Finding Constants A, B, C, And D In A Polynomial Identity

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In the realm of algebra, polynomial identities play a crucial role in simplifying expressions and solving equations. When faced with a complex rational expression, the task of decomposing it into simpler fractions often arises. This process, known as partial fraction decomposition, involves expressing a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. In this article, we embark on a journey to find the values of the constants A, B, C, and D in the given identity, showcasing the power and elegance of partial fraction decomposition.

The Challenge: Decomposing a Rational Expression

Our mission is to determine the constants A, B, C, and D that satisfy the following identity:

[\frac{x^3 - x^2 - x - 3}{x(x - 1)} \equiv Ax + B + \frac{C}{x} + \frac{D}{x-1}]

This identity presents us with a rational expression on the left-hand side, which we aim to decompose into a sum of simpler fractions on the right-hand side. The constants A, B, C, and D are the key to unlocking this decomposition. By finding their values, we can rewrite the complex rational expression in a more manageable form, paving the way for further analysis and manipulation.

The Strategy: A Step-by-Step Approach

To find the values of A, B, C, and D, we will employ a systematic approach that combines algebraic manipulation and strategic substitution. Our strategy involves the following steps:

  1. Multiply both sides of the identity by the common denominator: This step eliminates the fractions, transforming the identity into a polynomial equation.
  2. Expand the right-hand side: Expanding the right-hand side allows us to group terms with like powers of x.
  3. Equate coefficients: By equating the coefficients of corresponding powers of x on both sides of the equation, we obtain a system of linear equations in the unknowns A, B, C, and D.
  4. Solve the system of equations: We solve the system of equations to determine the values of the constants A, B, C, and D.

Step 1: Multiplying by the Common Denominator

The first step in our journey is to eliminate the fractions by multiplying both sides of the identity by the common denominator, which is x(x - 1). This gives us:

x^3 - x^2 - x - 3 \equiv (Ax + B)x(x - 1) + C(x - 1) + Dx

This transformation removes the fractions, making the equation easier to work with. We now have a polynomial equation that we can manipulate further.

Step 2: Expanding the Right-Hand Side

Next, we expand the right-hand side of the equation to group terms with like powers of x:

x^3 - x^2 - x - 3 \equiv Ax^3 - Ax^2 + Bx^2 - Bx + Cx - C + Dx

Expanding the right-hand side allows us to rearrange the terms and collect those with the same powers of x. This step is crucial for the next stage, where we equate coefficients.

Step 3: Equating Coefficients

Now, we equate the coefficients of corresponding powers of x on both sides of the equation. This yields the following system of linear equations:

  • Coefficient of x^3: A = 1
  • Coefficient of x^2: -A + B = -1
  • Coefficient of x: -B + C + D = -1
  • Constant term: -C = -3

This system of equations provides us with a set of relationships between the constants A, B, C, and D. By solving this system, we can determine the values of these constants.

Step 4: Solving the System of Equations

We can solve the system of equations using various methods, such as substitution or elimination. From the first equation, we directly obtain A = 1. From the last equation, we get C = 3. Substituting A = 1 into the second equation, we get -1 + B = -1, which gives us B = 0. Finally, substituting B = 0 and C = 3 into the third equation, we get 0 + 3 + D = -1, which gives us D = -4.

Therefore, the values of the constants are A = 1, B = 0, C = 3, and D = -4.

The Solution: Unveiling the Constants

After meticulously following our step-by-step approach, we have successfully found the values of the constants A, B, C, and D. They are:

  • A = 1
  • B = 0
  • C = 3
  • D = -4

These values satisfy the given identity, allowing us to rewrite the original rational expression as a sum of simpler fractions:

\frac{x^3 - x^2 - x - 3}{x(x - 1)} \equiv x + \frac{3}{x} - \frac{4}{x-1}

This decomposition provides a more insightful representation of the rational expression, making it easier to analyze and manipulate.

The Significance: Partial Fraction Decomposition in Action

The technique we have employed, partial fraction decomposition, is a powerful tool in algebra and calculus. It allows us to break down complex rational expressions into simpler fractions, which can be easier to integrate, differentiate, or analyze in other ways. This technique finds applications in various fields, including:

  • Calculus: Evaluating integrals of rational functions.
  • Differential Equations: Solving linear differential equations with constant coefficients.
  • Control Systems: Analyzing the stability and behavior of control systems.
  • Circuit Analysis: Determining the currents and voltages in electrical circuits.

