If `a^3 + a^2 - a - 1 = 0`, then which one of the following could be the value of a?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( a^3 + a^2 - a - 1 = 0 \), we can follow these steps:

### Step 1: Group the terms
We start with the equation:
\[ a^3 + a^2 - a - 1 = 0 \]
We can group the terms as follows:
\[ (a^3 + a^2) + (-a - 1) = 0 \]

### Step 2: Factor by grouping
Next, we can factor out common terms from each group:
\[ a^2(a + 1) - 1(a + 1) = 0 \]
Now, we can factor out \( (a + 1) \):
\[ (a^2 - 1)(a + 1) = 0 \]

### Step 3: Further factor the quadratic
The expression \( a^2 - 1 \) can be factored further using the difference of squares:
\[ (a - 1)(a + 1)(a + 1) = 0 \]

### Step 4: Set each factor to zero
Now, we set each factor equal to zero:
1. \( a - 1 = 0 \) → \( a = 1 \)
2. \( a + 1 = 0 \) → \( a = -1 \)

### Step 5: List the possible values
The possible values for \( a \) are:
\[ a = 1 \quad \text{or} \quad a = -1 \]

### Conclusion
From the given options, the value of \( a \) that satisfies the equation is \( a = 1 \).

### Final Answer
Thus, the correct answer is option \( b \): \( a = 1 \).

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