In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (6x^2-14x-27)/(x+2) (x − 3)^2

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1. Identify the form of the rational expression: \( \frac{6x^2 - 14x - 27}{(x+2)(x-3)^2} \).
2. Recognize that the denominator consists of a linear factor \((x+2)\) and a repeated linear factor \((x-3)^2\).
3. Set up the partial fraction decomposition: \( \frac{6x^2 - 14x - 27}{(x+2)(x-3)^2} = \frac{A}{x+2} + \frac{B}{x-3} + \frac{C}{(x-3)^2} \).
4. Understand that \(A\), \(B\), and \(C\) are constants that need to be determined.
5. Note that solving for these constants involves multiplying through by the common denominator and equating coefficients, but this step is not required for this exercise.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of this question.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The decomposition involves breaking down the rational expression based on the factors of the denominator, which can include linear and irreducible quadratic factors.

Polynomial Degree and Factorization

The degree of a polynomial is the highest power of the variable in the expression. In partial fraction decomposition, it is essential to factor the denominator completely and understand the degrees of the factors to set up the correct form of the decomposition. This includes recognizing repeated factors, which will affect the structure of the resulting partial fractions.