Introduction

The Relevance of the Topic

Polynomials: Girard's Relations is not just an abstract theme in mathematics, but a concept applied in various real-world contexts. These relations are essential tools in the analysis and resolution of polynomials. They allow unveiling hidden information about a polynomial, such as its degree and coefficients, even without knowing all the terms of the polynomial. Understanding these relations enhances students' ability to manipulate polynomials and deepens their knowledge of the structure of polynomials.

Contextualization

In the third year of High School, after having acquired basic knowledge about polynomials, the study of Girard's Relations implies a more advanced conceptual leap within the theme of polynomials. While the initial years focused on more elementary aspects, this content challenges students to explore polynomials beyond the superficial concepts of terms and degrees.

Girard's Relations are part of the Algebra block, being essential for understanding more advanced topics in the Exact Sciences, such as Differential Calculus and Analytical Geometry. Similarly, they are a structural basis for understanding various phenomena in Biological Sciences and Engineering. Therefore, this topic will serve as a solid preparation for future studies and will be used as a tool to solve concrete problems in various fields of expertise.

Theoretical Development

Components

• Polynomials: They are algebraic expressions formed by one or more variables, with non-negative integer exponents and real coefficients. The study of Girard's relations focuses on the use of these expressions and seeks to understand the behavior of their roots.

• Roots of a Polynomial: They are the numbers that, when substituted into the polynomial expression, result in zero. The study of Girard's relations allows relating the roots to the coefficients of the polynomial.

• Girard's Relations: They are mathematical formulas that relate the roots of a polynomial to its coefficients. There are three Girard's relations, each relating a different number of roots to the coefficients of the polynomial. These relations are crucial for polynomial resolution, even when the roots are not known.

Key Terms

• Degree of a Polynomial: It is the highest exponent among all variables of the polynomial. Girard's relations are used to determine the degree of a polynomial when the roots are known.

• Discriminant: It is a measure of the nature of the roots of a second-degree polynomial. In Girard's relation of second-order polynomials, the discriminant is used as a term in the formula.

• Coefficients: They are the numbers that multiply the variables in the terms of a polynomial. Girard's relations provide an excellent example of how coefficients can be used to obtain information about the roots of a polynomial.

Examples and Cases

• Girard's Relation for Second-Degree Polynomials: The Girard's relation for second-order polynomials allows calculating the roots of a polynomial from its coefficients. For a polynomial of the form ax² + bx + c, the roots are given by (-b ± √(b² - 4ac)) / (2a). Note the presence of the discriminant in the calculation of the roots.

• Girard's Relation for Third-Degree Polynomials: For a third-degree polynomial of the form ax³ + bx² + cx + d, there is a Girard's relation that links the roots to the coefficients. It is: s¹ + s² + s³ = -b/a, where s¹, s², and s³ are the roots of the polynomial.

• Practical Application of Girard's Relations: For example, if an engineer needs to design a bridge with a parabolic arch, they will need to understand Girard's relations, as the roots of the parabolic arch equation are the points where it touches the x-axis. Knowing these points, the engineer can adjust the bridge design as needed. Therefore, Girard's relations are an essential tool in real-world situations.

Detailed Summary

Key Points

• Composition of Polynomials: A polynomial is a mathematical expression that involves sums and multiplications, without the presence of other operations. They can be represented in the general form of the sum of terms of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂xⁿ⁻² + ... + ax + a₀, where aₙ, aₙ₋₁, ... , a₀ are the coefficients and x, x², ... , xⁿ are variables raised to a power (exponents).

• Roots of a Polynomial: A root of a polynomial is a numerical value that, when substituted into the polynomial expression, results in zero.

• Girard's Relations: They are mathematical formulas that establish connections between the roots and coefficients of a polynomial. They are fundamental for solving high-degree equations and unveiling information about the polynomial without necessarily knowing the roots.

• Girard's Relation for Second-Degree Polynomials: This relation allows calculating the roots of a second-degree polynomial from its coefficients. For a polynomial of the form ax² + bx + c, the roots are: ( -b ± √(b² - 4ac) ) / 2a.

• Girard's Relation for Third-Degree Polynomials: For a third-degree polynomial of the form ax³ + bx² + cx + d, there is a Girard's relation that relates the roots to the coefficients: s¹ + s² + s³ = -b/a, where s¹, s², s³ are the roots.

Conclusions

• Understanding Girard's Relations is essential in the study of algebra, as they allow the resolution of polynomials without necessarily knowing the roots.

• Each Girard's relation is applicable to a specific degree polynomial and provides information about all its roots from the coefficients.

• Girard's relations have notable practical application in various fields, such as engineering and physics, by allowing the calculation of unknown values from established relationships.

Exercises

1. Exercise 1: Calculate the roots of the polynomial 2x² - 5x + 2 using Girard's relation for second-degree polynomials.

2. Exercise 2: Find the roots of the polynomial x³ - 6x² + 11x - 6. Verify if the sum of the roots is equal to the quotient between the coefficient of the term of degree 2 and the coefficient of the highest degree term, as predicted in Girard's relation.

3. Exercise 3: Discuss the practical application of Girard's relations. How could an engineer use this theory in designing a bridge with a parabolic arch, for example?