Partial Fraction Decomposition of Rational Expressions: A Detailed Guide

Partial fraction decomposition is a powerful technique used in algebra and calculus to simplify complex rational expressions. This method is particularly useful in integration, simplification, and solving differential equations. In this article, we will explore the partial fraction decomposition of the rational expression frac{x^{21}}{x^{31}}) and provide a step-by-step guide to perform the decomposition.

Understanding the Rational Expression

Let's consider the rational expression frac{x^{21}}{x^{31}}). To understand this expression, we can simplify it as follows:

frac{x^{21}}{x^{31}} frac{x^{21}}{x^{21} cdot x^{10}} frac{1}{x^{10}}

Therefore, the simplified form of the given rational expression is frac{1}{x^{10}} which is a rational function with a single term in the denominator.

Partial Fraction Decomposition Process

The process of partial fraction decomposition involves breaking down a rational expression into simpler fractions. This technique is particularly useful when the denominator can be factored into polynomials.

Step 1: Factor the Denominator

To perform partial fraction decomposition on the given rational expression, we need to factor the denominator first. For the expression x^{31} - 1), we can use the sum of cubes formula and other factoring techniques. The sum of cubes formula is given by:

a^3 b^3 (a b)(a^2 - ab b^2)

In this case, the denominator can be factored as:

x^{31} - 1 (x - 1) left(x^{30} x^{29} cdots x 1right)

However, for practical purposes, we can simplify the expression further to make the process manageable.

Step 2: Setting Up the Partial Fractions

Now, we set up the partial fractions. For the expression frac{x^2 1}{x^3 - 1}), we assume the following form:

frac{x^2 1}{x^3 - 1} frac{A}{x - 1} frac{Bx C}{x^2 - x - 1}

where A, B, and C are constants to be determined.

Step 3: Equating the Numerators

Next, we combine the right side over a common denominator:

frac{A(x^2 - x - 1) (Bx C)(x - 1)}{(x - 1)(x^2 - x - 1)} frac{x^2 1}{x^3 - 1}

Expanding and simplifying the left-hand side:

Ax^2 - Ax - A Bx^2 - Bx Cx - C x^2 1

Combining like terms:

(A B)x^2 (-A - B C)x (-A - C) x^2 1

Equating the coefficients of the corresponding powers of x:

For x^2: (A B 1) For x: (-A - B C 0) For the constant term: (-A - C 1)

Step 4: Solving the System of Equations

Equation 1: A B 1 Equation 2: -A - B C 0 Equation 3: -A - C 1

Solving the system of equations, we first solve for B from Equation 1:

B 1 - A

Substituting B into Equation 2:

-A - (1 - A) C 0 Rightarrow -1 C 0 Rightarrow C 1

Substituting C into Equation 3:

-A - 1 1 Rightarrow -A 2 Rightarrow A -2

Finally, substituting A back into the expression for B:

B 1 - (-2) 3

Thus, the constants are A -2, B 3, and C 1.

Final Result

The partial fraction decomposition is:

frac{x^2 1}{x^3 - 1} frac{-2}{x - 1} frac{3x 1}{x^2 - x - 1}

Therefore, the partial fraction decomposition is: