Resolving (11-3x)/(x^2+2x-3) Into Partial Fractions A Step-by-Step Guide

In the realm of algebra and calculus, the technique of resolving rational expressions into partial fractions is invaluable. It simplifies complex fractions, making them easier to integrate, differentiate, or manipulate in various mathematical contexts. This article delves into the process of resolving the rational expression (11-3x)/(x^2+2x-3) into its partial fractions, providing a step-by-step guide that is both comprehensive and easy to follow. Understanding partial fraction decomposition is not just an academic exercise; it is a practical skill that finds applications in fields ranging from engineering to physics. The ability to break down complex rational functions into simpler components allows for easier analysis and problem-solving, especially when dealing with integrals and inverse Laplace transforms. This method transforms complicated expressions into a sum of simpler fractions, each with a linear or irreducible quadratic denominator. By mastering this technique, you'll be equipped to tackle a wide array of mathematical challenges.

Understanding Partial Fractions

Before we dive into the specifics of resolving (11-3x)/(x^2+2x-3), it's crucial to grasp the underlying principles of partial fractions. Partial fraction decomposition is the process of expressing a rational function (a fraction where both the numerator and denominator are polynomials) as a sum of simpler fractions. The premise is that any rational function can be broken down into a sum of fractions whose denominators are factors of the original denominator. This technique is particularly useful when dealing with integrals, as it can transform a complex integral into a sum of simpler integrals that are easier to solve. For instance, integrating a complex rational function directly can be cumbersome, but integrating its partial fraction decomposition often yields a straightforward solution. The same principle applies to other mathematical operations, such as finding inverse Laplace transforms in engineering applications. The core idea is to reverse the process of adding fractions; instead of combining fractions into a single one, we decompose a single fraction into its constituent parts. This decomposition is not just a mathematical trick; it reflects a fundamental property of rational functions and provides a powerful tool for simplifying complex expressions. In essence, partial fraction decomposition is like dissecting a complex machine into its individual components, making it easier to understand and work with.

Step 1: Factor the Denominator

The first critical step in resolving (11-3x)/(x^2+2x-3) into partial fractions is to factor the denominator, which is x^2+2x-3. Factoring the denominator is crucial because the factors will form the denominators of the partial fractions. This process involves finding two binomials that, when multiplied together, yield the original quadratic expression. In this case, we are looking for two numbers that multiply to -3 and add to 2. These numbers are 3 and -1. Therefore, we can factor the denominator as (x+3)(x-1). This factorization is the foundation upon which the partial fraction decomposition will be built. Each factor in the denominator will correspond to a partial fraction. For example, the factor (x+3) will give rise to a fraction of the form A/(x+3), where A is a constant to be determined. Similarly, the factor (x-1) will lead to a fraction of the form B/(x-1), where B is another constant. The ability to factor the denominator correctly is essential for the success of the entire process. An incorrect factorization will lead to incorrect partial fractions and an incorrect final result. Once the denominator is factored, the problem is significantly simplified, transforming from a single complex fraction into a sum of simpler fractions. This step is not just about finding the factors; it's about laying the groundwork for the rest of the process.

Step 2: Set Up the Partial Fraction Decomposition

With the denominator factored as (x+3)(x-1), we can now set up the partial fraction decomposition. This involves expressing the original fraction as a sum of fractions, each with one of the factors as its denominator. Since we have two linear factors, (x+3) and (x-1), we will have two partial fractions. Each partial fraction will have a constant in the numerator. We can represent these constants as A and B. Therefore, we can write the partial fraction decomposition as follows:

(11-3x)/(x^2+2x-3) = A/(x+3) + B/(x-1)

This equation is the core of the partial fraction decomposition process. It expresses the original complex fraction as a sum of two simpler fractions. The next step will involve solving for the constants A and B. The setup of the decomposition is a critical step because it defines the structure of the solution. If the setup is incorrect, the rest of the process will lead to an incorrect result. The number of partial fractions and the form of the denominators are determined by the factors of the original denominator. In this case, because we have two distinct linear factors, we have two partial fractions with those factors as denominators. The numerators are constants because the factors are linear. If we had repeated factors or irreducible quadratic factors, the setup would be slightly different, involving higher powers of the factors or linear expressions in the numerators. However, in this case, the setup is straightforward, laying the foundation for the subsequent steps in the partial fraction decomposition process.

Step 3: Clear the Denominators

To solve for the constants A and B, we need to clear the denominators in the equation:

(11-3x)/(x^2+2x-3) = A/(x+3) + B/(x-1)

We do this by multiplying both sides of the equation by the original denominator, which is (x+3)(x-1). This step eliminates the fractions, making it easier to solve for the unknowns. When we multiply both sides by (x+3)(x-1), we get:

11-3x = A(x-1) + B(x+3)

This equation is now free of fractions and is much easier to work with. The process of clearing denominators is a standard technique in algebra for solving equations involving fractions. It involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is simply the product of the two linear factors, (x+3)(x-1). This step is crucial because it transforms the equation into a form where we can directly compare coefficients or substitute values to solve for the constants. The resulting equation, 11-3x = A(x-1) + B(x+3), is a polynomial equation. This means that the coefficients of the corresponding terms on both sides of the equation must be equal. This property is the basis for one of the methods we can use to solve for A and B. Clearing the denominators is not just a mechanical step; it's a strategic move that simplifies the equation and sets the stage for solving for the unknown constants in the partial fraction decomposition.

