In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x³-4x²+9x-5)/(x² -2x+3)²

Verified step by step guidance

1. Identify the form of the denominator. Here, the denominator is \( (x^2 - 2x + 3)^2 \), which is a repeated irreducible quadratic factor.
2. Set up the partial fraction decomposition with terms corresponding to each power of the repeated quadratic. Since the quadratic is squared, write two terms: one over \( x^2 - 2x + 3 \) and one over \( (x^2 - 2x + 3)^2 \). Each numerator should be a linear polynomial because the denominator is quadratic and irreducible. So, write \( \frac{Ax + B}{x^2 - 2x + 3} + \frac{Cx + D}{(x^2 - 2x + 3)^2} \).
3. Multiply both sides of the equation by the denominator \( (x^2 - 2x + 3)^2 \) to clear the fractions. This will give an equation involving polynomials on both sides.
4. Expand and simplify the right-hand side polynomial expression, then collect like terms of powers of \( x \).
5. Equate the coefficients of corresponding powers of \( x \) from both sides to form a system of linear equations. Solve this system to find the values of \( A, B, C, \) and \( D \).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate and simplify these expressions is essential before performing operations like partial fraction decomposition.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions with denominators that are factors of the original denominator. This technique is useful for integration and solving equations involving rational expressions.

Repeated Quadratic Factors

When the denominator contains repeated irreducible quadratic factors, the decomposition includes terms with increasing powers of these quadratics in the denominator, each with linear numerators. Recognizing and handling these repeated factors is crucial for correct decomposition.