# Precalculus 2: Polynomials and rational functions

## What you'll 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.

## Requirements

- High-school mathematics, mainly arithmetic
- Precalculus 1: Basic notions (mainly the concept of function and related concepts; sets; logic)
- You will get a very brief introduction to *systems of linear equations* in this course (in Section 4), just enough for the needs of our applications (methods by undetermined coefficients, e.g. for partial fraction decomposition)
- You will get a very brief introduction to *complex numbers* in this course (in Section 5), just enough for the needs of our applications (factoring polynomials)
- You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

## Description

Precalculus 2: Polynomials and rational functions

*Mathematics from high school to university*

Chapter 1: Polynomials

S1. Introduction to the course

You will learn: about this course: its content and the optimal way of studying.

S2. A general presentation with the large picture and some spoilers

You will learn: why polynomials are important and why they are lovable; you will also get some general information about polynomials and rational functions, which will help you build up some important intuitions around the subject of the course.

S3. Powers, expressions, and polynomials

You will learn: about powers with natural, integer, and rational exponents and the computation rules holding for them (the product rules, the quotient rules, the power rule); basic terminology concerning polynomials (term, degree, monomial, binomial, trinomial, monic polynomial); polynomial arithmetic (addition, subtraction, scaling, multiplication), composition of polynomials.

S4. Linear equations and systems of equations

You will learn: how to solve *n*-by-*n* systems of linear equations and why you need it for your works with polynomials and rational functions.

S5. Second degree polynomials

You will learn: solving quadratic equations by using qualified guesses for factoring, completing the square, and the quadratic formula; plotting parabolas by finding the coordinates of the vertex and transforming the parabolas y=x^2 and y=ax^2 to this vertex; Vieta's formula with proof and some applications.

S6. Factoring polynomials is the same as finding zeros of polynomials

You will learn: polynomial divisibility; polynomial division, various methods: long division (two different notations), division with help of undetermined coefficients, Ruffini-Horner Scheme for division by monic binomials; consequences of the Fundamental Theorem of Algebra; Vieta's formulas; methods of finding rational zeros of polynomials with integer coefficients; Cauchy's Bound for zeros.

S7. Factoring polynomials: school versus reality

You will learn: that the reality is not as nice as school.

S8. Polynomial equations and inequalities

You will learn: solve polynomial equations and inequalities by factoring polynomials and analysing the signs (with help of the table or a sketch); you will also gain a geometrical understanding of the solution sets (graphically). Factoring of polynomials is omitted in this section, because this was the topic of the previous section, but at school you will have to factor polynomials in order to solve polynomial equations and inequalities.

S9. *Intermezzo*: Some topics from Calculus

You will learn: what it means that a function is continuous and that polynomials are continuous functions; the concept of the derivative; compute the derivatives of polynomials; why the curves of polynomials are rounded while intersecting the *x*-axis in multiple zeros of the polynomials; limits in the infinities and infinite limits.

S10. Plotting (sketching) polynomials

You will learn: how to sketch graphs of polynomial functions: how to establish the domain, the range, the *x*- and *y*-intercepts, the intervals of monotonicity (increasing, decreasing), and local extremums (max, min).

S11. More advanced future topics on polynomials

You will learn: in what other domains you will enjoy your gained knowledge about polynomials; I will *not* teach you about this topics, I will just give you some information on where to find them.

Chapter 2: Rational functions

S12. Rational functions and their domains

You will learn: the definition of rational functions; how to determine their domains, their zeros, and *y*-intercepts.

S13. Rational equations and inequalities

You will learn: add, subtract, multiply and divide rational expressions; solve rational equations and inequalities and understand the link between rational and polynomial equations and inequalities.

S14. Asymptotes

You will learn: horizontal and vertical asymptotes (intuitively; the concepts come back in the Calculus class).

S15. Plotting (sketching) rational functions

You will learn: to sketch some simple graphs of rational functions using graph transformations of *y*=1/*x* and *y*=1/(*x*^2+1); understand the link to polynomial division and polynomial / rational equations and inequalities.

S16. Partial fraction decomposition

You will learn: how to perform partial fraction decomposition of rational functions.

S17. More advanced future topics on rational functions

You will learn: about significant terms for polynomials near zero and in the infinity: the huge difference between computing indefinite limits of rational functions in zero (like in Taylor approximations) and in the infinity (for plotting graphs of rational functions, finding asymptotes, etc); importance of partial fraction decomposition for integrating rational functions. I will *not* teach you this stuff, I will only prepare you for some future topics and motivate why you should study rational functions.

S18. Some words about power functions and algebraic functions

You will learn: the definition and examples of power functions and algebraic functions.

S19. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.

*Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.*

*A detailed description of the content of the course, with all the 211 videos and their titles, and with the texts of all the 160 problems solved during this course, is presented in the resource file *

*“001 List_of_all_Videos_and_Problems_Precalculus_2.pdf” *

*under video 1 ("Introduction to the course"). This content is also presented in video 1.*

## Who this course is for:

- Students who plan to study Algebra, Calculus or Real Analysis
- High school students curious about university mathematics; the course is intended for purchase by adults for these students
- Everybody who wants to brush up their high school maths and gain a deeper understanding of the subject
- College and university students studying advanced courses, who want to understand all the details (concerning polynomials or rational functions) they might have missed in their earlier education
- Students wanting to learn about polynomials, for example for their College Algebra class.

## Instructors

I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.

I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.

I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Torun (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Torun, which gave me a very solid foundation for everything else I have learned and taught later.

My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".

In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).

I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.

I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.