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Chapter 5: Problem 24

Write the partial fraction decomposition of each rational expression. $$\frac{2 x^{2}+8 x+3}{(x+1)^{3}}$$

Short Answer

Expert verified
The partial fraction decomposition of \( \frac{2 x^{2}+8 x+3}{(x+1)^{3}} \) is \( \frac{1}{x + 1} + \frac{2}{(x + 1)^2} \).

Step by step solution

01

Identifying fractions

The function is given by \( \frac{2 x^{2} + 8x + 3}{(x+1)^{3}} \). To decompose this into partial fractions, we need to set it equal to a sum of simpler fractions. Since the denominator is a cube, we will have three terms. The simplest form of fractions will be \( \frac{A}{x + 1} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)^3} \) where A, B and C are constants to be determined.
02

Equating coefficients

Multiplying both sides by the common denominator \( (x + 1)^3 \), we get \( 2x^2 + 8x + 3 = A(x + 1)^2 + B(x + 1) + C \). This equation must hold for all values of x. Choose convenient values for x to solve for A, B and C.
03

Solving for coefficients

Let's start with x = -1. This simplifies the right side to C and the left becomes \( 2x^2 + 8x + 3 = 0 \). Thus, C = 0. With C = 0, we now have two terms left on the right side. Substituting x = 0, we get \( A + B = 3 \). Selecting another value for x, let us take x = 1. Substituting x = 1, we get \( A + 2B = 9 \). Now solve this system of equations to find A and B. After solving the equations, we get A = 1 and B = 2.
04

Writing out the decomposition

Substitute the values of A, B, and C back into the fractions, one will get the partial fractions for the original rational expression: \( \frac{1}{x + 1} + \frac{2}{(x + 1)^2} \).

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