The expression `(sqrt(2x^2+1)+sqrt(2x^2-1))^6 + (2/(sqrt(2x^2+1)+sqrt(2x^2-1)))^6` is polynomial of degree

Options

A. 6
B. 8
C. 10
D. 12

Explanation

To solve the problem, we need to analyze the expression:

\[ f(x) = \left(\sqrt{2x^2 + 1} + \sqrt{2x^2 - 1}\right)^6 + \left(\frac{2}{\sqrt{2x^2 + 1} + \sqrt{2x^2 - 1}}\right)^6 \]

### Step 1: Simplify the expression

First, let's denote:

\[ a = \sqrt{2x^2 + 1} \quad \text{and} \quad b = \sqrt{2x^2 - 1} \]

So, we can rewrite the expression as:

\[ f(x) = (a + b)^6 + \left(\frac{2}{a + b}\right)^6 \]

### Step 2: Evaluate \( a + b \)

Next, we need to evaluate \( a + b \):

\[ a + b = \sqrt{2x^2 + 1} + \sqrt{2x^2 - 1} \]

### Step 3: Evaluate \( \frac{2}{a + b} \)

Now, we can find \( \frac{2}{a + b} \):

\[ \frac{2}{a + b} = \frac{2}{\sqrt{2x^2 + 1} + \sqrt{2x^2 - 1}} \]

To simplify this, we can multiply the numerator and denominator by \( \sqrt{2x^2 + 1} - \sqrt{2x^2 - 1} \):

\[ \frac{2(\sqrt{2x^2 + 1} - \sqrt{2x^2 - 1})}{(2x^2 + 1) - (2x^2 - 1)} = \frac{2(\sqrt{2x^2 + 1} - \sqrt{2x^2 - 1})}{2} \]

This simplifies to:

\[ \sqrt{2x^2 + 1} - \sqrt{2x^2 - 1} \]

### Step 4: Substitute back into \( f(x) \)

Now substituting back into \( f(x) \):

\[ f(x) = (a + b)^6 + (a - b)^6 \]

### Step 5: Apply the Binomial Theorem

Using the Binomial Theorem, we can expand both \( (a + b)^6 \) and \( (a - b)^6 \):

\[ (a + b)^6 = \sum_{k=0}^{6} \binom{6}{k} a^{6-k} b^k \]
\[ (a - b)^6 = \sum_{k=0}^{6} \binom{6}{k} a^{6-k} (-b)^k \]

Adding these two expansions, we notice that the odd powers of \( b \) will cancel out:

\[ f(x) = 2 \sum_{k \text{ even}} \binom{6}{k} a^{6-k} b^k \]

### Step 6: Determine the degree of \( f(x) \)

The maximum degree of \( f(x) \) will occur when \( k = 0 \) and \( k = 6 \):

- For \( k = 0 \): \( a^6 = (2x^2 + 1)^3 \) contributes \( 6 \) (degree).
- For \( k = 6 \): \( b^6 = (2x^2 - 1)^3 \) contributes \( 6 \) (degree).

Thus, the maximum degree of \( f(x) \) is:

\[ \text{Degree of } f(x) = 6 \]

### Final Answer

The expression is a polynomial of degree **6**.