Certainly! Let's simplify the given polynomial step-by-step:
1. Starting Expression:
[tex]\[
t^3 (8 + 9i) - (t^2 + 4)(t^2 - 3t)
\][/tex]
2. Expand Each Term:
- For the first term:
[tex]\[
t^3 (8 + 9i) = 8t^3 + 9it^3
\][/tex]
- For the second term, perform the multiplication separately:
[tex]\[
(t^2 + 4)(t^2 - 3t) = t^2 \cdot t^2 + t^2 \cdot (-3t) + 4 \cdot t^2 + 4 \cdot (-3t)
\][/tex]
Simplify the multiplication:
[tex]\[
= t^4 - 3t^3 + 4t^2 - 12t
\][/tex]
3. Combine All Terms Together:
[tex]\[
8t^3 + 9it^3 - t^4 + 3t^3 - 4t^2 + 12t
\][/tex]
4. Combine Like Terms:
- Combine the [tex]\( t^3 \)[/tex] terms:
[tex]\[
8t^3 + 9it^3 + 3t^3 = 11t^3 + 9it^3
\][/tex]
- The remaining terms are:
[tex]\[
- t^4 - 4t^2 + 12t
\][/tex]
5. Final Simplified Expression:
[tex]\[
-t^4 + (11 + 9i)t^3 - 4t^2 + 12t
\][/tex]
Now, let's evaluate each of the statements with the finalized expression:
The simplified expression is a polynomial.
- True: The simplified expression [tex]\(-t^4 + (11 + 9i)t^3 - 4t^2 + 12t\)[/tex] is indeed a polynomial.
The simplified expression has a constant term.
- False: There is no constant term (a term without [tex]\( t \)[/tex]) in the simplified expression.
The simplified expression is cubic.
- False: The polynomial is not cubic; it is quartic (the highest power of [tex]\( t \)[/tex] is 4).
The simplified expression is a trinomial.
- False: The expression has more than three terms.
The simplified expression has four terms.
- True: The simplified polynomial has four terms: [tex]\(-t^4\)[/tex], [tex]\((11 + 9i)t^3\)[/tex], [tex]\(-4t^2\)[/tex], and [tex]\(12t\)[/tex].
So, the correct statements are:
- The simplified expression is a polynomial.
- The simplified expression has four terms.
