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Chapter 7: Problem 38

Give the appropriate form of the partial fraction decomposition for the following functions. $$\frac{2}{x\left(x^{2}-6 x+9\right)}$$

Short Answer

Expert verified
Answer: The appropriate form of the partial fraction decomposition for the given function is $$\frac{2}{x\left(x^{2}-6 x+9\right)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{(x-3)^{2}}$$, where A, B, and C are constants.

Step by step solution

01

Factor the denominator

The first step in finding the appropriate form of the partial fraction decomposition is to factor the denominator. We are given the denominator as: $$x\left(x^{2}-6 x+9\right)$$ The quadratic expression can be factored as a perfect square, so we get: $$x\left((x-3)^{2}\right)$$
02

Set up the partial fraction decomposition

Now that we have factored the denominator, we can set up the partial fraction decomposition. The appropriate form for our decomposition will have a term for each factor in the denominator. In our case, we have factors of \(x\) and \((x-3)^{2}\). Our partial fraction decomposition will have the following form: $$\frac{2}{x\left((x-3)^{2}\right)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{(x-3)^{2}}$$ where A, B, and C are constants that we need to determine.
03

Find the general solution

Since we are only required to find the appropriate form of the partial fraction decomposition, we do not need to find the values for A, B, and C. Instead, we can present our general solution as follows: The appropriate form of the partial fraction decomposition for the given function is: $$\frac{2}{x\left(x^{2}-6 x+9\right)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{(x-3)^{2}}$$

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