Find the partial fraction decomposition for each rational expression. See Examples 1–4. x/(x^2 + 4x - 5)

Verified step by step guidance

1. Factor the denominator $x^2+4x-5$ into two linear factors.
2. The factored form of the denominator is $(x+5)(x-1)$.
3. Set up the partial fraction decomposition: $\frac{x}{(x+5)(x-1)}=\frac{A}{x+5}+\frac{B}{x-1}$.
4. Multiply through by the common denominator $(x+5)(x-1)$ to clear the fractions: $x=A(x-1)+B(x+5)$.
5. Expand and collect like terms to form an equation in terms of $x$, then solve for $A$ and $B$ by equating coefficients.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this context, the expression x/(x^2 + 4x - 5) is a rational expression that needs to be decomposed into simpler fractions.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the original fraction into components that are easier to work with, especially when the denominator can be factored.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler polynomial factors. This is essential in partial fraction decomposition, as the first step is to factor the denominator completely. For the expression x^2 + 4x - 5, identifying its factors will allow for the correct setup of the partial fractions.