If `(n)^3-(n)^2-n = n`, then the number of values of n that satisfy the given relation is :

Options

A. A) 1
B. B) 2
C. C) 3
D. D) CND

Answer

To solve the equation \( n^3 - n^2 - n = n \), we can follow these steps:

### Step-by-Step Solution:

1. **Rearrange the Equation:**
Start by moving all terms to one side of the equation:
\[ n^3 - n^2 - n - n = 0 \]
This simplifies to:
\[ n^3 - n^2 - 2n = 0 \]

2. **Factor Out Common Terms:**
Notice that \( n \) is a common factor in all terms:
\[ n(n^2 - n - 2) = 0 \]

3. **Set Each Factor to Zero:**
This gives us two cases to consider:
- Case 1: \( n = 0 \)
- Case 2: \( n^2 - n - 2 = 0 \)

4. **Solve the Quadratic Equation:**
For the quadratic equation \( n^2 - n - 2 = 0 \), we can factor it:
\[ (n - 2)(n + 1) = 0 \]
This gives us two additional solutions:
- \( n - 2 = 0 \) → \( n = 2 \)
- \( n + 1 = 0 \) → \( n = -1 \)

5. **List All Solutions:**
The solutions we found are:
- \( n = 0 \)
- \( n = 2 \)
- \( n = -1 \)

6. **Count the Number of Solutions:**
We have three values of \( n \) that satisfy the equation: \( 0, 2, -1 \).

### Final Answer:
The number of values of \( n \) that satisfy the given relation is **3**.

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