The value of `(1)/( a^(2) + ax + x^(2)) - (1)/( a^(2) - ax + x^(2)) + ( 2ax)/( a^(4) + a^(2) x^(2) + x^(4))` is

Options

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

Answer

To solve the expression

\[ \frac{1}{a^2 + ax + x^2} - \frac{1}{a^2 - ax + x^2} + \frac{2ax}{a^4 + a^2x^2 + x^4}, \]

we will simplify it step by step.

### Step 1: Find a common denominator for the first two fractions

The common denominator for the first two fractions is

\[ (a^2 + ax + x^2)(a^2 - ax + x^2). \]

### Step 2: Rewrite the first two fractions with the common denominator

We rewrite the first two fractions as:

\[ \frac{(a^2 - ax + x^2) - (a^2 + ax + x^2)}{(a^2 + ax + x^2)(a^2 - ax + x^2)}. \]

### Step 3: Simplify the numerator

Now, simplify the numerator:

\[ (a^2 - ax + x^2) - (a^2 + ax + x^2) = -2ax. \]

So, we have:

\[ \frac{-2ax}{(a^2 + ax + x^2)(a^2 - ax + x^2)}. \]

### Step 4: Add the third term

Now, we need to add the third term:

\[ \frac{-2ax}{(a^2 + ax + x^2)(a^2 - ax + x^2)} + \frac{2ax}{a^4 + a^2x^2 + x^4}. \]

### Step 5: Find a common denominator for all three terms

The common denominator for all three fractions is

\[ (a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4). \]

### Step 6: Rewrite all fractions with the common denominator

The first term becomes:

\[ \frac{-2ax(a^4 + a^2x^2 + x^4)}{(a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4)}. \]

The second term becomes:

\[ \frac{2ax(a^2 + ax + x^2)(a^2 - ax + x^2)}{(a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4)}. \]

### Step 7: Combine the fractions

Now we can combine the two fractions:

\[ \frac{-2ax(a^4 + a^2x^2 + x^4) + 2ax(a^2 + ax + x^2)(a^2 - ax + x^2)}{(a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4)}. \]

### Step 8: Simplify the numerator

Factor out \(2ax\) from the numerator:

\[ \frac{2ax\left[-(a^4 + a^2x^2 + x^4) + (a^2 + ax + x^2)(a^2 - ax + x^2)\right]}{(a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4)}. \]

### Step 9: Expand and simplify the expression inside the brackets

Now, we need to simplify the expression inside the brackets.

When you expand \((a^2 + ax + x^2)(a^2 - ax + x^2)\), you will find that it simplifies to \(a^4 + a^2x^2 + x^4\).

Thus,

\[ -(a^4 + a^2x^2 + x^4) + (a^4 + a^2x^2 + x^4) = 0. \]

### Final Result

This means that the entire expression simplifies to:

\[ \frac{0}{(a^2 + ax + x^2)(a^2 - ax + x^2)(a^4 + a^2x^2 + x^4)} = 0. \]

### Conclusion

Therefore, the value of the given expression is

\[ \boxed{0}. \]