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.
