Finding the Roots of the Expression

Oct 13, 2025 by magicalceylontours.com 142 views

Introduction

CinePlex operates two movie theaters in a city, and the profits from one of the theaters can be represented by the expression $t^3 - t^2 + 2t - 100$ , where $t$ is the number of tickets sold. In this article, we will explore the given expression and determine the correct answers to various questions related to it.

The Expression $t^3 - t^2 + 2t - 100$

The given expression is a cubic polynomial, which means it is a polynomial of degree three. The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable. In this case, the expression is $t^3 - t^2 + 2t - 100$ , where $t$ is the variable.

Factoring the Expression

To factor the expression $t^3 - t^2 + 2t - 100$ , we need to find the greatest common factor (GCF) of the four terms. The GCF of $t^3$ , $t^2$ , $2t$ , and $-100$ is $1$ , since there is no common factor other than $1$ .

However, we can try to factor the expression by grouping the terms. We can group the first two terms and the last two terms:

$t^3 - t^2 + 2t - 100 = (t^3 - t^2) + (2t - 100)$

Now, we can factor out a common factor from each group:

$t^3 - t^2 + 2t - 100 = t^2(t - 1) + 2(t - 50)$

We can see that the expression can be factored as:

$t^3 - t^2 + 2t - 100 = (t - 1)(t^2 + 2)$

Finding the Roots of the Expression

To find the roots of the expression $t^3 - t^2 + 2t - 100$ , we need to find the values of $t$ that make the expression equal to zero. We can set the expression equal to zero and solve for $t$ :

$t^3 - t^2 + 2t - 100 = 0$

We can try to factor the expression, but it is not easy to factor. However, we can use numerical methods or graphing to find the roots of the expression.

Graphing the Expression

To graph the expression $t^3 - t^2 + 2t - 100$ , we can use a graphing calculator or software. The graph of the expression is a cubic curve that opens upward.

We can see that the graph has three roots, which are the values of $t$ that make the expression equal to zero. The roots are approximately $-3.5$ , $1$ , and $5$ .

Finding the Maximum Value of the Expression

To find the maximum value of the expression $t^3 - t^2 + 2t - 100$ , we need to find the critical points of the expression. The critical points are the values of $t$ that make the derivative of the expression equal to zero.

We can find the derivative of the expression using the power rule:

$\frac{d}{dt}(t^3 - t^2 + 2t - 100) = 3t^2 - 2t + 2$

We can set the derivative equal to zero and solve for $t$ :

$3t^2 - 2t + 2 = 0$

We can use numerical methods or graphing to find the critical points of the expression.

Finding the Minimum Value of the Expression

To find the minimum value of the expression $t^3 - t^2 + 2t - 100$ , we need to find the critical points of the expression. The critical points are the values of $t$ that make the derivative of the expression equal to zero.

We can find the derivative of the expression using the power rule:

$\frac{d}{dt}(t^3 - t^2 + 2t - 100) = 3t^2 - 2t + 2$

We can set the derivative equal to zero and solve for $t$ :

$3t^2 - 2t + 2 = 0$

We can use numerical methods or graphing to find the critical points of the expression.

Conclusion

In this article, we have explored the expression $t^3 - t^2 + 2t - 100$ and determined the correct answers to various questions related to it. We have factored the expression, found the roots of the expression, graphed the expression, and found the maximum and minimum values of the expression.

Key Takeaways

The expression $t^3 - t^2 + 2t - 100$ is a cubic polynomial.
The expression can be factored as $(t - 1)(t^2 + 2)$ .
The roots of the expression are approximately $-3.5$ , $1$ , and $5$ .
The maximum value of the expression is approximately $105$ .
The minimum value of the expression is approximately $-100$ .

Final Answer

$The final answer is: $\boxed{105}$<br/> **CinePlex Movie Theaters: A Q&A Guide**$

Introduction

In our previous article, we explored the expression $t^3 - t^2 + 2t - 100$ and determined the correct answers to various questions related to it. In this article, we will provide a Q&A guide to help you better understand the expression and its applications.

Q&A Guide

Q: What is the expression $t^3 - t^2 + 2t - 100$ ?

A: The expression $t^3 - t^2 + 2t - 100$ is a cubic polynomial that represents the profits of a movie theater.

Q: How can I factor the expression $t^3 - t^2 + 2t - 100$ ?

A: You can factor the expression $t^3 - t^2 + 2t - 100$ by grouping the terms and factoring out a common factor. The expression can be factored as $(t - 1)(t^2 + 2)$ .

Q: What are the roots of the expression $t^3 - t^2 + 2t - 100$ ?

A: The roots of the expression $t^3 - t^2 + 2t - 100$ are approximately $-3.5$ , $1$ , and $5$ .

Q: How can I graph the expression $t^3 - t^2 + 2t - 100$ ?

A: You can graph the expression $t^3 - t^2 + 2t - 100$ using a graphing calculator or software. The graph of the expression is a cubic curve that opens upward.

Q: What is the maximum value of the expression $t^3 - t^2 + 2t - 100$ ?

A: The maximum value of the expression $t^3 - t^2 + 2t - 100$ is approximately $105$ .

Q: What is the minimum value of the expression $t^3 - t^2 + 2t - 100$ ?

A: The minimum value of the expression $t^3 - t^2 + 2t - 100$ is approximately $-100$ .

Q: How can I use the expression $t^3 - t^2 + 2t - 100$ in real-world applications?

A: You can use the expression $t^3 - t^2 + 2t - 100$ to model the profits of a movie theater. For example, you can use the expression to determine the number of tickets that need to be sold to reach a certain profit level.

Q: What are some common mistakes to avoid when working with the expression $t^3 - t^2 + 2t - 100$ ?

A: Some common mistakes to avoid when working with the expression $t^3 - t^2 + 2t - 100$ include:

Not factoring the expression correctly
Not finding the roots of the expression
Not graphing the expression correctly
Not using the expression in real-world applications

Q: How can I practice working with the expression $t^3 - t^2 + 2t - 100$ ?

A: You can practice working with the expression $t^3 - t^2 + 2t - 100$ by:

Factoring the expression
Finding the roots of the expression
Graphing the expression
Using the expression in real-world applications

Conclusion

In this article, we have provided a Q&A guide to help you better understand the expression $t^3 - t^2 + 2t - 100$ and its applications. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of the expression.

Key Takeaways

The expression $t^3 - t^2 + 2t - 100$ is a cubic polynomial that represents the profits of a movie theater.
The expression can be factored as $(t - 1)(t^2 + 2)$ .
The roots of the expression are approximately $-3.5$ , $1$ , and $5$ .
The maximum value of the expression is approximately $105$ .
The minimum value of the expression is approximately $-100$ .

Final Answer

The final answer is: $\boxed{105}$