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

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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 \(4\).

Rewrite the denominator as a product of two binomials: \(x^2 + 4x - 5 = (x + 5)(x - 1)\).

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.

Multiply both sides of the equation by the common denominator \((x + 5)(x - 1)\) to clear the fractions: \(x = A(x - 1) + B(x + 5)\).

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 \(B\).

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주요 개념

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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.

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