Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.