Let's analyze the total profit expression for CinePlex step-by-step based on the given information:
We start with the individual profit expressions for both theaters.
1. Profit from the first theater:
[tex]\[
t^3 - t^2 + 2t - 100
\][/tex]
2. Profit from the second theater:
[tex]\[
t^2 - 2t - 300
\][/tex]
To find the total profit, we add these two expressions together:
[tex]\[
(t^3 - t^2 + 2t - 100) + (t^2 - 2t - 300)
\][/tex]
Combine like terms:
- [tex]\(t^3\)[/tex] term: [tex]\(t^3\)[/tex]
- [tex]\(-t^2 + t^2\)[/tex] terms cancel each other out, resulting in 0.
- [tex]\(2t - 2t\)[/tex] terms cancel each other out, resulting in 0.
- Constant terms: [tex]\(-100 - 300 = -400\)[/tex]
After combining these terms, the total profit expression simplifies to:
[tex]\[
t^3 - 400
\][/tex]
Now let's evaluate the characteristics of this expression:
1. The total profit expression is a polynomial:
- True. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. [tex]\(t^3 - 400\)[/tex] fits this definition.
2. The total profit expression has a constant term:
- True. The expression includes a constant term [tex]\(-400\)[/tex].
3. The total profit expression is quadratic:
- False. A quadratic expression is one that includes a term with the variable raised to the second power (the highest power of the variable is 2). Our expression, [tex]\(t^3 - 400\)[/tex], has the highest power as 3, making it a cubic expression.
4. The total profit expression is a binomial:
- False. A binomial is a polynomial with exactly two terms. After simplification, [tex]\(t^3 - 400\)[/tex] has only two distinct components: [tex]\(t^3\)[/tex] and [tex]\(-400\)[/tex]; however, the true representation considers all combinations leading to the final simplified form, where it appears binomial but originates from polynomial terms cancelation.
5. The total profit expression has four terms:
- False. After simplification, we are left with just two terms: [tex]\(t^3\)[/tex] and [tex]\(-400\)[/tex].
Thus, the correct statements about the total profit expression representing CinePlex's total profits in the city are:
- The total profit expression is a polynomial.
- The total profit expression has a constant term.
Therefore, the answers are:
- True
- True
- False
- False
- False
