Decomposing Complex Rational Functions into Partial Fractions

In this article, we will explore the process of decomposing a complex rational function into partial fractions. We will walk through a detailed example where we decompose the expression (frac{x^4 x^3 - x - 1}{x - 1x^3 x}).

Introduction to Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify a rational function by expressing it as a sum of simpler rational functions. This technique is particularly useful in integration and solving complex algebraic equations. The process involves factoring the denominator and then determining the coefficients of the simpler fractions.

Step-by-Step Process of Decomposition

Let's begin with the given expression:

[ frac{x^4 x^3 - x - 1}{x - 1x^3 x} ]

Step 1: Factor the Denominator

First, we need to factor the denominator completely. The denominator is:

[ x - 1x^3 x x - 1x^2 1 ]

This can be further factored as:

[ x - 1x^2 1 x - 1(x^2 1) ]

Thus, the complete factorization of the denominator is:

[ x - 1x^2 1 ]

Step 2: Set Up the Partial Fraction Decomposition

We express the fraction as:

[ frac{x^4 x^3 - x - 1}{x - 1x^2 1} frac{A}{x - 1} frac{B}{x} frac{Cx D}{x^2 1} ]

Here, (A), (B), (C), and (D) are constants that we need to determine.

Step 3: Combine the Right Side

To combine the right-hand side into a single fraction, we find a common denominator:

[ frac{A}{x - 1} frac{B}{x} frac{Cx D}{x^2 1} frac{A cdot x cdot (x^2 1) B cdot (x - 1)(x^2 1) (Cx D)(x - 1)x}{(x - 1)(x^2 1)x} ]

This simplifies to:

[ frac{A cdot x^3 A cdot x B cdot x^3 - Bx^2 - Bx - B Cx^2 - Cx Dx - D}{(x - 1)x^2 1)} ]

Step 4: Expand and Collect Terms

Now we want to equate the numerators:

[ x^4 x^3 - x - 1 A cdot x^3 A cdot x B cdot x^3 - Bx^2 - Bx - B Cx^2 - Cx Dx - D ]

Expanding and combining like terms:

[ x^4 x^3 - x - 1 (A B)x^3 - Bx^2 (A - B - C)x (D - A - B - C) ]

Step 5: Equate Coefficients

Now we set the coefficients from the left side equal to those on the right side:

x^4: (0 0) x^3: (A B 1) x^2: (-B C 0) x: (A - B - C D -1) Constant: (D - A - B - C -1)

Step 6: Solve the System of Equations

From (-B C 0) we have:

[ C B ]

Substituting (C B) into the other equations:

(A B 1) (A - B - B D -1 rightarrow A - 2B D -1) (B - A - B - B -1 rightarrow -A - B -1 rightarrow A B 1)

Since (A B 1) and (A - 2B D -1), we can solve for (A) and (B):

From (A B 1), we have (A 1 - B).

Substituting (A 1 - B) into (A - 2B D -1):

[ (1 - B) - 2B D -1 rightarrow 1 - 3B D -1 rightarrow D 3B - 2 ]

From (A B 1) and (A 1 - B), we have:

(A frac{1}{2}) (B frac{1}{2}) (C frac{1}{2}) (D -frac{3}{2})

Step 7: Final Values and Decomposition

We have:

[ A frac{1}{2}, B frac{1}{2}, C frac{1}{2}, D -frac{3}{2} ]

Thus, the partial fraction decomposition is:

[ frac{x^4 x^3 - x - 1}{(x - 1)(x^2 1)} frac{1/2}{x - 1} frac{1/2}{x} frac{frac{1}{2}x - frac{3}{2}}{x^2 1} ]

Or written more clearly:

[ frac{1/2}{x - 1} frac{1/2}{x} frac{frac{1}{2}x - frac{3}{2}}{x^2 1} ]

Conclusion

By following these steps, we can decompose complex rational functions into simpler partial fractions, making them much easier to integrate or manipulate further in mathematical computations.