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Chapter 5: Problem 22

Write the partial fraction decomposition of each rational expression. $$\frac{x}{(x+1)^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of the rational expression \( \frac{x}{(x+1)^{2}} \) is \( \frac{1}{x+1} - \frac{1}{(x+1)^2} \).

Step by step solution

01

Set Up the Equation

Set up an equation of the original rational expression equal to its partial fractions: \( \frac{x}{(x+1)^{2}} = \frac{A}{x+1} + \frac{B}{(x+1)^2} \). In this form, A and B are the constants that we need to find.
02

Clearing the Denominator

Multiply both sides by the common denominator, \( (x+1)^2 \), to eliminate the fractions, which gives us \( x = A*(x+1) + B \).
03

Solving for A and B

First, set \( x = -1 \) in the above equation to find the value of B. Because the term with A goes to zero, we are left with \( B = -1 \). Then, to find A, equate the coefficients. The x-coefficient on the left is 1, and on the right, it's A, thus \( A = 1 \).
04

Writing the Partial Fraction Decomposition

Substitute the values of A and B back into the equation from Step 1 to get the partial fraction decomposition: \( \frac{x}{(x+1)^{2}} = \frac{1}{x+1} - \frac{1}{(x+1)^2} \).

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