In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

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 case, the expression 1/x(x-1) is a rational expression that needs to be decomposed into simpler fractions.

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the original expression into fractions whose denominators are the factors of the original denominator.

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential in partial fraction decomposition, as it allows us to identify the linear and irreducible quadratic factors in the denominator. In the expression 1/x(x-1), recognizing that x and (x-1) are factors helps in setting up the decomposition correctly.