Select all the correct answers.

Which statements are true about the result of simplifying this polynomial?

[tex]\[ t^3(8 + 9i) - (t^2 + 4)(t^2 - 3t) \][/tex]

A. The simplified expression is a polynomial.
B. The simplified expression has a constant term.
C. The simplified expression is cubic.
D. The simplified expression is a trinomial.
E. The simplified expression has four terms.



Answer :

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.