Sum of coefficients of terms of odd powers of in `(1+x - x^2 -x^3)^8`is

Options

A. a) 0
B. b) 1
C. c) 2
D. d) `-1`

Answer

To find the sum of the coefficients of the terms of odd powers in the expression \( (1 + x - x^2 - x^3)^8 \), we can use the property that the sum of the coefficients of the terms of odd powers can be calculated using the formula:

\[ \text{Sum of coefficients of odd powers} = \frac{f(1) - f(-1)}{2} \]

where \( f(x) = (1 + x - x^2 - x^3)^8 \).

### Step 1: Calculate \( f(1) \)

We start by substituting \( x = 1 \) into the function \( f(x) \):

\[ f(1) = (1 + 1 - 1^2 - 1^3)^8 \]

Calculating inside the parentheses:

\[ 1 + 1 - 1 - 1 = 0 \]

Thus,

\[ f(1) = 0^8 = 0 \]

### Step 2: Calculate \( f(-1) \)

Next, we substitute \( x = -1 \):

\[ f(-1) = (1 - 1 - (-1)^2 - (-1)^3)^8 \]

Calculating inside the parentheses:

\[ 1 - 1 - 1 + 1 = 0 \]

Thus,

\[ f(-1) = 0^8 = 0 \]

### Step 3: Use the formula to find the sum of coefficients of odd powers

Now we can substitute \( f(1) \) and \( f(-1) \) into the formula:

\[ \text{Sum of coefficients of odd powers} = \frac{f(1) - f(-1)}{2} = \frac{0 - 0}{2} = 0 \]

### Final Answer

The sum of the coefficients of the terms of odd powers in \( (1 + x - x^2 - x^3)^8 \) is \( \boxed{0} \).

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