To solve the given problem, [tex]\( x^4 - 81 \)[/tex], we need to factorize the polynomial completely. Let's start by recognizing the pattern in the polynomial.
1. Recognize the Structure: Difference of Squares
The expression [tex]\( x^4 - 81 \)[/tex] is a difference of squares because it can be written as:
[tex]\[
x^4 - 81 = (x^2)^2 - 9^2
\][/tex]
According to the difference of squares formula, [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex]:
[tex]\[
(x^2)^2 - 9^2 = (x^2 + 9)(x^2 - 9)
\][/tex]
2. Factorize Further: [tex]\( x^2 - 9 \)[/tex]
Next, we can further factorize [tex]\( x^2 - 9 \)[/tex]. Noting that [tex]\( x^2 - 9 \)[/tex] is again a difference of squares:
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
3. Handle [tex]\( x^2 + 9 \)[/tex]
The term [tex]\( x^2 + 9 \)[/tex] can also be factorized using complex numbers because it can be rewritten as:
[tex]\[
x^2 + 9 = x^2 + (3i)^2 = (x + 3i)(x - 3i)
\][/tex]
where [tex]\( i \)[/tex] is the imaginary unit satisfying [tex]\( i^2 = -1 \)[/tex].
4. Combine All Factors
Bringing together all these factors, the complete factorization of [tex]\( x^4 - 81 \)[/tex] is:
[tex]\[
x^4 - 81 = (x^2 + 9)(x^2 - 9) = (x + 3i)(x - 3i)(x + 3)(x - 3)
\][/tex]
This factorization matches the polynomial factorization provided in Option D:
[tex]\[
(x + 3)(x - 3)(x + 3i)(x - 3i)
\][/tex]
Therefore, the correct answer is
[tex]\[
\boxed{D}
\][/tex]
