Question

select all the correct answers

cineplex operates two movie theaters in a city. the profits from one theater can be represented by the expression $t^3 - t^2 + 2t - 100$, where $t$ is the number of tickets sold. the profits from the second theater can be represented by the expression $t^2 - 2t - 300$
which statements are true about the expression representing cineplex’s total profits in that city?

• the total profit expression is a polynomial.
• the total profit expression is a binomial
• the total profit expression has a constant term.
• the total profit expression has four terms.
• the total profit expression is quadratic.

Explanation:

Step1: Find total profit expression

Add the two profit expressions: \((t^3 - t^2 + 2t - 100) + (t^2 - 2t - 300)\)
Simplify: \(t^3 - t^2 + t^2 + 2t - 2t - 100 - 300 = t^3 - 400\)

Step2: Analyze each statement

• "The total profit expression is a polynomial": A polynomial is an expression with non - negative integer exponents. \(t^3 - 400\) has exponents 3 and 0 (for the constant term), so it is a polynomial. True.
• "The total profit expression is a binomial": A binomial has two terms. \(t^3 - 400\) has two terms (\(t^3\) and \(-400\)), so it is a binomial. True.
• "The total profit expression has a constant term": The constant term is \(-400\), so it has a constant term. True.
• "The total profit expression has four terms": After simplification, it has two terms (\(t^3\) and \(-400\)), so this is False.
• "The total profit expression is quadratic": A quadratic has degree 2. The expression \(t^3 - 400\) has degree 3, so it is not quadratic. False.

Answer:

• The total profit expression is a polynomial.
• The total profit expression is a binomial.
• The total profit expression has a constant term.