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

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

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Addition", "Combining Like Terms", "Polynomial Classification" ], "new_concepts": [], "current_concepts": [ "Polynomial Addition", "Combining Like Terms", "Polynomial Classification" ] } ## Sum the two profit expressions $$ \begin{aligned} P_{\text{total}}(t) &= (t^3 - t^2 + 2t - 100) + (t^2 - 2t - 300) \ &= t^3 + (-t^2 + t^2) + (2t - 2t) + (-100 - 300) \ &= t^3 - 400 \end{aligned} $$ ## Analyze the resulting expression $$ \begin{aligned} &\text{Expression: } t^3 - 400 \ &\text{Number of terms: } 2 \quad (\text{binomial}) \ &\text{Classification: } \text{cubic polynomial with a constant term of } -400 \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{"The total profit expression has four terms."} \quad ightarrow \text{False (it has two terms)} \ &\text{"The total profit expression is a polynomial."} \quad ightarrow \text{True} \ &\text{"The total profit expression has a constant term."} \quad ightarrow \text{True (constant is } -400) \ &\text{"The total profit expression is quadratic."} \quad ightarrow \text{False (it is cubic, degree 3)} \ &\text{"The total profit expression is a binomial."} \quad ightarrow \text{True (it has exactly two terms)} \end{aligned} $$

The total profit expression has four terms.

The total profit expression is a polynomial.

The total profit expression has a constant term.

The total profit expression is quadratic.

The total profit expression is a binomial.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Addition" ] }

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Explanation:

Sum the two profit expressions

Analyze the resulting expression

Evaluate the given statements

Answer: