Precalculus 2: Polynomials and rational functions
The Algebra 2 topics of polynomials and rational functions, preparing for studies of Calculus
What you will learn
How to solve problems concerning polynomials or rational functions (illustrated with 160 solved problems) and why these methods work.
Definition and basic terminology for polynomials: variable, coefficient, degree; a brief repetition about powers with rational exponents, and main power rules.
Arithmetical operations (addition, subtraction, multiplication) on polynomials; the polynomial ring R[x].
Completing the square for solving second degree equations and plotting parabolas; derivation of the quadratic formula.
Polynomial division: quotient and remainder; three methods for performing the division: factoring out the dividend, long division, undetermined coefficients.
Vieta's formulas for quadratic and cubic polynomials; Binomial Theorem (proof will be given in Precalculus 4) as a special case of Vieta's formulas.
The Remainder Theorem and The Factor Theorem with many applications; the proofs, based on the Division Theorem (proven in an article), are presented.
Ruffini-Horner Scheme for division by monic binomials of degree one, with many examples of applications; the derivation of the method is presented.
Factoring polynomials, its applications for solving polynomial equations and inequalities, and its importance for Calculus.
Polynomials as functions: their domain, range, zeros, intervals of monotonicity, and graphs (just rough sketches).
Behaviour of polynomials near to zero and in both infinities, and why it is important to understand these topics (Taylor polynomials); limits in the infinities.
Rational functions: their definition, domain, zeros, (y-intercept), intervals of monotonicity, asymptotes (infinite limits), and graphs (just rough sketches).
Application of factoring polynomials for solving *rational* equations and inequalities, and its importance for Calculus.
Partial fraction decomposition and its importance for Integral Calculus; some simple examples of integration.
Derivatives and antiderivatives of polynomials are polynomials; a brief introduction to derivatives.
Derivatives of rational functions are rational functions; antiderivatives can also involve inverse tangent (arctan) and logarithm.
Why take this course?
๐ Precalculus 2: Polynomials and Rational Functions ๐
Course Description:
Chapter 1: Polynomials
๐ S1. Introduction to the Course ๐
- What this course is all about: its content, structure, and optimal study methods.
๐ฌ S2. A General Overview ๐
- The significance of polynomials and rational functions in mathematics and beyond.
- An introduction to the key concepts you'll encounter throughout this course.
๐งฎ S3. Powers, Expressions, and Polynomials ๐งฐ
- Mastering powers with various exponents, polynomial operations (addition, subtraction, scaling, multiplication), and the rules that govern them.
- Learning terminology: monomial, binomial, trinomial, and understanding monic polynomials.
- Solving linear systems and grasping their importance in polynomial manipulation.
๐ S4. Second Degree Polynomials ๐ฑ
- Conquering quadratic equations through different methods: qualified guesses, completing the square, factoring, and graphing.
๐ค S11. Rational Functions and Their Domains ๐
- Understanding the definition of rational functions and how to identify their domains, zeros, and y-intercepts.
๐ S13. Rational Equations and Inequalities ๐
- Manipulating and solving rational expressions with precision.
๐โโ๏ธ S14. Asymptotes ๐โ๐จ๏ธ
- Intuitive understanding of horizontal and vertical asymptotes.
๐ชก S16. Partial Fraction Decomposition ๐
- Learning the technique to break down complex rational functions into simpler components.
๐ค S17. Beyond Polynomials & Rational Functions ๐
- Exploring the future topics where your newfound knowledge will shine.
๐ ๏ธ S18. Power and Algebraic Functions ๐
- Definitions and examples of power and algebraic functions to expand your mathematical toolkit.
Chapter 2: Rational Functions
๐ง S12. Domains and Transformations ๐
- Identifying the domain of rational functions and understanding their transformations.
โฐ S15. Plotting Rational Functions ๐
- Sketching graphs using rational expressions and understanding their connection to polynomial functions.
๐ S18. A Closer Look at Power and Algebraic Functions ๐ง
- Exploring the world of power functions, which are a type of algebraic function, and learning their significance.
Extras & Resources ๐โจ
- Comprehensive list of all course videos and problems solved, available in the resource file "001 List_of_all_Videos_and_Problems_Precalculus_2.pdf".
- Detailed course description including all video titles and texts for problems. (Located in video 1.)
Important Note:
๐ฃ "Make sure that you check with your professor what parts of the course you will need for your final exam."
- The content needed for your final exam can vary, so it's crucial to align with your instructor's requirements.
Your Journey Awaits! ๐ฃ๏ธ
Embark on this mathematical adventure with confidence, as you navigate through the world of polynomials and rational functions. With clear explanations, engaging video content, and a wealth of problems to solve, you'll gain a deep understanding of these fundamental concepts that are crucial for your mathematical journey.
๐ Dive into Precalculus 2 now and transform your mathematical skills! ๐
Screenshots
Our review
๐ Course Review: Precalculus - Mastering the Fundamentals
Overview: The Precalculus course offered by our platform stands out with a global rating of 4.97, reflecting an overwhelmingly positive reception from recent students. The course is designed to provide a comprehensive understanding of precalculus concepts, with a focus on clarity, practical application, and real-world problem-solving.
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