Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.