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Chapter 5: Problem 33

Write the partial fraction decomposition of each rational expression. $$\frac{x+4}{x^{2}\left(x^{2}+4\right)}$$

Short Answer

Expert verified
The partial fraction decomposition of \(\frac{x+4}{x^{2}(x^{2}+4)}\) is \(\frac{1}{x^{2}} + \frac{x+1}{x^{2}+4}\).

Step by step solution

01

Identify the form of the denominator

The given expression is of the form \(\frac{A}{x^{2}} + \frac{Bx+C}{x^{2}+4}\). This is because the denominator \(x^2(x^{2}+4)\) contains two separate terms, namely \(x^2\) and \(x^2 + 4\). Each of these terms contributes to a component in the decomposition.
02

Write the partial fraction decomposition

Next, write down the form the decomposition must take: \(\frac{x+4}{x^{2}(x^{2}+4)} = \frac{A}{x^{2}} + \frac{Bx+C}{x^{2}+4}\).
03

Clear the fractions

Subtract \(A/(x^2)\) and \((Bx+C)/(x^2+4)\) from both sides and multiply every term by \(x^2(x^2 + 4)\) to clear the fractions: \(x+4 = A(x^2 + 4) + Bx(x^2) + Cx^2\).
04

Solve the resulting equation

Solving the resulting equation gives A = 1, B = 0, C = 1.
05

Substituting A, B, and C into the decomposition

Substitute the values of A, B, and C to the decomposition, we get \(\frac{x+4}{x^{2}(x^{2}+4)} = \frac{1}{x^2} + \frac{x+1}{x^2+4}\).

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