Precalculus 2: polynomials and rational functions

A total of 42 hours of lectures

Due to the size of precalculus, it is divided into four parts. This page describes the second of those four parts.



Prerequisites

High-school maths, mainly arithmetics. Precalculus 1: Basic notions (mainly the concept of function and related concepts; sets; logic).



Curriculum

Make sure that you check with your professor what parts of the course you will need for your 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.

Precalculus: polynomials and rational functions

Get the outline

A detailed list of all the lectures in part 2 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

Get Precalculus 2: polynomials and rational functions on Udemy

When you buy the course on Udemy, you get access to it for life. There is just a one-time fee. The prices do vary a lot on Udemy, but if you use our link by clicking on this panel, you will get the best current price. See also our page on “coupon codes” in the menu (the current code is TPOT_JUL25).

Course Objectives & Outcomes

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.

Adrian

Udemy student

Another exceptional course from Hania, building on the very solid foundation laid out in her precalc 1 course.

She delivers the material with her usual finesse and easy to follow style and explanations, always going above and beyond to make sure the concepts are clear.

Without a question the most in depth course on the topic on the udemy platform, I can only recommend it and hope that anyone who wants to expand their knowledge or fill gaps in their understanding comes across it sooner rather than later!

Almir B.

Udemy student

This is my 5th course by Hania, as always , I am enjoying it very much.

Vlad M.

Udemy student

The course absolutely exceeded my expectations: starting from fundamental concepts, it quickly advances to examine polynomials & rational functions in a thorough manner. Constant learning by doing along with detailed instructor’s explanations ensures understanding of even such non-trivial topics as Pascal’s triangle, polynomial long division and asymptotes. Questions which are asked in a Q&A section are carefully answered by instructor in a matter of hours.

Danielle

Udemy student

Excellent! Excellent! Excellent! I feel very confident in my knowledge of polynomials now after watching all the videos. I hit the jackpot finding Dr. Uscka-Wehlou’s tutorials!

Antonio S.

Udemy student

This amazing polynomials course could not be more perfect!!

Miguel Ángel G.

Udemy student

Hania really has the ability to state everything she explains with absolute precision. In my opinion she is the best math teacher I have found on the internet for all these years. Thank you very much, I look forward to the next courses of this magnificent teacher.

Francisco O.

Udemy student

Hania’s precalculus 2 course is a remarkable math course, because she took both theory and practice in account. Furthermore Hania often used geometrical expositions to clarify hard math concepts

Bernard X.

Udemy student

J’ai pris beaucoup de plaisir à suivre ce cours.

merci

Wanda W.

Udemy student

Great course, very detailed. The lecturer is very good and exercises helpful. Everything from the computation rules to applications of polynomials is explained.

Devin F.

Udemy student

This course is my life saver

Richard B.

Mathematics lecturer

The four standard arithmetic operations, when recursively applied to a single numerical input x in a computational scheme, compute what is called a rational function of x, defining in this way the smallest class of functions containing the identity, id(x)=x, the unit, f(x)=1, and closed for arithmetic. These functions are basic to all mathematics, obviously, both in the formal logical sense and for general literacy, but they often are seen too complex to fully discuss in school and too cumbersome to quickly explain at university. Hania fills the gap beautifully, presenting everything in full, illustrating richly with pictures and examples, always in simple language. I don’t know of any better presentation of this material at this level.

Tetyana M.

Mathematics lecturer

A next great course! Useful for students. A very good example for teachers.

Nicholas N.

Udemy student

YOU’RE THE BEST ! I haven’t had an Instructor like you so detailed, knowledgeable with lots of follow though in her work !

Michael M.

Udemy student

Highly anticipated. The first course covers everything you need to know from basic algebra to discrete math to geometry. I can’t wait to start this second Precalculus course.

Maciej K.

Udemy student

Great course, as usually. Thank you, Hania!

Andrzej P.

Udemy student

Fantastic course, as always! The lectures cover a wide range of topics, providing a complete overview of all relevant concepts in a clear way. I appreciate all the solved problems, they help in better understanding the covered subjects and show how to handle different exercises. The lecturer explains everything in an easy to follow manner.

Adrian

Udemy student

Another exceptional course from Hania, building on the very solid foundation laid out in her precalc 1 course.

She delivers the material with her usual finesse and easy to follow style and explanations, always going above and beyond to make sure the concepts are clear.

Almir B.

Udemy student

This is my 5th course by Hania, as always , I am enjoying it very much.

Vlad M.

Udemy student

Danielle

Udemy student

Excellent! Excellent! Excellent! I feel very confident in my knowledge of polynomials now after watching all the videos. I hit the jackpot finding Dr. Uscka-Wehlou’s tutorials!