In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x²-6x+3)/(x − 2)³

Verified step by step guidance

1. Identify the form of the rational expression: the numerator is a polynomial of degree 2, and the denominator is \( (x - 2)^3 \), a repeated linear factor.
2. Since the denominator is \( (x - 2)^3 \), set up the partial fraction decomposition with terms for each power of \( (x - 2) \): \( \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} \), where A, B, and C are constants to be determined.
3. Write the equation: \[ \frac{x^2 - 6x + 3}{(x - 2)^3} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} \]. Multiply both sides by \( (x - 2)^3 \) to clear the denominators, resulting in \[ x^2 - 6x + 3 = A(x - 2)^2 + B(x - 2) + C \].
4. Expand the right-hand side: first expand \( (x - 2)^2 \) to get \( x^2 - 4x + 4 \), then multiply by A and add the other terms to express the right side as a polynomial in standard form.
5. Equate the coefficients of corresponding powers of \( x \) from both sides to form a system of equations for A, B, and C. Solve this system to find the values of A, B, and C, completing the partial fraction decomposition.

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 irreducible quadratic factors.

Repeated Linear Factors

When the denominator contains repeated linear factors, such as (x - 2)³, the partial fraction decomposition includes terms for each power of the factor up to its multiplicity. For example, terms with denominators (x - 2), (x - 2)², and (x - 2)³ are included, each with its own constant numerator.

Polynomial Division

If the degree of the numerator is equal to or greater than the degree of the denominator, polynomial division is performed first to rewrite the expression as a polynomial plus a proper fraction. This step ensures the rational expression is in a form suitable for partial fraction decomposition.