Contextualization

Polynomials are mathematical functions present in various practical applications, from physics to economics. A polynomial is a mathematical expression composed of coefficients and variables, each raised to a non-negative integer power. In our lesson today, we will focus our attention on the properties of these versatile functions.

The properties of polynomials are intrinsic characteristics that these functions possess and are key to understanding and manipulating them effectively. These properties include: the property of addition, subtraction, multiplication, and division of polynomials; the property of the product of polynomials with the same root; and the property of the product and sum of polynomials with their roots.

It is important to emphasize that the polynomial function is widely used in mathematical modeling, a powerful tool that helps us understand and interpret real-world phenomena through mathematics. Polynomial functions can, for example, be used to model the trajectory of a rocket, or to predict the growth of a population. Thus, the study of the properties of polynomials is not just something abstract, but has concrete and significant applications in different contexts.

This connection between polynomials and the world around us gives a new meaning to our study. For example, polynomial functions are essential in many areas of engineering for modeling and solving technical problems. In economics, economists use polynomials to model various functions, such as demand, supply, production, and consumption.

Therefore, it is essential that we understand the properties of polynomials so that we can apply them properly in our study of mathematics or any other field that requires their use.

For a deeper dive into the topic, we suggest the following resources:

• Book "Fundamentos de Matemática Elementar: Polinômios - Gelson Iezzi e Carlos Murakami", this is an excellent book to understand the theory behind polynomials and their properties.
• Website "Só Matemática", in the Polynomials section you will find detailed content on the topic, as well as solved and proposed exercises.
• Videos from the channel "Me Salva" on YouTube about polynomials and their properties, the channel focuses on teaching mathematics for High School and College Entrance Exam, being an excellent tool for understanding and revision.

Practical Activity

Activity Title: Polynomials and their properties: a practical approach

Project Objective

This project aims to reinforce students' knowledge about polynomials and their properties. The main focus is to apply these properties in problem-solving and modeling real-world situations using polynomials. The project also aims to develop teamwork, communication, time management, and creative thinking skills.

Detailed Project Description

Students will be divided into groups of 3 to 5 members. Each group will receive a list of problems to be solved and a real-world situation to be modeled using polynomials.

The problems are mainly centered on the use of polynomial properties, while the real-world situation is designed to demonstrate the practical application of polynomials.

Each group must submit a comprehensive report describing the solution to the problems and the modeling of the real-world situation.

Required Materials

• Textbooks or online resources for reference.
• Pencils, erasers, paper, or math apps to perform the necessary calculations.
• Word processing software to write the report.

Detailed Step-by-Step Guide for the Activity

1. Group Formation: Students will be divided into groups of 3 to 5 members.

2. Study and Research: Groups should study and research about the properties of polynomials. They can use textbooks, online resources, videos, etc.

3. Problem Solving: Groups must solve the provided list of problems. Each solution must be well-founded, clearly explaining how the properties of polynomials were used.

4. Modeling the Real-World Situation: Groups must model the real-world situation using polynomials. They should clearly explain their modeling process, including how they chose the polynomials and applied their properties.

5. Report Creation: Finally, each group must write a detailed report as outlined below in the "Report Details" section.

Report Details

The report should have the following topics:

1. Introduction: Students should contextualize the theme, its relevance, and real-world application, as well as the objective of this project.

2. Development: Students should explain the theory behind the properties of polynomials, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained. Here, students should show the solutions to the problems and the modeling of the real-world situation, always discussing how the properties of polynomials were used.

3. Conclusion: Students should conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.

4. Bibliography: Students should indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Students will have one week to complete the project. Each student should dedicate between two and four hours to its completion.