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. Here, the denominator is \( (x - 1)(x^2 + 1) \), which consists of a linear factor \( (x - 1) \) and an irreducible quadratic factor \( (x^2 + 1) \).
2. Set up the partial fraction decomposition with unknown constants. For the linear factor, use a constant numerator \( A \), and for the quadratic factor, use a linear numerator \( Bx + C \). So, write: \[ \frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1} \]
3. Multiply both sides of the equation by the denominator \( (x - 1)(x^2 + 1) \) to clear the fractions: \[ 5x^2 - 6x + 7 = A(x^2 + 1) + (Bx + C)(x - 1) \]
4. Expand the right-hand side by distributing \( A \) and then expanding \( (Bx + C)(x - 1) \). This will give a polynomial expression in terms of \( x \): \[ A x^2 + A + Bx^2 - Bx + Cx - C \]
5. Group like terms (powers of \( x \)) on the right side and equate the coefficients of corresponding powers of \( x \) from both sides to form a system of equations. Solve this system 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 expressions into manageable parts.

Factoring the Denominator

Factoring the denominator involves expressing it as a product of linear and/or irreducible quadratic factors. In this problem, the denominator is already factored as (x − 1)(x² + 1), which guides the form of the partial fractions to include terms over each factor.

Form of Partial Fractions for Linear and Quadratic Factors

For linear factors like (x − 1), the partial fraction takes the form A/(x − 1). For irreducible quadratic factors like (x² + 1), the partial fraction is expressed as (Bx + C)/(x² + 1). This ensures the decomposition accounts for all possible numerators matching the degree of the factors.