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 denominator and its factors. Here, the denominator is \( (x^{2} - 2x + 3)^{2} \), which is a repeated irreducible quadratic factor since \( x^{2} - 2x + 3 \) cannot be factored further over the reals.
2. Set up the form of the partial fraction decomposition. For a repeated irreducible quadratic factor \( (ax^{2} + bx + c)^{2} \), the decomposition includes terms with linear numerators over each power of the quadratic factor. So, write:
3. \[ \frac{x^{3} - 4x^{2} + 9x - 5}{(x^{2} - 2x + 3)^{2}} = \frac{Ax + B}{x^{2} - 2x + 3} + \frac{Cx + D}{(x^{2} - 2x + 3)^{2}} \]
4. Multiply both sides of the equation by the denominator \( (x^{2} - 2x + 3)^{2} \) to clear the fractions:
5. \[ x^{3} - 4x^{2} + 9x - 5 = (Ax + B)(x^{2} - 2x + 3) + (Cx + D) \]
6. Expand the right-hand side and then collect like terms by powers of \( x \). This will give you a polynomial equation where the coefficients of corresponding powers of \( x \) on both sides must be equal. From this, you can set up a system of equations to solve for \( A, B, C, \) and \( D \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex rational expression into a sum of fractions with simpler denominators, typically linear or quadratic factors.

Repeated Quadratic Factors in Denominators

When the denominator contains a repeated irreducible quadratic factor, such as (x² - 2x + 3)², the decomposition includes terms with the quadratic factor raised to increasing powers. Each term has a numerator that is a linear polynomial, reflecting the irreducible quadratic nature.

Degree Comparison Between Numerator and Denominator

Before decomposing, ensure the degree of the numerator is less than the degree of the denominator. If not, perform polynomial division first. In this problem, the numerator is degree 3 and the denominator degree 4, so decomposition can proceed directly.