In Exercises 9–42, write the partial fraction decomposition of each rational expression. 5x² -6x+7/(x − 1) (x² + 1)

Verified step by step guidance

1. Identify the form of the denominator: it is a product of a linear factor (x - 1) and an irreducible quadratic factor (x² + 1).
2. Set up the partial fraction decomposition with unknown constants: for the linear factor, use A/(x - 1); for the quadratic factor, use (Bx + C)/(x² + 1). So, write the expression as \( \frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1} \).
3. Multiply both sides of the equation by the denominator \( (x - 1)(x^2 + 1) \) to clear the fractions, resulting in \( 5x^2 - 6x + 7 = A(x^2 + 1) + (Bx + C)(x - 1) \).
4. Expand the right-hand side by distributing A and then expanding \( (Bx + C)(x - 1) \), combining like terms to express the right side as a polynomial in terms of x.
5. Equate the coefficients of corresponding powers of x on both sides to form a system of equations, which you can solve to find the values of A, B, and C.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factoring Denominators

Factoring the denominator into linear and irreducible quadratic factors is essential for setting up the correct form of the partial fractions. In this problem, the denominator factors as (x - 1)(x² + 1), where (x - 1) is linear and (x² + 1) is an irreducible quadratic.

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of partial fractions with unknown coefficients, you multiply both sides by the common denominator and equate coefficients of corresponding powers of x. Solving the resulting system of equations determines the values of these coefficients.