The coefficient of `x^(4)` in the expansion of `(1-x-2x^(2))^(8)` is

Options

A. 144
B. `-144`
C. `154`
D. `-154`

Answer

To find the coefficient of \( x^4 \) in the expansion of \( (1 - x - 2x^2)^8 \), we can use the multinomial theorem. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the terms in the multinomial expansion
The expression can be rewritten as:
\[ (1 + (-x) + (-2x^2))^8 \]
Let \( a = 1 \), \( b = -x \), and \( c = -2x^2 \). We want to find the coefficient of \( x^4 \) in the expansion of \( (a + b + c)^8 \).

### Step 2: Use the multinomial theorem
The general term in the expansion is given by:
\[ T = \frac{n!}{\alpha! \beta! \gamma!} a^\alpha b^\beta c^\gamma \]
where \( n = 8 \) and \( \alpha + \beta + \gamma = 8 \).

### Step 3: Determine the conditions for \( x^4 \)
We need to find combinations of \( \beta \) and \( \gamma \) such that the total power of \( x \) equals 4:
- The term \( b^\beta = (-x)^\beta \) contributes \( \beta \) to the power of \( x \).
- The term \( c^\gamma = (-2x^2)^\gamma \) contributes \( 2\gamma \) to the power of \( x \).

Thus, we need:
\[ \beta + 2\gamma = 4 \]

### Step 4: Solve for possible values of \( \beta \) and \( \gamma \)
We also have the equation:
\[ \alpha + \beta + \gamma = 8 \]

Now we can analyze different cases based on the values of \( \gamma \):

#### Case 1: \( \gamma = 0 \)
- Then \( \beta + 2(0) = 4 \) implies \( \beta = 4 \).
- Thus, \( \alpha = 8 - 4 - 0 = 4 \).
- The term is:
\[ T = \frac{8!}{4!4!0!} (1)^4 (-x)^4 (-2x^2)^0 = \frac{8!}{4!4!} (-1)^4 x^4 = \frac{8!}{4!4!} x^4 \]

#### Case 2: \( \gamma = 1 \)
- Then \( \beta + 2(1) = 4 \) implies \( \beta = 2 \).
- Thus, \( \alpha = 8 - 2 - 1 = 5 \).
- The term is:
\[ T = \frac{8!}{5!2!1!} (1)^5 (-x)^2 (-2x^2)^1 = \frac{8!}{5!2!1!} (-1)^2 (-2) x^4 = -2 \cdot \frac{8!}{5!2!1!} x^4 \]

#### Case 3: \( \gamma = 2 \)
- Then \( \beta + 2(2) = 4 \) implies \( \beta = 0 \).
- Thus, \( \alpha = 8 - 0 - 2 = 6 \).
- The term is:
\[ T = \frac{8!}{6!0!2!} (1)^6 (-x)^0 (-2x^2)^2 = \frac{8!}{6!0!2!} (4) x^4 = 4 \cdot \frac{8!}{6!2!} x^4 \]

### Step 5: Calculate the coefficients
Now we calculate the coefficients from each case:

1. **Case 1:**
\[ \frac{8!}{4!4!} = 70 \quad \text{(coefficient is } 70) \]

2. **Case 2:**
\[ -2 \cdot \frac{8!}{5!2!1!} = -2 \cdot \frac{8 \cdot 7}{2 \cdot 1} = -56 \quad \text{(coefficient is } -112) \]

3. **Case 3:**
\[ 4 \cdot \frac{8!}{6!2!} = 4 \cdot \frac{8 \cdot 7}{2 \cdot 1} = 112 \quad \text{(coefficient is } 112) \]

### Step 6: Combine the coefficients
Now, we sum the coefficients from all cases:
\[ 70 - 112 + 112 = 70 \]

### Final Answer
The coefficient of \( x^4 \) in the expansion of \( (1 - x - 2x^2)^8 \) is \( 70 \).