Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)}$

Verified step by step guidance

1. Identify the denominator factors: here, the denominator is \( (x - 1)(x^2 + 1) \). The first factor \( (x - 1) \) is linear, and the second factor \( (x^2 + 1) \) is an irreducible quadratic.
2. For the linear factor \( (x - 1) \), assign a constant numerator: \( \frac{A}{x - 1} \), where \( A \) is a constant to be determined.
3. For the irreducible quadratic factor \( (x^2 + 1) \), assign a linear numerator: \( \frac{Bx + C}{x^2 + 1} \), where \( B \) and \( C \) are constants to be determined.
4. Write the partial fraction decomposition as the sum of these two fractions: \[ \frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1} \]
5. This is the form of the partial fraction decomposition. The next step (not required here) would be to multiply both sides by the denominator and solve for \( A \), \( B \), 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.

Types of Factors in Denominators

Denominators can have linear factors (like x - 1) or irreducible quadratic factors (like x² + 1). Each type requires a different form in the decomposition: linear factors correspond to constants in the numerator, while irreducible quadratics require linear expressions in the numerator.

Formulating the Decomposition Without Solving Constants

When asked to write the form of the partial fraction decomposition without solving for constants, you set up the sum of fractions with unknown coefficients in the numerators according to the factor types. This step focuses on structure rather than finding specific values.