Polynomials and their Roots - Professor Raymond Flood

TL;DR

This lecture explores the concept of polynomials and their roots, discussing the degree of polynomials and the coefficients involved. It also delves into the history of finding formulas for solving polynomial equations, specifically focusing on the search for a formula that can solve equations of degree 5 or higher.

Transcript

Russian College presents shaping modern mathematics polynomials and their roots my professor Raymond fled Gresham professor of geometry well again and this is the second of my series of lectures this academic year of shipping modern mathematics and last time I considered analysis and for those of you who were here last time I'm very pleased to see ... Read More

Key Insights

• 😑 Polynomials are expressions in a variable with coefficients and powers of the variable.
• 🫚 The roots of a polynomial are the values of the variable that make the polynomial equal to zero.
• ✋ While there are formulas for finding the roots of quadratic, cubic, and quartic equations, there is no general formula for quintic equations and higher.
• ❓ The fundamental theorem of algebra states that every polynomial has at least one complex solution.
• #️⃣ Complex numbers, represented as pairs of real numbers, are crucial for understanding the roots of polynomial equations.
• 🥺 The quest for formulas to solve polynomial equations led to the development of modern algebra and the study of mathematical structures called groups.

Questions & Answers

Q: What is a polynomial and what are its roots?

A polynomial is an expression in a variable, usually denoted as X, where different powers of X and coefficients are involved. The roots of a polynomial are the values of X that make the polynomial equal to zero.

Q: Can you provide an example of a polynomial and its roots?

Sure. Let's take the polynomial X^2 - 4X + 3. To find its roots, we set the polynomial equal to zero and solve for X. In this case, the roots of the polynomial are X = 1 and X = 3.

Q: Is there a formula for finding the roots of all polynomial equations?

No, there is not a general formula for finding the roots of polynomial equations of degree 5 or higher. This was proven by mathematicians Abel and Galois in the 19th century.

Q: What is the significance of the fundamental theorem of algebra?

The fundamental theorem of algebra states that every polynomial equation of degree n has n complex solutions, taking into account both real and imaginary numbers. This theorem assures us that every polynomial has at least one root.

Summary & Key Takeaways

• The lecture explains the definition of polynomials and their roots, highlighting the degree of polynomials and the coefficients involved.

• Examples of polynomials of different degrees are provided, from linear to quintic.

• The lecture also discusses the quest to find a formula for the roots of polynomial equations, specifically focusing on equations of degree 5 or higher and the impossibility of finding a general formula.