Divide :
`x^(6) - 8` by `x^(2) - 2`

Answer

To divide \( x^6 - 8 \) by \( x^2 - 2 \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We want to divide \( x^6 - 8 \) by \( x^2 - 2 \). We can write this as:

\[ \frac{x^6 - 8}{x^2 - 2} \]

### Step 2: Divide the leading terms
The leading term of the dividend \( x^6 \) is divided by the leading term of the divisor \( x^2 \):

\[ \frac{x^6}{x^2} = x^4 \]

### Step 3: Multiply and subtract
Now, multiply \( x^4 \) by the entire divisor \( x^2 - 2 \):

\[ x^4 \cdot (x^2 - 2) = x^6 - 2x^4 \]

Now subtract this from the original polynomial:

\[ (x^6 - 8) - (x^6 - 2x^4) = 2x^4 - 8 \]

### Step 4: Repeat the process
Now, we will divide the new leading term \( 2x^4 \) by \( x^2 \):

\[ \frac{2x^4}{x^2} = 2x^2 \]

Multiply \( 2x^2 \) by the divisor \( x^2 - 2 \):

\[ 2x^2 \cdot (x^2 - 2) = 2x^4 - 4x^2 \]

Now subtract this from \( 2x^4 - 8 \):

\[ (2x^4 - 8) - (2x^4 - 4x^2) = 4x^2 - 8 \]

### Step 5: Repeat again
Now, divide \( 4x^2 \) by \( x^2 \):

\[ \frac{4x^2}{x^2} = 4 \]

Multiply \( 4 \) by the divisor \( x^2 - 2 \):

\[ 4 \cdot (x^2 - 2) = 4x^2 - 8 \]

Now subtract this from \( 4x^2 - 8 \):

\[ (4x^2 - 8) - (4x^2 - 8) = 0 \]

### Final Result
Since the remainder is \( 0 \), we have completed the division. The result of dividing \( x^6 - 8 \) by \( x^2 - 2 \) is:

\[ x^4 + 2x^2 + 4 \]

### Summary
The final answer is:

\[ \frac{x^6 - 8}{x^2 - 2} = x^4 + 2x^2 + 4 \]