Question

Find the partial fraction decomposition of each rational expression. $$ \frac{x^{3}+1}{x^{5}-x^{4}} $$

Short answer

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

Step-by-step solution

- Factor the Denominator

Factor the denominator first: \[x^{5} - x^{4} = x^{4}(x - 1)\]

- Set Up Partial Fractions

Since the denominator is factored as \(x^{4}(x - 1)\), set up the partial fractions as follows: \[\frac{x^{3}+1}{x^{4}(x-1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}} + \frac{D}{x^{4}} + \frac{E}{x-1}\]

- Clear the Denominator

Multiply both sides by \(x^{4}(x-1)\) to clear the denominator: \[x^{3} + 1 = A x^{3}(x - 1) + B x^{2}(x - 1) + C x(x - 1) + D(x - 1) + E x^{4}\]

- Expand and Combine Like Terms

Expand the right side and combine like terms to get: \[x^{3} + 1 = A x^{4} - A x^{3} + B x^{3} - B x^{2} + C x^{2} - C x + D x - D + E x^{4}\] Combine terms: \[x^{3} + 1 = (A+E)x^{4} + (B - A)x^{3} + (C - B)x^{2} + (D - C)x - D\]

- Set Up a System of Equations

Equate the coefficients of like terms on both sides: \[\begin{cases} A + E = 0 \ B - A = 1 \ C - B = 0 \ D - C = 0 \ -D = 1 \ \text{Note: there is no } x^4 \text{ term on the left side which means } A + E = 0 \end{cases}\]

- Solve the System of Equations

Solve the system: \[\begin{cases} D = -1 \ C = -1 \ B = -1 \ A = -2 \ E = 2 \end{cases}\]

- Write the Partial Fractions

Plug the values back into the partial fraction form: \[\frac{x^{3}+1}{x^{4}(x-1)} = \frac{-2}{x} + \frac{-1}{x^{2}} + \frac{-1}{x^{3}} + \frac{-1}{x^{4}} + \frac{2}{x-1}\]

Key concepts

factorization

Factorization is the process of breaking down an expression into a product of simpler factors. In this exercise, we need to factorize the denominator of the rational expression \(\frac{x^{3}+1}{x^{5}-x^{4}}\). The denominator \(x^{5} - x^{4}\) can be factored as \(x^{4}(x - 1)\).

This step is crucial as it helps simplify the expression and makes it easier to decompose into partial fractions. By identifying the factors, we establish the basis for setting up partial fractions later. Factorization plays a key role in simplifying complex rational functions, making them more manageable.

rational expressions

A rational expression is a fraction whose numerator and denominator are both polynomials. For instance, in the given problem, \(\frac{x^{3}+1}{x^{5}-x^{4}}\), both the numerator \(x^{3} + 1\) and the denominator \(x^{5} - x^{4}\) are polynomials.

Working with rational expressions involves operations such as addition, subtraction, multiplication, and division, similar to numerical fractions. However, because they contain variables, these operations often require additional steps like finding a common denominator or factoring polynomials.

Understanding how to manipulate rational expressions is critical in algebra, especially when dealing with complex fractions and partial fractions.

system of equations

A system of equations is a set of equations with multiple variables. To solve a system, we look for values of the variables that satisfy all equations simultaneously. In partial fraction decomposition, we often set up a system of equations based on the coefficients of like terms.

For instance, in our exercise, we obtained the following system:

• \(A + E = 0\)
• \(B - A = 1\)
• \(C - B = 0\)
• \(D - C = 0\)
• \(-D = 1\)

We solve this system by substitution or elimination methods to find values of A, B, C, D, and E. This process allows us to determine the coefficients for the partial fractions in the decomposition.

partial fractions

Partial fraction decomposition is the process of breaking down a complex rational expression into a sum of simpler fractions called partial fractions. This technique is particularly useful for integrating complex rational expressions.

In our exercise, after factorizing the denominator and equating the partial fractions, we set up the form:
\( \frac{x^{3}+1}{x^{4}(x-1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}} + \frac{D}{x^{4}} + \frac{E}{x-1} \), where A, B, C, D, and E are constants to be determined.

After clearing the common denominator and expanding both sides, we equate coefficients of like terms to set up a system of equations. Solving this system gives us the coefficients A, B, C, D, and E. Plugging these values back into the partial fraction form, we get: \( \frac{x^{3}+1}{x^{4}(x-1)} = \frac{-2}{x} + \frac{-1}{x^{2}} + \frac{-1}{x^{3}} + \frac{-1}{x^{4}} + \frac{2}{x-1} \).