The sum of the coefficients of all odd degree terms in the expansion of `(x+sqrt(x^(3)-1))^(5)+(x-sqrt(x^(3)-1))^(5), (xgt1)` is :

Options

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

Answer

To solve the problem, we need to find the sum of the coefficients of all odd degree terms in the expansion of the expression:

\[ (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5 \]

### Step 1: Expand the expression using the Binomial Theorem
We can use the Binomial Theorem to expand both parts of the expression. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

Applying this to both expansions:

1. For \( (x + \sqrt{x^3 - 1})^5 \):
\[ (x + \sqrt{x^3 - 1})^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (\sqrt{x^3 - 1})^k \]

2. For \( (x - \sqrt{x^3 - 1})^5 \):
\[ (x - \sqrt{x^3 - 1})^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (-\sqrt{x^3 - 1})^k \]

### Step 2: Combine the two expansions
Adding these two expansions together, we notice that the terms with odd \( k \) will cancel out because they will have opposite signs, while the terms with even \( k \) will add up.

Thus, we only need to consider the even \( k \) terms:

\[ (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5 = 2 \sum_{k \text{ even}} \binom{5}{k} x^{5-k} (\sqrt{x^3 - 1})^k \]

### Step 3: Identify the even \( k \) values
The even values of \( k \) from 0 to 5 are \( k = 0, 2, 4 \).

### Step 4: Calculate the contributions from the even \( k \) values
1. For \( k = 0 \):
\[ 2 \cdot \binom{5}{0} x^5 = 2 \cdot 1 \cdot x^5 = 2x^5 \]

2. For \( k = 2 \):
\[ 2 \cdot \binom{5}{2} x^3 (x^3 - 1) = 2 \cdot 10 x^3 (x^3 - 1) = 20x^3 (x^3 - 1) = 20x^6 - 20x^3 \]

3. For \( k = 4 \):
\[ 2 \cdot \binom{5}{4} x (x^3 - 1)^2 = 2 \cdot 5 x (x^3 - 1)^2 = 10x (x^6 - 2x^3 + 1) = 10x^7 - 20x^4 + 10x \]

### Step 5: Combine all contributions
Now, we combine all the contributions:

\[ 2x^5 + (20x^6 - 20x^3) + (10x^7 - 20x^4 + 10x) \]

This simplifies to:

\[ 10x^7 + 20x^6 - 20x^4 + 2x^5 - 20x^3 + 10x \]

### Step 6: Identify the odd degree terms
The odd degree terms in the final expression are:
- \( -20x^3 \)
- \( 10x \)

### Step 7: Sum the coefficients of the odd degree terms
Now, we sum the coefficients of the odd degree terms:

\[ -20 + 10 = -10 \]

Thus, the sum of the coefficients of all odd degree terms in the expansion is:

\[ \boxed{-10} \]