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Chapter 5: Problem 37

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

Short Answer

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

Step by step solution

01

Identify the form of the denominator

Since the denominator is \((x^{2}+2)^{2}\), the partial fraction decomposition of the rational expression will be in the form \(A/(x^{2}+2) + Bx/(x^{2}+2)^{2}\). We will need to solve for the values of A and B.
02

Equate the rational expression and fractions

We now set the given expression equal to the sum of partial fractions, then multiply through by the common denominator to clear the fractions: \(x^{3} + x^{2} + 2 = A(x^{2} + 2) + Bx\). This equation should hold for all values of x, so let's choose some x values to solve for A and B.
03

Find the values of A and B

First, let's set x=0: 2=A*2 or A=1. \n Next, differentiate both sides of the equation \(x^{3} + x^{2} + 2 = A(x^{2} + 2) + Bx\) to get: \n \(3x^{2} + 2x = 2Ax + B\). Now we also set x=0 to solve for B: 0 = 2B so B=0.

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