Find the number of partial fractions of `(x+2)/(x^(2)(x^(2)-1))`

Answer

To find the number of partial fractions for the expression \(\frac{x+2}{x^2(x^2-1)}\), we will follow a systematic approach.

### Step 1: Factor the Denominator
The first step is to factor the denominator completely. The denominator is given as:
\[ x^2(x^2 - 1) \]
We can factor \(x^2 - 1\) further:
\[ x^2 - 1 = (x - 1)(x + 1) \]
Thus, the complete factorization of the denominator is:
\[ x^2(x - 1)(x + 1) \]

### Step 2: Identify the Types of Factors
Now, we identify the types of factors in the denominator:
1. \(x^2\) - This is a repeated linear factor.
2. \(x - 1\) - This is a linear factor.
3. \(x + 1\) - This is also a linear factor.

### Step 3: Write the Form of Partial Fractions
According to the rules of partial fractions, we can express the fraction as follows:
- For the factor \(x^2\), we will have:
\[ \frac{A}{x} + \frac{B}{x^2} \]
- For the factor \(x - 1\), we will have:
\[ \frac{C}{x - 1} \]
- For the factor \(x + 1\), we will have:
\[ \frac{D}{x + 1} \]

Thus, the complete form of the partial fractions will be:
\[ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1} + \frac{D}{x + 1} \]

### Step 4: Count the Number of Partial Fractions
Now, we can count the number of distinct partial fractions we have:
1. \(\frac{A}{x}\)
2. \(\frac{B}{x^2}\)
3. \(\frac{C}{x - 1}\)
4. \(\frac{D}{x + 1}\)

In total, we have **4 partial fractions**.

### Final Answer
The number of partial fractions for the expression \(\frac{x + 2}{x^2(x^2 - 1)}\) is **4**.

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