If ` a + a^(2) + a^(3) - 1 =0 ` then what is the value of ` a^(3) + (1//a)` ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( a + a^2 + a^3 - 1 = 0 \) and find the value of \( a^3 + \frac{1}{a} \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ a + a^2 + a^3 - 1 = 0 \]
Rearranging gives us:
\[ a + a^2 + a^3 = 1 \tag{1} \]

**Hint:** Rearranging the equation helps isolate the terms we need to work with.

### Step 2: Express \( a^3 + \frac{1}{a} \)
We need to find \( a^3 + \frac{1}{a} \). We can express \( \frac{1}{a} \) in terms of \( a \) using the equation we derived:
\[ \frac{1}{a} = \frac{1 - (a + a^2)}{a} \]

**Hint:** Use the equation from Step 1 to express \( \frac{1}{a} \) in terms of \( a \).

### Step 3: Substitute \( \frac{1}{a} \) into the expression
Now, substituting \( \frac{1}{a} \) back into \( a^3 + \frac{1}{a} \):
\[ a^3 + \frac{1}{a} = a^3 + \frac{1 - (a + a^2)}{a} \]
This simplifies to:
\[ a^3 + \frac{1}{a} = a^3 + \frac{1}{a} - (1 + a + a^2) = a^3 + 1 - (a + a^2) \]

**Hint:** Simplifying the expression will help in finding the final value.

### Step 4: Use equation (1) to simplify further
From equation (1), we know:
\[ a + a^2 = 1 - a^3 \]
Substituting this back gives:
\[ a^3 + \frac{1}{a} = a^3 + 1 - (1 - a^3) = 2a^3 \]

**Hint:** This step involves substituting back the known values from previous steps.

### Step 5: Find \( a^3 \)
From equation (1), we can also express \( a^3 \):
\[ a^3 = 1 - (a + a^2) \]
We can substitute \( a + a^2 \) from equation (1) into this expression.

### Step 6: Final value
Now we can evaluate \( a^3 + \frac{1}{a} \) directly:
\[ a^3 + \frac{1}{a} = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]