Contextualization

Polynomials are algebraic expressions that hold significant importance in Mathematics and its applications. They are composed of terms, which can contain variables and constants, raised to a natural number, including zero. Through polynomials, it is possible to perform various operations such as addition, subtraction, multiplication, and division, and understand fundamental aspects of Algebra.

Polynomials and their roots play a crucial role in solving many mathematical problems. The roots of a polynomial, also known as zeros, are the values of the variable that make the polynomial equal to zero. They determine the solution of polynomial equations, which are essential tools for solving practical problems in various areas such as Physics, Engineering, and Economics.

The study of polynomials and their roots also brings up other important concepts, such as the Fundamental Theorem of Algebra, which states that every polynomial of degree n has exactly n complex roots, and Vieta's Theorem, which establishes a relationship between the coefficients of a polynomial and its roots.

Polynomials are used in a wide variety of real-world applications. This includes the mathematical modeling of physical phenomena and engineering systems, as well as data manipulation in statistics and computer science. The roots of polynomials are particularly important in applications involving the analysis of dynamic systems, such as predicting the behavior of a system over time.

Furthermore, as we will see in this project, the manipulation and resolution of polynomials help develop essential skills in logical thinking and problem-solving. This also allows us to enhance our understanding of other mathematical topics, such as Complex Numbers, which will be integrated into our study.

To deepen their knowledge of polynomials and their roots, students can consult the following resources:

• Book "Fundamentos de Matemática Elementar" by Gelson Iezzi and Carlos Murakami. This book is an excellent source to understand the fundamental principles of algebra and polynomials.
• Khan Academy (link): Free online platform with a variety of videos and interactive exercises on the topic. The Algebra section is full of resources on polynomials and their roots.
• YouTube channel "Matemática Rio" (link): Channel focused on teaching mathematics in a light and fun way, with a series of videos covering polynomials.

Practical Activity

Activity Title: "Enigma of Polynomial Roots: An Interdisciplinary Journey"

Project Objective:

Through an interactive challenge involving mathematics, physics, and programming, students will have the mission to solve a fictional enigma that involves the calculation and application of polynomial roots. The project aims to stimulate a deep understanding of the topic, as well as promote collaboration, teamwork, and the practical application of the concepts learned.

Detailed Project Description:

The project will be divided into three interconnected stages:

1. In-Depth Theoretical Study on Polynomials and their Roots: Students will conduct research and detailed studies on the topic, including concepts such as the Fundamental Theorem of Algebra and Vieta's Theorem. A group presentation containing all the learned concepts will be created.

2. Solve the Enigma: A fictional problem will be proposed where the calculation and applications of polynomial roots will be necessary to solve it. The problem will involve a free fall situation, in which it will be necessary to identify the time it takes for an object to fall, using knowledge of physics and mathematics.

3. Programming the Enigma: Students will use a programming language (Python recommended for being more accessible to beginners) to simulate the proposed enigma, to confirm the conclusions obtained analytically.

Required Materials:

• High school Mathematics and Physics books
• Internet access for research
• Computers with Python installed for programming tasks

Detailed Step-by-Step for Activity Completion:

1. Form groups of 3 to 5 students. This large project should be divided among team members and will require more than 12 hours of work for each student.

2. In the first stage of the project, each group will conduct detailed research and in-depth study on polynomials and their roots. Each group member should prepare a part of the presentation covering a specified subtopic, ensuring that all members have a solid understanding of the concepts.

3. After the elaboration and presentation of the theoretical concepts, a fictional problem will be proposed. The situation will involve simulating the free fall of an object, and students will need to determine when this object will hit the ground. To do this, they must use their mathematical skills to model the situation with a polynomial and then find its roots.

4. In the final stage, students will need to program the situation in Python to validate their calculations. They will need to create a program that simulates the object's fall and verifies the correspondence with the calculations made in the previous stage.

5. Finally, students will write a detailed report on the project. The report should contain four main sections: Introduction, Development, Conclusions, and Bibliography.

• In the Introduction, students should contextualize the theme of polynomials and their roots, mentioning their relevance and application in the real world.

• In the Development, students should present the theory learned about polynomials and their roots, explain the activity in detail, the methodology used, and the results obtained.

• In the Conclusions, students should expose the learnings obtained, the difficulties encountered, and the proposed solutions, as well as summarize the impact of this project on their understanding of the concepts of polynomials and their roots.

• In the Bibliography, students should list all research sources that assisted them in carrying out the project.

Project Deliverables:

• Presentation on Polynomials and their Roots
• Solution to the proposed problem and mathematical confirmation
• Python program that simulates the enigma
• Final report containing Introduction, Development, Conclusions, and Bibliography.

Each of these deliverables should reflect the collective and collaborative effort of the group to understand and apply the concepts of polynomials and their roots. The presentation and the final report will serve as a tangible record of the students' progress and learning throughout the project.