A Deeper Dive into Partial Fraction Decomposition

Partial fraction decomposition is a fundamental technique in algebra that allows us to express a rational function as a sum of simpler fractions. This process is particularly useful when dealing with complex rational expressions that are difficult to work with directly. By breaking down these expressions into simpler components, we can often simplify calculations, solve equations, and gain a better understanding of the underlying functions.

The Essence of the Technique

The core idea behind partial fraction decomposition is to reverse the process of adding fractions with different denominators. When we add fractions, we typically find a common denominator and combine the numerators. Partial fraction decomposition does the opposite: it starts with a single fraction and decomposes it into a sum of fractions with simpler denominators.

The General Approach

The general approach to partial fraction decomposition involves the following steps:

  1. Factor the denominator: The first step is to factor the denominator of the rational function into linear and irreducible quadratic factors. This factorization is crucial for determining the form of the partial fraction decomposition.
  2. Set up the partial fraction decomposition: Based on the factors in the denominator, we set up the partial fraction decomposition as a sum of fractions with unknown constants in the numerators. The form of the decomposition depends on the type of factors in the denominator.
  3. Solve for the constants: We solve for the unknown constants by multiplying both sides of the equation by the original denominator, expanding the resulting expression, and equating coefficients of like terms. This gives us a system of linear equations that we can solve to find the constants.
  4. Write the partial fraction decomposition: Once we have found the constants, we can write the partial fraction decomposition by substituting the values of the constants into the expression we set up in step 2.

Different Cases of Partial Fraction Decomposition

The specific form of the partial fraction decomposition depends on the factors in the denominator. Here are some common cases:

  • Distinct Linear Factors: If the denominator has distinct linear factors, then the partial fraction decomposition will have a term of the form A/(x - a) for each linear factor (x - a), where A is a constant.
  • Repeated Linear Factors: If the denominator has a repeated linear factor (x - a)^n, then the partial fraction decomposition will have terms of the form A1/(x - a), A2/(x - a)^2, ..., An/(x - a)^n, where A1, A2, ..., An are constants.
  • Irreducible Quadratic Factors: If the denominator has an irreducible quadratic factor (ax^2 + bx + c), then the partial fraction decomposition will have a term of the form (Ax + B)/(ax^2 + bx + c), where A and B are constants.

Applications of Partial Fraction Decomposition

Partial fraction decomposition has numerous applications in mathematics, engineering, and other fields. Some of the most common applications include:

  • Integration: Partial fraction decomposition is often used to integrate rational functions. By breaking down the rational function into simpler fractions, we can often find antiderivatives more easily.
  • Laplace Transforms: Partial fraction decomposition is used to find inverse Laplace transforms, which are used in solving differential equations.
  • Series Expansions: Partial fraction decomposition can be used to find series expansions of rational functions.
  • Circuit Analysis: Partial fraction decomposition is used in circuit analysis to find the response of a circuit to a given input.

Common Mistakes to Avoid

While partial fraction decomposition is a powerful technique, it's important to be aware of some common mistakes that can occur. Here are a few to watch out for:

  • Forgetting to factor the denominator: The first step in partial fraction decomposition is to factor the denominator. If you forget to do this, you won't be able to set up the correct partial fraction decomposition.
  • Using the wrong form of the decomposition: The form of the partial fraction decomposition depends on the factors in the denominator. Make sure you use the correct form for each type of factor.
  • Making algebraic errors: Solving for the constants in the partial fraction decomposition can involve a fair amount of algebra. Be careful to avoid algebraic errors, as these can lead to incorrect results.

By understanding the principles of partial fraction decomposition and avoiding common mistakes, you can effectively use this technique to solve a wide range of problems.

Conclusion: A Triumph of Algebraic Techniques

In this article, we have successfully navigated the challenge of finding the constants A, B, C, and D in the given identity. By employing a systematic approach that combines algebraic manipulation and strategic substitution, we have unveiled the values of these constants and rewritten the complex rational expression in a simpler form. This exercise highlights the power and elegance of partial fraction decomposition, a technique that finds widespread applications in mathematics, engineering, and beyond. The journey of finding these constants not only reinforces our understanding of algebraic principles but also showcases the beauty and utility of mathematical tools in simplifying complex problems.