Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x + 1)/(x + 2)^3

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Identify the form of the denominator. Since the denominator is $ (x + 2)^3 $, which is a repeated linear factor, the partial fraction decomposition will include terms with denominators $ (x + 2) $, $ (x + 2)^2 $, and $ (x + 2)^3 $.

Set up the partial fraction decomposition as follows: \[ \frac{2x + 1}{(x + 2)^3} = \frac{A}{x + 2} + \frac{B}{(x + 2)^2} + \frac{C}{(x + 2)^3} \] where $ A $, $ B $, and $ C $ are constants to be determined.

Multiply both sides of the equation by the common denominator $ (x + 2)^3 $ to clear the denominators: \[ 2x + 1 = A(x + 2)^2 + B(x + 2) + C \].

Expand the right-hand side by first expanding $ (x + 2)^2 $ to $ x^2 + 4x + 4 $, then distribute $ A $, and combine like terms to express the right side as a polynomial in $ x $.

Equate the coefficients of corresponding powers of $ x $ from both sides of the equation to form a system of equations. Solve this system to find the values of $ A $, $ B $, and $ C $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or solving equations. It involves breaking down a fraction into components with simpler denominators, often linear or 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.

Setting Up and Solving Equations for Coefficients

To find the unknown numerators in the decomposition, multiply both sides by the common denominator to clear fractions, then equate coefficients of corresponding powers of x. This results in a system of linear equations that can be solved to determine the constants in the partial fractions.

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