Degree of the polynomial `(x^(2)-sqrt(1-x^(3)))^(4)+(x^(2)+sqrt(1-x^(3)))^(4)` is

Options

A. 7
B. 8
C. 9
D. 12

Answer

To find the degree of the polynomial \( (x^2 - \sqrt{1 - x^3})^4 + (x^2 + \sqrt{1 - x^3})^4 \), we can follow these steps:

### Step 1: Identify the components of the polynomial
Let:
- \( a = x^2 \)
- \( b = \sqrt{1 - x^3} \)

Then, we can rewrite the polynomial as:
\[ (a - b)^4 + (a + b)^4 \]

### Step 2: Expand both terms using the Binomial Theorem
Using the Binomial Theorem, we can expand both \( (a - b)^4 \) and \( (a + b)^4 \).

The expansion of \( (a - b)^4 \) is:
\[ (a - b)^4 = \sum_{k=0}^{4} \binom{4}{k} a^{4-k} (-b)^k = \binom{4}{0} a^4 + \binom{4}{1} a^3 (-b) + \binom{4}{2} a^2 b^2 + \binom{4}{3} a (-b)^3 + \binom{4}{4} (-b)^4 \]

The expansion of \( (a + b)^4 \) is:
\[ (a + b)^4 = \sum_{k=0}^{4} \binom{4}{k} a^{4-k} b^k = \binom{4}{0} a^4 + \binom{4}{1} a^3 b + \binom{4}{2} a^2 b^2 + \binom{4}{3} a b^3 + \binom{4}{4} b^4 \]

### Step 3: Combine the expansions
Now, we add the two expansions:
\[ (a - b)^4 + (a + b)^4 = 2a^4 + 2\binom{4}{2} a^2 b^2 \]
The odd-powered terms will cancel out because they will have opposite signs.

### Step 4: Substitute back the values of \( a \) and \( b \)
Substituting back \( a = x^2 \) and \( b = \sqrt{1 - x^3} \):
\[ = 2(x^2)^4 + 2 \cdot \binom{4}{2} (x^2)^2 (1 - x^3) \]
Calculating \( \binom{4}{2} = 6 \):
\[ = 2x^8 + 12x^4(1 - x^3) \]
Expanding \( 12x^4(1 - x^3) \):
\[ = 12x^4 - 12x^7 \]

### Step 5: Combine all terms
Now, we combine all the terms:
\[ = 2x^8 - 12x^7 + 12x^4 \]

### Step 6: Determine the degree of the polynomial
The highest degree term in the polynomial \( 2x^8 - 12x^7 + 12x^4 \) is \( 2x^8 \). Therefore, the degree of the polynomial is:
\[ \text{Degree} = 8 \]

### Final Answer
The degree of the polynomial is \( 8 \).