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Chapter 8: Problem 60

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x+4}{x^{2}(3 x-1)^{2}}$$

Short Answer

Expert verified
\(\frac{x+4}{x^{2}(3x-1)^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{3x-1} + \frac{D}{(3x-1)^{2}}\)

Step by step solution

01

Identify the distinct factors

In this case, the denominator of the fraction is \(x^{2}(3x-1)^{2}\). Here, \(x\) and \(3x-1\) are distinct factors.
02

Determine the general form for repeated factors

The general form for repeated factors is \(\frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{3x-1} + \frac{D}{(3x-1)^{2}}\), where A, B, C, and D are constants to be defined.
03

Compose the final form

Putting it all together, the expression can be written as \(\frac{x+4}{x^{2}(3x-1)^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{3x-1} + \frac{D}{(3x-1)^{2}}\). This is the form of the partial fraction decomposition. It is not necessary to solve for the constants A, B, C, and D, since the problem doesn't ask for it.

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