In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 5x²-6x+7 /(x − 1) (x² + 1)

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 like addition, subtraction, and decomposition. In this context, the expression 5x² - 6x + 7 / ((x - 1)(x² + 1)) is a rational expression that needs to be decomposed into simpler fractions.

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 for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, based on the factors of the denominator.

Polynomial Degree and Factorization

The degree of a polynomial is the highest power of the variable in the polynomial expression. In partial fraction decomposition, the degree of the numerator must be less than the degree of the denominator. Additionally, understanding how to factor polynomials, such as recognizing (x - 1) and (x² + 1) as factors, is essential for setting up the correct form of the partial fractions.