If ` a + (1)/(a) + 1 = 0 (a ne 0)` then the value of ` (a^(4) - a)` is

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To solve the equation \( a + \frac{1}{a} + 1 = 0 \) and find the value of \( a^4 - a \), we can follow these steps:

### Step 1: Rearrange the given equation
Start with the equation:
\[ a + \frac{1}{a} + 1 = 0 \]
Rearranging gives:
\[ a + \frac{1}{a} = -1 \]

### Step 2: Cube both sides
Next, we will cube both sides of the equation. Recall the identity:
\[ \left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3\left(a + \frac{1}{a}\right) \]
Substituting \( k = a + \frac{1}{a} \):
\[ k^3 = a^3 + \frac{1}{a^3} + 3k \]
Substituting \( k = -1 \):
\[ (-1)^3 = a^3 + \frac{1}{a^3} + 3(-1) \]
This simplifies to:
\[ -1 = a^3 + \frac{1}{a^3} - 3 \]
Thus:
\[ a^3 + \frac{1}{a^3} = -1 + 3 = 2 \]

### Step 3: Express \( a^4 \) in terms of \( a^3 \)
We know that:
\[ a^4 = a \cdot a^3 \]
So we can express \( a^4 - a \) as:
\[ a^4 - a = a \cdot a^3 - a = a(a^3 - 1) \]

### Step 4: Substitute \( a^3 \)
From our previous result, we have \( a^3 + \frac{1}{a^3} = 2 \). This implies:
\[ a^3 = 2 - \frac{1}{a^3} \]
However, we already know \( a^3 = 1 \) is a valid solution since \( a^3 + \frac{1}{a^3} = 2 \) when \( a^3 = 1 \).

### Step 5: Calculate \( a^4 - a \)
Substituting \( a^3 = 1 \):
\[ a(a^3 - 1) = a(1 - 1) = a \cdot 0 = 0 \]

### Final Answer
Thus, the value of \( a^4 - a \) is:
\[ \boxed{0} \]