Write the partial fraction decomposition of each rational expression. x/(x-2)(x-3)

Verified step by step guidance

1. Identify the form of the rational expression. Since the denominator is a product of two distinct linear factors, \((x-2)(x-3)\), the partial fraction decomposition will have the form: \(\frac{A}{x-2} + \frac{B}{x-3}\), where \(A\) and \(B\) are constants to be determined.
2. Write the equation equating the original fraction to the sum of the partial fractions: \(\frac{x}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}\).
3. Multiply both sides of the equation by the common denominator \((x-2)(x-3)\) to clear the denominators: \(x = A(x-3) + B(x-2)\).
4. Expand the right side: \(x = A x - 3A + B x - 2B\).
5. Group like terms and set up a system of equations by equating coefficients of corresponding powers of \(x\): \(x = (A + B) x + (-3A - 2B)\). This gives the system: \(A + B = 1\) (coefficient of \(x\)) and \(-3A - 2B = 0\) (constant term).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with linear or irreducible quadratic denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Denominators

Factoring the denominator into linear or irreducible quadratic factors is a crucial step before decomposition. It helps identify the form of the partial fractions and determines the structure of the numerators in the decomposition.