Summary Tradisional | Polynomials: Girard's Relations

Contextualization

Vieta's formulas, often referred to as Vieta's relations, are important mathematical tools that link the roots of a polynomial with its coefficients. Introduced by the French mathematician François Viète during the 16th century, these relations form a basic part of our study in various mathematical subjects, from algebra to analysis. They enable us to find the sum and product of the roots of a polynomial without having to solve the entire equation.

Their usefulness goes beyond just theoretical mathematics. In diverse fields like engineering, physics, and economics, these formulas find practical application. For instance, in engineering, they help in calculating the natural frequencies of vibrating systems, while in economics, they contribute to modelling and predicting market trends. A good grasp of Vieta's relations equips students with a handy tool to tackle complex problems in a straightforward and efficient manner.

To Remember!

Definition of Vieta's Relations

Vieta's relations are formulas that create a direct link between the roots of a polynomial and the coefficients of its terms. They are derived from expanding the product of the polynomial’s roots. Essentially, these relations help in computing the sum and the product of the roots of a polynomial without having to solve the equation step by step.

For a polynomial of degree n in the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁*x + a₀, these formulas connect the coefficients aₙ, aₙ₋₁, ... , a₀ with the roots of the polynomial (which are the solutions of the equation P(x) = 0). For example, in a quadratic polynomial P(x) = ax² + bx + c, the sum of the roots is –b/a and their product is c/a.

This basic understanding is incredibly valuable when simplifying polynomial equations or when analysing more complex situations in areas like engineering and physics. Moreover, these formulas are often used in solving systems of equations and studying time series in algebra.

• They create a connection between the roots and coefficients of polynomials.

• They come from the expansion of the product of the roots of a typical polynomial.

• They simplify the process of solving polynomial equations and are useful in various applications.

First Vieta's Relation

The first Vieta's relation explains how the sum of the roots of a polynomial is determined. For a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁*x + a₀, the sum of the roots is found by taking the coefficient of xⁿ⁻¹, changing its sign, and dividing it by the coefficient of the highest degree term. In other words, if the roots are r₁, r₂, ... , rₙ, then r₁ + r₂ + ... + rₙ equals –aₙ₋₁/aₙ.

This is particularly useful as it gives us the sum without needing to find each root individually. For instance, take the polynomial P(x) = x² – 5x + 6; here, the sum of the roots turns out to be 5, considering the coefficient of x is –5 and the coefficient of x² is 1. This relation is commonly used in solving algebraic problems as well as in various practical scenarios.

Beyond its practical benefits, the first Vieta's relation provides a deeper understanding of the structure of polynomials, clearly showing how the roots are inherently linked to the coefficients.

• It deals with the sum of the roots of a polynomial.

• It is computed from the coefficient of xⁿ⁻¹ with the opposite sign and then divided by the highest degree term's coefficient.

• It helps to calculate the sum of the roots without finding them one by one.

Second Vieta's Relation

The second Vieta's relation deals with the product of the roots of a polynomial. In a polynomial of degree n written as P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁*x + a₀, the product of the roots is determined by the constant term a₀ divided by the coefficient of the highest degree term aₙ. The sign of the product depends on whether n is even or odd: for odd n, the product is –a₀/aₙ, and for even n, it is a₀/aₙ.

For example, consider the quadratic polynomial P(x) = x² – 5x + 6 (degree 2, even); the product of its roots is 6. Another example is the cubic polynomial Q(x) = x³ – 6x² + 11x – 6 (degree 3, odd) where the product of the roots also results in 6 after considering the sign change.

This relation is immensely helpful when verifying the correctness of calculated roots or when solving optimisation problems that involve polynomials. It is also often used in engineering to study the stability of systems or when analysing dynamic characteristics in physics.

• It explains how to find the product of the roots of a polynomial.

• The product is given by the constant term divided by the leading coefficient, with the sign depending on the degree of the polynomial.

• It is useful for verifying calculated roots and in various optimisation challenges.

Application of Vieta's Relations in Problems

Using Vieta's relations in problem-solving involves applying these formulas to both solve and check the roots of polynomial equations. They are particularly effective when dealing with polynomials of different degrees, as they provide a more direct path to obtaining the roots and understanding their relationships.

For example, in solving a second-degree polynomial such as P(x) = x² – 5x + 6, one can quickly see that the sum of the roots is 5 and the product is 6, which assists in confirming that the roots (2 and 3) are indeed correct. In cases involving higher degree polynomials like Q(x) = x³ – 6x² + 11x – 6, Vieta's relations help compute not only the sum of the roots but also the sum of the products of the roots taken two at a time, along with their overall product.

These relations are not only a shortcut in solving these equations but also find applications in various sectors such as engineering, where they might be used to determine natural frequencies, in physics to explore dynamic systems, and in economics to model and predict market trends.

• They simplify the process of solving and checking polynomial roots.

• They are effective for polynomials of different degrees.

• They have practical applications in fields like engineering, physics, and economics.

Key Terms

• Polynomial: An algebraic expression consisting of variables and coefficients, involving operations like addition, subtraction, multiplication and non-negative integer exponents.

• Vieta's Relations: Formulas that directly connect the roots of a polynomial with its coefficients.

• Roots of Polynomials: The values which make the polynomial equal to zero when substituted into it.

• Coefficients: The numerical factors multiplying the variables in a polynomial.

• Vieta's Formulas: Another term for Vieta's relations, named in honour of François Viète.

• Algebra: A branch of mathematics focused on symbols and the rules for manipulating these symbols.

Important Conclusions

Vieta's relations, also known as Vieta's formulas, are indispensable tools that link a polynomial’s roots to its coefficients. They allow us to effortlessly compute the sum and product of the roots without having to solve the entire polynomial equation, thereby simplifying the analysis of complex problems. This understanding is particularly beneficial not only in solving polynomial equations but also in advanced applications in fields like engineering and physics.

The first relation focuses on finding the sum of the roots, while the second one deals with the product, taking into account the parity (even or odd) of the polynomial's degree. These formulas are very effective in analysing and manipulating polynomial equations and have widespread use across various disciplines including engineering, physics and economics.

Grasping and applying these relations equips students with a robust tool for solving complicated problems in an intuitive and effective manner. Moreover, it provides profound insights into the underlying structure of polynomials. We encourage students to explore this topic further and practice using these relations in diverse scenarios to fully consolidate their learning.

Study Tips

• Review solved examples in class and attempt similar problems to reinforce your understanding of Vieta's relations.

• Make use of additional resources such as textbooks and online tutorials to deepen both theoretical and practical knowledge of these formulas.

• Form study groups to discuss and work through problems involving Vieta's relations, which can help in sharing diverse approaches and insights.