Find the partial fraction decomposition for each rational expression. See Examples 1–4. x/(x² + 4x - 5)

Verified step by step guidance

1. Start by factoring the denominator of the rational expression. The denominator is \(x^2 + 4x - 5\). To factor it, find two numbers that multiply to \(-5\) and add to \$4$.
2. Rewrite the denominator as a product of two binomials: \(x^2 + 4x - 5 = (x + 5)(x - 1)\).
3. Set up the partial fraction decomposition form. Since the denominator factors into two distinct linear factors, express the fraction as \(\frac{x}{(x + 5)(x - 1)} = \frac{A}{x + 5} + \frac{B}{x - 1}\), where \(A\) and \(B\) are constants to be determined.
4. Multiply both sides of the equation by the common denominator \((x + 5)(x - 1)\) to clear the fractions: \(x = A(x - 1) + B(x + 5)\).
5. Expand the right side and collect like terms: \(x = A x - A + B x + 5 B = (A + B) x + (-A + 5 B)\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations: For \(x\) terms, \$1 = A + B\(; for constants, \)0 = -A + 5 B\(. Solve this system to find \)A\( and \)B$.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex fraction into simpler terms with linear or quadratic denominators.

Factoring Quadratic Expressions

Factoring quadratic expressions means rewriting a quadratic polynomial as a product of two binomials. For example, x^2 + 4x - 5 factors into (x + 5)(x - 1). This step is essential to identify the denominators in partial fractions.

Setting Up and Solving Equations for Coefficients

After expressing the rational function as a sum of partial fractions, you set up equations by equating numerators. Solving these equations for unknown coefficients allows you to find the values that complete the decomposition.