Sure, let's go through the factorization process for the expression [tex]\( t^4 - 81 \)[/tex].
### Step-by-Step Solution:
1. Identify the Given Polynomial:
The expression we need to factorize is [tex]\( t^4 - 81 \)[/tex].
2. Recognize the Form:
Notice that [tex]\( t^4 - 81 \)[/tex] can be seen as a difference of squares. Recall that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
3. Apply the Difference of Squares Formula:
We can rewrite [tex]\( t^4 - 81 \)[/tex] as [tex]\( (t^2)^2 - 9^2 \)[/tex]. This is in the form of [tex]\( a^2 - b^2 \)[/tex], where [tex]\( a = t^2 \)[/tex] and [tex]\( b = 9 \)[/tex].
Hence,
[tex]\[
(t^2)^2 - 9^2 = (t^2 - 9)(t^2 + 9)
\][/tex]
4. Further Factorize:
Notice that [tex]\( t^2 - 9 \)[/tex] is itself a difference of squares:
[tex]\[
t^2 - 9 = (t - 3)(t + 3)
\][/tex]
So, we can continue factorizing:
[tex]\[
t^4 - 81 = (t^2 - 9)(t^2 + 9) = (t - 3)(t + 3)(t^2 + 9)
\][/tex]
5. Identify the Correct Option:
Comparing the fully factorized form [tex]\( (t - 3)(t + 3)(t^2 + 9) \)[/tex] to the given options:
- Option A: [tex]\((t-3)(t+3)^2\)[/tex]
- Option B: [tex]\((t-3)(t+3)\left(t^2+9\right)\)[/tex]
- Option C: [tex]\(\left(r^2-9\right)\left(x^2-9\right)\)[/tex]
- Option D: [tex]\((t-3)^2(t+3)^2\)[/tex]
The correct factorization [tex]\( (t - 3)(t + 3)(t^2 + 9) \)[/tex] matches Option B.
Therefore, the correct answer is:
B. [tex]\((t-3)(t+3)\left(t^2+9\right)\)[/tex]
