Partial Fraction Decomposition: How To Solve 6x+17/(x+1)^2?

by Alex Johnson 60 views

Understanding partial fraction decomposition is a crucial skill in calculus and algebra. If you've ever wondered how to break down a complex rational expression into simpler fractions, you're in the right place. In this article, we'll dive deep into the process of partial fraction decomposition, specifically focusing on the expression $\frac{6x+17}{(x+1)^2}$. By the end, you’ll not only understand the theory but also be able to apply it to solve similar problems. So, let’s get started and unravel the mystery behind this mathematical technique.

What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down a rational function (a fraction where both the numerator and denominator are polynomials) into simpler fractions. This is especially useful when integrating rational functions or solving differential equations. Imagine you have a complicated fraction that's hard to deal with directly. Partial fraction decomposition helps you rewrite it as a sum of simpler fractions, making it easier to work with. The basic idea is to reverse the process of adding fractions with different denominators. We start with the combined fraction and break it down into its original components. This method is widely used in calculus to simplify integrals and is a fundamental tool in various areas of mathematics and engineering. By mastering this technique, you'll be able to tackle more complex problems with greater ease and efficiency. So, whether you're a student grappling with calculus or a professional needing to solve intricate equations, understanding partial fraction decomposition is an invaluable asset.

Why Use Partial Fraction Decomposition?

There are several reasons why partial fraction decomposition is such a valuable technique in mathematics. First and foremost, it simplifies complex rational expressions. When you have a fraction with a high-degree polynomial in the denominator, it can be challenging to work with directly. Breaking it down into simpler fractions makes the expression more manageable. Another key reason is its application in calculus, particularly in integration. Many integrals involving rational functions are difficult to solve in their original form. By decomposing the fraction, you can often rewrite the integral as a sum of simpler integrals that are easier to evaluate. This technique is also crucial in solving differential equations. Many differential equations involve rational functions, and partial fraction decomposition can be used to find solutions by simplifying the expressions involved. Furthermore, partial fraction decomposition is used in Laplace transforms, which are essential in engineering and physics for solving linear differential equations. In essence, partial fraction decomposition is a versatile tool that bridges algebra and calculus, providing a systematic way to handle rational functions in various contexts. Its ability to transform complex expressions into simpler ones makes it an indispensable technique for anyone working with mathematical problems in science and engineering.

Key Concepts and Prerequisites

Before diving into the specifics of decomposing $\frac{6x+17}{(x+1)^2}$, it’s important to have a solid grasp of some fundamental concepts. First, you should be comfortable with polynomials – what they are, how to add, subtract, multiply, and divide them. Understanding polynomial factorization is also crucial, as it helps in identifying the factors in the denominator of the rational function. Next, you need to know the basics of rational expressions, which are fractions where the numerator and denominator are polynomials. You should be able to simplify rational expressions and perform operations like addition, subtraction, multiplication, and division. A key concept in partial fraction decomposition is understanding the different types of factors that can appear in the denominator, such as linear factors (e.g., x + 1) and irreducible quadratic factors (e.g., x² + 1). You also need to know how to deal with repeated factors, which are factors that appear multiple times (e.g., ( x + 1)²). Finally, a basic understanding of algebraic equations and how to solve them is essential, as you’ll need to solve for unknown coefficients in the decomposed fractions. With these prerequisites in mind, you’ll be well-prepared to tackle partial fraction decomposition and apply it effectively to various mathematical problems.

Decomposing $\frac{6 x+17}{(x+1)^2}$: A Step-by-Step Guide

Now, let's get to the heart of the matter and decompose the given expression, $\frac{6x+17}{(x+1)^2}$. This will involve a methodical approach, breaking down each step to ensure clarity and understanding. Remember, the goal is to rewrite the complex fraction as a sum of simpler fractions. Let's dive in!

Step 1: Identify the Type of Factors in the Denominator

The first step in partial fraction decomposition is to analyze the denominator of the rational expression. In our case, the denominator is (x+1)2(x+1)^2. This is a repeated linear factor, meaning the factor (x+1)(x+1) appears twice. Recognizing the type of factors is crucial because it dictates the form of the partial fractions we’ll use. For each distinct linear factor, we'll have a term of the form $\frac{A}{x+1}$, and for each repeated linear factor, we'll have terms for each power of the factor up to the repetition. Since we have a repeated linear factor (x+1)2(x+1)^2, we’ll need two terms in our decomposition: one for (x+1)(x+1) and another for (x+1)2(x+1)^2. This initial assessment is the foundation for the entire process, so it’s vital to correctly identify the factors and their nature. By understanding the structure of the denominator, we can set up the correct form for the partial fractions and proceed with the decomposition effectively.

