Decomposition of Rational Functions: A Step-by-Step Guide to Partial Fractions

The method of decomposing rational functions into partial fractions is a foundational technique in algebra and calculus. This guide explains how to decompose the expression (frac{x^{23}}{x-1 cdot x^{22}}) into a sum of simpler fractions, illustrating the process with detailed steps and clarifying each mathematical operation.

Problem Statement

The expression we need to decompose is:

[ frac{x^{23}}{x-1 cdot x^{22}} ]

Initial Decomposition

We can decompose the given rational function into partial fractions in the form:

[ frac{x^{23}}{x-1 cdot x^{22}} frac{A}{x-1} frac{B}{x} frac{C}{x^{22}} ]

Expanding the right-hand side, we get:

[ frac{A}{x-1} frac{B}{x} frac{C}{x^{22}} frac{1}{3} left( 1 - frac{1}{x^{22}} - frac{7}{x-1} right) ]

Determining the Constants A, B, and C

The constants (A), (B), and (C) can be found by comparing coefficients of like terms on both sides of the equation.

Step 1: Solve for (A)

To find (A), we set (x 1):

[ A(1-1) B(1) C(1^{22}) 1^{23} cdot (1-1 cdot 1^{22})^{-1} ]

This simplifies to:

[ 2B C 1 cdot (-1)^{-1} ]

Given that (2A B C 1), we set (x -1) to find (A):

[ A(-1-1) B(-1) C(-1^{22}) 1(-1)^{23} cdot ((-1)-1 cdot (-1)^{22})^{-1} ]

Simplifying further:

[ 3A 1 - 3 - 5 ]

[ 3A -7 ]

[ A -frac{7}{3} ]

Step 2: Solve for (B)

To find (B), we compare the coefficients of (x^2) terms:

[ A B 1 ]

Substituting (A -frac{7}{3}):

[ -frac{7}{3} B 1 ]

[ B 1 frac{7}{3} frac{10}{3} ]

Step 3: Solve for (C)

To find (C), we set (x 0):

[ 2A C -5 ]

Substituting (A -frac{7}{3}):

[ 2(-frac{7}{3}) C -5 ]

[ -frac{14}{3} C -5 ]

[ C -5 frac{14}{3} -frac{15}{3} frac{14}{3} -frac{1}{3} ]

Verification

Substituting the values of (A), (B), and (C) back into the decomposed expression, we verify if it matches the original function:

[ frac{x^{23}}{x-1 cdot x^{22}} frac{-frac{7}{3}}{x-1} frac{frac{10}{3}}{x} frac{-frac{1}{3}}{x^{22}} ]

Which simplifies to:

[ frac{1}{3} left( 1 - frac{1}{x^{22}} - frac{7}{x-1} right) ]

Conclusion

The decomposed form of the rational function is:

[ frac{x^{23}}{x-1 cdot x^{22}} frac{1}{3} left( 1 - frac{1}{x^{22}} - frac{7}{x-1} right) ]

Where the constants are:

[ A -frac{7}{3}, B frac{10}{3}, C -frac{1}{3} ]

Understanding how to decompose rational functions into partial fractions is crucial for simplifying complex expressions and solving integrals and other calculus problems.