Select all the correct answers.

CinePlex operates two movie theaters in a city. The profits from one theater can be represented by the expression [tex]t^3 - t^2 + 2t - 100[/tex], where [tex]t[/tex] is the number of tickets sold. The profits from the second theater can be represented by the expression [tex]t^2 - 2t - 300[/tex].

Which statements are true about the expression representing CinePlex's total profits in that city?

A. The total profit expression is a binomial.
B. The total profit expression is quadratic.
C. The total profit expression has four terms.
D. The total profit expression has a constant term.
E. The total profit expression is a polynomial.



Answer :

Let's analyze the problem step by step to determine the true statements about the total profit expression.

Given:
- Profits from the first theater: [tex]\( t^3 - t^2 + 2t - 100 \)[/tex]
- Profits from the second theater: [tex]\( t^2 - 2t - 300 \)[/tex]

To get the total profit, we need to sum these two expressions:

[tex]\[ \text{Total Profit} = (t^3 - t^2 + 2t - 100) + (t^2 - 2t - 300) \][/tex]

Simplifying the expression by combining like terms:

[tex]\[ \text{Total Profit} = t^3 - t^2 + t^2 + 2t - 2t - 100 - 300 \][/tex]
[tex]\[ \text{Total Profit} = t^3 - 400 \][/tex]

So, the total profit expression is [tex]\( t^3 - 400 \)[/tex].

Now, let's verify each of the given statements about this expression:

1. The total profit expression is a binomial.
- A binomial is a polynomial with exactly two terms. In this case, [tex]\( t^3 - 400 \)[/tex] has two terms. Therefore, this statement is true.

2. The total profit expression is quadratic.
- A quadratic expression is a polynomial of degree 2. The degree of the polynomial [tex]\( t^3 - 400 \)[/tex] is 3. Therefore, this statement is false.

3. The total profit expression has four terms.
- The total profit expression [tex]\( t^3 - 400 \)[/tex] consists of two terms, not four. Therefore, this statement is false.

4. The total profit expression has a constant term.
- The expression [tex]\( t^3 - 400 \)[/tex] includes the constant term -400. Therefore, this statement is true.

5. The total profit expression is a polynomial.
- A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. The term [tex]\( t^3 - 400 \)[/tex] fits this definition. Therefore, this statement is true.

The correct statements about the total profit expression are:
- The total profit expression is a binomial.
- The total profit expression has a constant term.
- The total profit expression is a polynomial.