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 denominator. Here, the denominator is \( (x - 2)^3 \), which is a repeated linear factor raised to the third power.
2. Set up the partial fraction decomposition with terms for each power of the repeated factor. Since the denominator is \( (x - 2)^3 \), the decomposition will have three terms: \( \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} \), where \(A\), \(B\), and \(C\) are constants to be determined.
3. Write the equation equating the original rational expression to the sum of the partial fractions: \[ \frac{x^2 - 6x + 3}{(x - 2)^3} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} \]
4. Multiply both sides of the equation by \( (x - 2)^3 \) to clear the denominators, resulting in: \[ x^2 - 6x + 3 = A(x - 2)^2 + B(x - 2) + C \]
5. Expand the right side and then equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\), \(B\), and \(C\).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including factoring and simplifying, is essential before performing 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 Linear Factors in Denominators

When the denominator has repeated linear factors, such as (x − 2)³, the decomposition includes terms with increasing powers of that factor in the denominator. Each power from 1 up to the highest exponent must be included as separate terms in the decomposition.