Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{6x^2-14x-27}{(x+2)(x-3{)}^2}$

Verified step by step guidance

1. Identify the denominator factors and their types. Here, the denominator is \((x+2)(x-3)^2\), which consists of a linear factor \((x+2)\) and a repeated linear factor \((x-3)^2\).
2. Write a separate term for each factor in the denominator. For the linear factor \((x+2)\), the corresponding term in the partial fraction decomposition is \(\frac{A}{x+2}\), where \(A\) is a constant to be determined.
3. For the repeated linear factor \((x-3)^2\), write terms for each power up to the multiplicity. This means you write \(\frac{B}{x-3} + \frac{C}{(x-3)^2}\), where \(B\) and \(C\) are constants to be determined.
4. Combine all terms to write the full form of the partial fraction decomposition as: \(\frac{A}{x+2} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\).
5. Note that you do not need to solve for \(A\), \(B\), and \(C\); just write the form of the decomposition as shown.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions whose denominators are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factorization of the Denominator

Understanding how to factor the denominator into linear and repeated factors is essential. In this problem, the denominator is already factored as (x + 2)(x - 3)^2, indicating a linear factor and a repeated linear factor, which affects the form of the partial fractions.

Form of Partial Fractions for Repeated Factors

When the denominator contains repeated linear factors like (x - 3)^2, the partial fraction decomposition includes terms for each power of the repeated factor. Specifically, you write separate fractions with denominators (x - 3) and (x - 3)^2, each with its own constant numerator.