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

Options

A. 1
B. 2
C. 3
D. can't be determined

Answer

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

### Step 1: Rearranging the Equation
First, we need to move all terms to one side of the equation:

\[ n^3 - n^2 - n - n = 0 \]

This simplifies to:

\[ n^3 - n^2 - 2n = 0 \]

### Step 2: Factoring Out Common Terms
Next, we can factor out \( n \) from the equation:

\[ n(n^2 - n - 2) = 0 \]

### Step 3: Factoring the Quadratic
Now, we need to factor the quadratic \( n^2 - n - 2 \). We look for two numbers that multiply to \(-2\) (the constant term) and add to \(-1\) (the coefficient of \( n \)). The numbers \(-2\) and \(1\) work:

\[ n^2 - n - 2 = (n - 2)(n + 1) \]

### Step 4: Setting Each Factor to Zero
Now we set each factor equal to zero:

1. \( n = 0 \)
2. \( n - 2 = 0 \) → \( n = 2 \)
3. \( n + 1 = 0 \) → \( n = -1 \)

### Step 5: Listing the Solutions
The solutions to the equation are:

- \( n = 0 \)
- \( n = 2 \)
- \( n = -1 \)

### Conclusion
Thus, the number of values of \( n \) that satisfy the given relation is **3**.

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