in partial fraction decomposition. It allows us to systematically determine the unknown constants by leveraging the polynomial equality. By comparing the coefficients of corresponding terms, we can create a set of linear equations that can be solved using standard algebraic methods. This approach is particularly useful when dealing with higher-degree polynomials or more complex decompositions, as it provides a structured way to break down the problem into manageable steps. Understanding and mastering the technique of equating coefficients is essential for successful partial fraction decomposition and its applications in calculus and engineering.
  4. Solve the system of equations. We now have a system of linear equations. From the first equation, we know A = 6. Substitute this into the second equation: 2(6) + B = 7, which simplifies to 12 + B = 7. Solving for B, we get B = -5. And there you have it! We've found the values of A and B, completing our partial fraction decomposition. This step-by-step approach to solving for the unknown constants is a cornerstone of partial fraction decomposition. By carefully multiplying through by the common denominator, expanding the resulting expression, and equating coefficients, we can systematically determine the values that make the decomposition valid. This process not only allows us to express the original rational function as a sum of simpler fractions but also reinforces our understanding of polynomial algebra and equation solving. With practice, this method becomes second nature, enabling us to tackle even more complex partial fraction decomposition problems with confidence.
  5. Write the final decomposition. Substitute the values of A and B back into our form: (6x + 7) / (x + 2)^2 = 6 / (x + 2) - 5 / (x + 2)^2.

Key Takeaways

  • Repeated linear factors in the denominator require a term for each power of the factor.
  • The correct form of the partial fraction decomposition is crucial for solving the problem correctly.
  • Equating coefficients is a powerful technique for finding the unknown constants.

Wrapping Up

So, there you have it! We've successfully identified the correct form for the partial fraction decomposition of (6x + 7) / (x + 2)^2 and even walked through how to solve for the unknown constants. Partial fraction decomposition is a fundamental technique, and understanding it well will definitely help you in your math journey. Keep practicing, and you'll become a decomposition master in no time! You've got this! This journey through partial fraction decomposition highlights the importance of understanding the underlying principles and applying them systematically. By recognizing the structure of the denominator, setting up the correct form, and employing techniques like equating coefficients, we can effectively break down complex rational functions into simpler, more manageable pieces. This skill is not only valuable in calculus but also in various other fields, such as engineering and physics, where rational functions frequently arise. The ability to decompose fractions allows us to simplify complex problems, making them easier to analyze and solve. So, keep exploring, keep practicing, and keep pushing your mathematical boundaries – you never know what exciting discoveries await you!