The number of partial fractions of `(x^(3)-3x^(2)+3x)/((x-1)^(5))` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the number of partial fractions of the expression \(\frac{x^3 - 3x^2 + 3x}{(x-1)^5}\), we can follow these steps:

### Step 1: Identify the Denominator
The denominator is \((x-1)^5\).

### Step 2: Determine the Form of Partial Fractions
For a denominator of the form \((x-a)^n\), the partial fraction decomposition will have the following form:
\[ \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \frac{A_3}{(x-a)^3} + \ldots + \frac{A_n}{(x-a)^n} \]
where \(A_1, A_2, \ldots, A_n\) are constants.

### Step 3: Count the Number of Terms
Since our denominator is \((x-1)^5\), we will have:
- One term for \((x-1)^1\)
- One term for \((x-1)^2\)
- One term for \((x-1)^3\)
- One term for \((x-1)^4\)
- One term for \((x-1)^5\)

This gives us a total of 5 terms.

### Step 4: Conclusion
Thus, the number of partial fractions for the expression \(\frac{x^3 - 3x^2 + 3x}{(x-1)^5}\) is **5**.

### Final Answer
The number of partial fractions is **5**.

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