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Chapter 12: Problem 11

Write the partial fraction decomposition for the expression. $$ \frac{8 x^{2}+15 x+9}{(x+1)^{3}} $$

Short Answer

Expert verified
The partial fraction decomposition of the given expression is \[ \frac{-6}{x + 1} + \frac{22}{(x + 1)^2} + \frac{8}{(x + 1)^3} \]

Step by step solution

01

Set up the equation

Everyone can write out the partial fractions decomposition form as follows: \[ \frac{8 x^{2}+15 x+9}{(x+1)^{3}} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \]
02

Clear the denominator

Multiply through by the common denominator \((x+1)^3\) to eliminate the fractions. This results in the following equation: \[ 8x^2 + 15x + 9 = A(x+1)^2 + B(x+1) + C \]
03

Solve for A, B, and C

The parameters can be determined by picking values for \(x\). A convenient choice is \(x = -1\), as this will cancel out terms to solve for \(C\). This gives \(C=8\). Next, differentiate the equation twice to solve for \(A\) and \(B\). Differentiating the original expression yields \[ 16x + 15 = 2A(x + 1) + B, \] and setting \(x = -1\) gives \(A = -6\). Differentiating a second time gives \[16 = 2A,\] and substititing \(A = -6\) gives \(B = 22\).
04

Write out the partial fractions decomposition

Now that the parameters \(A\), \(B\), and \(C\) have been found, one can substitute them into the decomposition from step 1 to obtain the result: \[ \frac{-6}{x + 1} + \frac{22}{(x + 1)^2} + \frac{8}{(x + 1)^3} \]

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Most popular questions from this chapter

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{1}^{2} \frac{1}{x} d x, n=4 $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{3} \frac{1}{2-2 x+x^{2}} d x, n=6 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} \frac{1}{x+1} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=4 $$

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x \sqrt{x+4} d x $$
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