To find the completely factored form of [tex]\( t^4 - 16 \)[/tex], let's go through it step-by-step.
1. Recognize the given polynomial: [tex]\( t^4 - 16 \)[/tex].
2. Identify if there are any patterns or factorizations that we can apply:
- Notice that [tex]\( t^4 - 16 \)[/tex] can be written as a difference of squares: [tex]\( t^4 - 16 = (t^2)^2 - 4^2 \)[/tex].
3. Apply the difference of squares formula: [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
- Using [tex]\( a = t^2 \)[/tex] and [tex]\( b = 4 \)[/tex], we can rewrite the expression as:
[tex]\[
t^4 - 16 = (t^2 - 4)(t^2 + 4)
\][/tex]
4. Factor the quadratic term [tex]\( t^2 - 4 \)[/tex] further:
- Notice that [tex]\( t^2 - 4 \)[/tex] is also a difference of squares: [tex]\( t^2 - 4 = (t - 2)(t + 2) \)[/tex].
5. Combine all factored terms:
- Now we have factored [tex]\( t^2 - 4 \)[/tex] into [tex]\( (t - 2)(t + 2) \)[/tex] and multiply by the remaining factor [tex]\( t^2 + 4 \)[/tex]. So we have:
[tex]\[
t^4 - 16 = (t - 2)(t + 2)(t^2 + 4)
\][/tex]
The completely factored form of [tex]\( t^4 - 16 \)[/tex] is:
[tex]\[
(t - 2)(t + 2)(t^2 + 4)
\][/tex]
So, the correct answer is:
[tex]\[
\left(t^2 + 4\right)(t + 2)(t - 2)
\][/tex]
