In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

Verified step by step guidance

1. Identify the form of the partial fraction decomposition. Since the denominator is (x - 3)(x + 2), the decomposition will be of the form $\frac{A}{x-3}+\frac{B}{x+2}$.
2. Set up the equation: $\frac{x}{(x-3)(x+2)}=\frac{A}{x-3}+\frac{B}{x+2}$.
3. Multiply through by the common denominator $(x-3)(x+2)$ to clear the fractions: $x=A(x+2)+B(x-3)$.
4. Expand the right side: $x=Ax+2A+Bx-3B$.
5. Combine like terms and equate coefficients: $x=(A+B)x+(2A-3B)$.

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 such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is often necessary for integration or solving equations.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful when integrating rational expressions, as it allows for easier manipulation and integration of each term. The process involves breaking down the expression based on the factors of the denominator.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be linear or irreducible quadratic expressions. This is essential in partial fraction decomposition, as the form of the factors in the denominator determines how the rational expression can be decomposed into simpler fractions.