Step 2: Set Up the Partial Fraction Decomposition

Once we’ve identified the type of factors in the denominator, the next step is to set up the partial fraction decomposition. For our expression, $\frac{6x+17}{(x+1)^2}$, since we have a repeated linear factor (x+1)2(x+1)^2, we set up the decomposition as follows:

6x+17(x+1)2=Ax+1+B(x+1)2\frac{6x+17}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}

Here, A and B are constants that we need to determine. The key idea is to express the original fraction as a sum of simpler fractions, each with a denominator that is a factor of the original denominator. The constants A and B represent the numerators of these simpler fractions. This setup is critical because it provides the framework for solving for the unknown constants. By setting up the equation correctly, we pave the way for the subsequent steps, where we’ll solve for A and B and ultimately find the partial fraction decomposition. Ensuring the correct setup is crucial for a smooth and accurate decomposition process.

Step 3: Clear the Denominators

After setting up the partial fraction decomposition, the next step is to clear the denominators. This involves multiplying both sides of the equation by the original denominator, which in our case is (x+1)2(x+1)^2. So, we multiply both sides of the equation:

6x+17(x+1)2=Ax+1+B(x+1)2\frac{6x+17}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}

by (x+1)2(x+1)^2. This gives us:

(x+1)2â‹…6x+17(x+1)2=(x+1)2â‹…(Ax+1+B(x+1)2)(x+1)^2 \cdot \frac{6x+17}{(x+1)^2} = (x+1)^2 \cdot \left(\frac{A}{x+1} + \frac{B}{(x+1)^2}\right)

Simplifying this, we get:

6x+17=A(x+1)+B6x + 17 = A(x+1) + B

Clearing the denominators is a crucial step because it transforms the equation into a more manageable form, eliminating the fractions. This makes it easier to solve for the unknown constants, A and B. By removing the denominators, we create a polynomial equation that we can work with directly, setting the stage for the next steps in the decomposition process. This simplification is key to making the problem solvable and progressing towards the final solution.

Step 4: Solve for the Unknown Constants

With the denominators cleared, we now have the equation $6x + 17 = A(x+1) + B$. The next step is to solve for the unknown constants, A and B. There are a couple of methods we can use to do this.

Method 1: Substitution

One effective method is substitution. We can choose specific values of x that will simplify the equation and allow us to solve for the constants. A particularly useful value to substitute is x = -1, because it will eliminate the term with A:

6(−1)+17=A(−1+1)+B6(-1) + 17 = A(-1+1) + B

−6+17=A(0)+B-6 + 17 = A(0) + B

11=B11 = B

So, we find that B = 11.

Now that we have B, we can substitute another value for x. Let’s use x = 0:

6(0)+17=A(0+1)+116(0) + 17 = A(0+1) + 11

17=A+1117 = A + 11

A=6A = 6

Thus, we find that A = 6.

Method 2: Equating Coefficients

Another method is to expand the equation and equate the coefficients of like terms. Expanding the equation gives us:

6x+17=Ax+A+B6x + 17 = Ax + A + B

Now, we equate the coefficients of the x terms and the constant terms:

  • Coefficient of x: 6 = A
  • Constant term: 17 = A + B

From the first equation, we immediately get A = 6. Substituting this into the second equation gives us:

17=6+B17 = 6 + B

B=11B = 11

So, using either method, we find that A = 6 and B = 11. Solving for these constants is a pivotal step, as it allows us to rewrite the original fraction in its decomposed form. With these values in hand, we can now express the partial fraction decomposition.

Step 5: Write the Partial Fraction Decomposition

Now that we have found the values of the constants, A = 6 and B = 11, we can write the partial fraction decomposition of the original expression. Recall that we set up the decomposition as:

6x+17(x+1)2=Ax+1+B(x+1)2\frac{6x+17}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}

Substituting the values of A and B into this equation, we get:

6x+17(x+1)2=6x+1+11(x+1)2\frac{6x+17}{(x+1)^2} = \frac{6}{x+1} + \frac{11}{(x+1)^2}

This is the partial fraction decomposition of $\frac{6x+17}{(x+1)^2}$. We have successfully broken down the complex rational expression into two simpler fractions. This final step completes the process, providing us with a form that is often much easier to work with in further calculations, such as integration. By rewriting the original fraction in this decomposed form, we’ve made it more accessible and manageable for various mathematical operations.

Conclusion

In conclusion, we've successfully decomposed the rational expression $\frac{6x+17}{(x+1)^2}$ into its partial fractions, which are $\frac{6}{x+1} + \frac{11}{(x+1)^2}$. This process involved several key steps, from identifying the type of factors in the denominator to solving for the unknown constants and writing out the final decomposition. Understanding and applying partial fraction decomposition is a valuable skill in calculus and algebra, enabling you to simplify complex rational expressions and make them easier to work with. Whether you're integrating functions, solving differential equations, or tackling other mathematical problems, this technique provides a powerful tool for simplifying your work.

For further exploration and practice, you can check out resources on websites like Khan Academy's Partial Fraction Decomposition section. Keep practicing, and you'll master this essential mathematical technique!