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 cineplexs total profits in that city?

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

Explanation:

Sum the profit expressions

$$ P_{\text{total}}(t) = (t^3 - t^2 + 2t - 100) + (t^2 - 2t - 300) $$

$$ P_{\text{total}}(t) = t^3 + (-t^2 + t^2) + (2t - 2t) + (-100 - 300) $$

$$ P_{\text{total}}(t) = t^3 - 400 $$

Analyze the resulting expression

• The expression \(t^3 - 400\) is a polynomial.
• It has two terms (a binomial), so it is not a binomial is false? Wait, a binomial is a polynomial with exactly two terms. Thus, "is a binomial" is true.
• It has a constant term, which is \(-400\).
• It is a cubic polynomial (degree 3), so it is not quadratic.
• It has two terms, so "has four terms" is false.

Evaluate the statements

• "The total profit expression has a constant term." -> True (constant is \(-400\)).
• "The total profit expression is a binomial." -> True (it has exactly two terms: \(t^3\) and \(-400\)).
• "The total profit expression is a polynomial." -> True (all binomials are polynomials).
• "The total profit expression has four terms." -> False (it has two terms).
• "The total profit expression is quadratic." -> False (it is cubic, degree 3).

Answer:

• The total profit expression has a constant term. (Correct answer)
• The total profit expression is a binomial. (Correct answer)
• The total profit expression is a polynomial. (Correct answer)
• The total profit expression has four terms.
• The total profit expression is quadratic.