Step 4: Solve for the Constants A and B

There are two common methods for solving for the constants A and B: the substitution method and the equating coefficients method. Let's explore both.

Substitution Method

The substitution method involves choosing values of x that will eliminate one of the constants, allowing us to solve for the other. From the equation:

11-3x = A(x-1) + B(x+3)

We can choose x=1 to eliminate A:

11-3(1) = A(1-1) + B(1+3)

8 = 4B

B = 2

Next, we can choose x=-3 to eliminate B:

11-3(-3) = A(-3-1) + B(-3+3)

20 = -4A

A = -5

The substitution method is particularly effective when the denominator factors are linear and distinct. By choosing values of x that make one of the factors zero, we can directly solve for the corresponding constant. This method is often quicker and more intuitive than the equating coefficients method, especially for simpler problems. The key to using the substitution method effectively is to choose values of x that will make the factors zero. This eliminates one of the terms on the right-hand side of the equation, leaving a simple equation in one unknown. However, the substitution method may not be as straightforward when dealing with repeated factors or irreducible quadratic factors. In such cases, the equating coefficients method may be more suitable.

Equating Coefficients Method

The equating coefficients method involves expanding the equation and equating the coefficients of like terms on both sides. Starting with:

11-3x = A(x-1) + B(x+3)

Expand the right side:

11-3x = Ax - A + Bx + 3B

Group like terms:

11-3x = (A+B)x + (-A+3B)

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

-3 = A + B

11 = -A + 3B

This gives us a system of two linear equations in two unknowns. We can solve this system using substitution or elimination. From the first equation, A = -3 - B. Substitute this into the second equation:

11 = -(-3-B) + 3B

11 = 3 + B + 3B

8 = 4B

B = 2

Now, substitute B=2 back into A = -3 - B:

A = -3 - 2

A = -5

The equating coefficients method is a more systematic approach that works for all partial fraction decomposition problems, regardless of the nature of the denominator factors. It involves expanding the equation, grouping like terms, and then equating the coefficients of the corresponding terms on both sides. This results in a system of linear equations, which can be solved using standard techniques such as substitution, elimination, or matrix methods. The equating coefficients method is particularly useful when dealing with repeated factors or irreducible quadratic factors, where the substitution method may not be as straightforward. The key to using the equating coefficients method effectively is to carefully expand the equation and group like terms. Then, the coefficients of the corresponding terms must be equated to form the system of linear equations. This method may involve more algebraic manipulation than the substitution method, but it is a reliable and versatile technique for solving for the constants in partial fraction decomposition.

Step 5: Write the Partial Fraction Decomposition

Now that we have found A = -5 and B = 2, we can write the partial fraction decomposition:

(11-3x)/(x^2+2x-3) = -5/(x+3) + 2/(x-1)

This is the final answer. We have successfully resolved the original rational expression into its partial fractions. Writing the final partial fraction decomposition is the culmination of all the previous steps. It involves substituting the values of the constants that were solved for in the previous step back into the original partial fraction decomposition setup. In this case, we found A = -5 and B = 2, so we substitute these values into the equation:

(11-3x)/(x^2+2x-3) = A/(x+3) + B/(x-1)

Which gives us:

(11-3x)/(x^2+2x-3) = -5/(x+3) + 2/(x-1)

This final expression represents the partial fraction decomposition of the original rational expression. It expresses the original complex fraction as a sum of two simpler fractions, each with a linear denominator. This decomposition is useful for various mathematical operations, such as integration, differentiation, and finding inverse Laplace transforms. The final partial fraction decomposition is not just a result; it's a new representation of the original expression that is often easier to work with. It's like taking a complex puzzle and breaking it down into smaller, more manageable pieces. This step completes the partial fraction decomposition process and provides a valuable tool for further mathematical analysis.

Conclusion

Resolving (11-3x)/(x^2+2x-3) into partial fractions involves factoring the denominator, setting up the decomposition, clearing denominators, solving for the constants, and writing the final result. This technique is a fundamental tool in calculus and algebra, simplifying complex rational expressions for various applications. Mastering partial fraction decomposition opens doors to solving a wide range of problems in mathematics and engineering, making it an essential skill for anyone pursuing these fields. The process of partial fraction decomposition may seem complex at first, but with practice, it becomes a straightforward and powerful technique. Each step in the process has a specific purpose, and understanding these purposes is key to mastering the technique. From factoring the denominator to solving for the constants, each step builds upon the previous one, leading to the final partial fraction decomposition. This decomposition is not just a mathematical trick; it's a way of understanding the structure of rational functions and expressing them in a more manageable form. So, whether you're a student tackling calculus problems or an engineer working with complex systems, partial fraction decomposition is a valuable tool to have in your arsenal. By following the steps outlined in this article and practicing regularly, you can develop the skills and confidence needed to tackle any partial fraction decomposition problem.