`x^3-512`

Answer

To factor the expression \( x^3 - 512 \), we can follow these steps:

### Step 1: Recognize the expression as a difference of cubes
The expression \( x^3 - 512 \) can be recognized as a difference of cubes. We can rewrite \( 512 \) as \( 8^3 \):
\[ x^3 - 512 = x^3 - 8^3 \]

### Step 2: Apply the difference of cubes formula
The difference of cubes can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In this case, let \( a = x \) and \( b = 8 \).

### Step 3: Substitute \( a \) and \( b \) into the formula
Using the values of \( a \) and \( b \):
\[ x^3 - 8^3 = (x - 8)(x^2 + x \cdot 8 + 8^2) \]

### Step 4: Simplify the second factor
Now we simplify the second factor:
\[ x^2 + 8x + 64 \]

### Step 5: Write the final factored form
Putting it all together, we have:
\[ x^3 - 512 = (x - 8)(x^2 + 8x + 64) \]

Thus, the final answer is:
\[ \boxed{(x - 8)(x^2 + 8x + 64)} \]

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