|
Lecturer(s)
|
-
Dofková Radka, doc. PhDr. Ph.D.
-
Zdráhal Tomáš, doc. RNDr. CSc.
|
|
Course content
|
Polynomial Algebra - Polynomials in One Variable: Definitions and basic operations (addition, subtraction, multiplication, division); degrees of polynomials and coefficients. - Divisibility of Polynomials: Criteria for the divisibility of polynomials; factorization of polynomials into irreducible factors. - Roots of Polynomials: Definition of the roots of polynomials; Vieta's formulas and relationships between roots and coefficients; finding and analyzing roots. - Polynomials with Integer Coefficients: Specific properties and divisibility in the ring of integer coefficients; application of Eisenstein's criterion for irreducibility. - Derivatives of Polynomials: Definitions and calculations of polynomial derivatives; applications of derivatives in analyzing the behavior of polynomials. - Polynomials in Multiple Variables: Basic operations and properties of polynomials with multiple variables; factorization and analysis of polynomials in multiple variables; symmetric polynomials: definitions and properties; applications of symmetric polynomials in analyzing polynomials with multiple variables; examples and exercises on symmetric polynomials. Algebraic Equations - Solving Binomial Equations: Methods and examples. - Quadratic Equations over complex numbers C: Solving quadratic equations; use of the discriminant; geometric interpretation and applications. - Cubic Equations over complex numbers V:: Methods for solving cubic equations; Cardano's method and its applications; examples and analytical methods. - Reciprocal Equations: Definitions and methods for solving equations where variables appear in the denominators; analysis and applications. - Numerical Solutions of Equations: Methods for numerical solutions of equations (e.g., bisection method, Newton's method); applications and examples.
|
|
Learning activities and teaching methods
|
Lecture, Monologic Lecture(Interpretation, Training)
- Homework for Teaching
- 50 hours per semester
|
|
Learning outcomes
|
The objective of the course is to provide students with a deep understanding of both fundamental and advanced algebraic concepts and their applications. Students will become familiar with various aspects of polynomial algebra and algebraic equations and develop the skills necessary for effective algebra instruction at different educational levels. Upon completion of the course, students will be able to: - Analyze and manipulate polynomials in one and multiple variables: Students will learn to define and perform basic operations with polynomials, determine their degrees and coefficients, and understand the specific properties of polynomials with integer coefficients. They will be able to apply polynomial derivatives and recognize and analyze symmetric polynomials and their uses. - Address problems related to divisibility and factorization of polynomials: Students will acquire skills in identifying polynomial divisibility, factoring polynomials into irreducible factors, and applying Eisenstein's criterion for irreducibility. - Work with polynomial roots: They will be able to use Vieta's formulas, analyze the relationships between roots and coefficients, and effectively find and interpret the roots of polynomials. - Solve algebraic equations of various degrees: Students will master solving binomial equations, quadratic equations over the complex numbers, cubic equations, and equations with variables in the denominators. They will also learn to use numerical methods for solving equations, such as the bisection method and Newton's method. - Apply theoretical knowledge in practice: They will be able to utilize algebraic methods and techniques to solve practical problems and tasks, and effectively apply them in mathematics instruction. The course will provide students with both theoretical foundations and practical skills needed for teaching algebra and applying algebraic methods in educational practice. The aim is understanding of algebraic solvability of algebraic equations.
Understanding the phenomenon of polynomilal and algebraic equation.
|
|
Prerequisites
|
Prerequisites To successfully complete the course, the following prerequisites are required: - Knowledge of Secondary School Mathematics: Students should have a solid foundation in secondary school mathematics, including knowledge of polynomials, functions, equations, and basic algebraic operations. - Knowledge from Algebra 1 and Algebra 2: Students must have knowledge from Algebra 1 and Algebra 2 courses. - Basic Knowledge of Mathematical Analysis: Basic knowledge of mathematical analysis will also be useful, particularly in the areas of derivatives and fundamental functions. These prerequisites will ensure that students have the necessary background to understand and apply the advanced algebraic concepts covered in the course.
|
|
Assessment methods and criteria
|
Student performance, Dialog, Seminar Work
Active knowledge of the subject matter. Specific requirements (including individual assignments for term papers) are listed in the respective team on MS Teams. Regarding active physical attendance, students in both full-time and part-time study programs are required to attend at least 49% of classes, while students studying under an Individual Study Plan (ISP) must attend at least 20%. (The exact requirements for ISP students must be clarified with the instructor at the beginning of the semester.) ctive knowledge of subject matter.
|
|
Recommended literature
|
-
BLAŽEK, J. a kol. Algebra a teoretická aritmetika 2. Praha: SPN 1985..
-
Blažek, J, a kol.: Algebra a teoretická aritmetika 2, Praha, SPN, 1985; Emanovský, P.: Cvičení z algebry (polynomy a algebraické rovnice). Miniskriptum, PdF UP Olomouc, 1998; Kopecký, M., Emanovský, P.: Sbírka řešených příkladů z algebry. Skriptum, olomouc, UP, 1990.
-
EMANOVSKÝ, P.: Algebra 3. Olomouc. VUP 2005. ISBN 80-244-0490-7.
|