`x^(6)-9^(3)`=______

Options

A. `(x^(2)+9)(x^(4)-9x^(2)-81)`
B. `(x^(2)-9)(x^(4)-9x^(2)+81)`
C. `(x+3)(x-3)(x^(4)+9x^(2)+81)`
D. `(x^(2)+9)(x^(4)+9x^(2)+81)`

Answer

To factorize the expression \( x^6 - 9^3 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^6 - 9^3 \]
We can rewrite \( x^6 \) as \( (x^2)^3 \) and \( 9^3 \) as \( (3^2)^3 = 27 \). Thus, we have:
\[ (x^2)^3 - (3)^3 \]

### Step 2: Apply the difference of cubes formula
We can use the difference of cubes formula, which states that:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, let \( a = x^2 \) and \( b = 3 \). Applying the formula, we get:
\[ (x^2 - 3)(x^4 + 3x^2 + 9) \]

### Step 3: Factor the first part further
The first factor \( x^2 - 3 \) can be recognized as a difference of squares:
\[ x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3}) \]
However, since we are looking for integer factors, we will leave it as \( x^2 - 3 \).

### Step 4: Final expression
Thus, the fully factored form of the expression \( x^6 - 9^3 \) is:
\[ (x^2 - 3)(x^4 + 3x^2 + 9) \]

### Summary
The final answer is:
\[ (x^2 - 3)(x^4 + 3x^2 + 9) \]