In Exercises 9–42, write the partial fraction decomposition of each rational expression. 5x^2 -6x+7/(x − 1) (x² + 1)

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1. Step 1: Identify the form of the partial fraction decomposition. Since the denominator is (x - 1)(x^2 + 1), the partial fraction decomposition will be of the form $\frac{A}{x-1}+\frac{Bx+C}{x^2+1}$.
2. Step 2: Set up the equation by equating the original expression to the partial fraction decomposition: $\frac{5x^2-6x+7}{(x-1)(x^2+1)}=\frac{A}{x-1}+\frac{Bx+C}{x^2+1}$.
3. Step 3: Clear the denominators by multiplying through by $(x-1)(x^2+1)$ to obtain: $5x^2-6x+7=A(x^2+1)+(Bx+C)(x-1)$.
4. Step 4: Expand the right-hand side: $A(x^2+1)=A{x}^2+A$ and $(Bx+C)(x-1)=B{x}^2-Bx+Cx-C$.
5. Step 5: Combine like terms and equate coefficients from both sides of the equation to solve for A, B, and C.

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 the focus of this question.

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 process involves breaking down the rational expression based on the factors of the denominator.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. In the context of partial fraction decomposition, recognizing the factors of the denominator is essential, as it determines how the rational expression can be split into simpler fractions. This includes identifying linear and irreducible quadratic factors.