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.
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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.
Reviews (51)
Shyamsunder R.
Udemy student
GOOD
John H.
Udemy student
Excellent in that it leaves no gaps, doesn't gloss over any part of the subject
Ian H.
Udemy student
I really appreciate these courses! Hania doesn't assume that anything is known by the student, so everything is gone over very thoroughly. So often difficulty in mathematics comes from a fundamental point that got glossed over in a course that tries to move too quickly. Here you can choose to dive deep into a topic, or if you are already comfortable with it, you can move ahead with the help of Hania's great outlines. It's a really good way to teach.
Carlos S.
Udemy student
I learned a lot of mathematics with Hania. Thank God I found this course. I’m preparing myself to a master degree, and for those which plan to take calculus classes, I’d strongly recommend taking the pre calculus classes before.
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Udemy student
Above excellent if not defined yet.
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Udemy student
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Udemy student
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Udemy student
Amazing teacher, with outstanding dedication.
Vladislav V.
Udemy student
Great
Odell H.
Udemy student
Good instructions
Felipe V.
Udemy student
Thank you, professor. As with all your courses, this one is also pure gold
Paiin
Udemy student
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Debbie C.
Udemy student
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Anonymized User
Udemy student
excellent content
Nuwra A.
Udemy student
thorough.
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Udemy student
Brilliant course with tons of example exercises/problems to drive the concepts home. Awesome course!
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Udemy student
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for computer science . Great work!
for computer science . Great work!
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Udemy student
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Udemy student
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Udemy student
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Udemy student
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Udemy student
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Really incredible teacher with incredibly well done courses! 👏
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Udemy student
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i love you
Ludwig F.
Udemy student
Very good course, detailed and precise with lots of practical problem solving and prospects for further use in calculus and linear algebra.
Paul L.
Udemy student
excellent so far.
Sarbojit G.
Udemy student
absolutely marvelous, a lot of concepts were cleared.
Bradley M.
Udemy student
I took the first course in the series and enjoyed it very much. Hania is a great teacher and explains the matterial well. So, far I'm reviewing information that I learned some 40 plus years ago. I'm looking forward to taking the higher level courses.
Andrew D.
Udemy student
The Instructor has a perfect speed in pronunciation. Her explanations are clear and concise. I could not ask for more. Well done!
Guillaume L.
Udemy student
great!
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.
picha W.
Udemy student
all your courses are superb!
Pierpaolo D.
Udemy student
The course is wonderful!!!
Every topic is thoroughly explained and there aren't missing links in the reasoning. Every step is motivated and leads to the next topic. All the exercises are clearly explained step by step. Moreover, Hania always answers quickly and in-depth all the questions!
Highly recommended!!
Every topic is thoroughly explained and there aren't missing links in the reasoning. Every step is motivated and leads to the next topic. All the exercises are clearly explained step by step. Moreover, Hania always answers quickly and in-depth all the questions!
Highly recommended!!
Danielle D.
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 R.
Udemy student
This amazing polynomials course could not be more perfect!!
Gabi O.
Udemy student
As always, great stuff coming from Hania!
Junwoo L.
Udemy student
It is just more than a math.
Nico K.
Udemy student
Just finished pt.2 of pre-calc after the 1st part was so convincing to me the 2nd course didn’t change anything about my opinion. Looking forward to calc 1 for my exam in September.
Almir B.
Udemy student
This is my 5th course by Hania, as always , I am enjoying it very much.
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
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Udemy student
This course is my life saver
Miguel 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. (sorry for may English)
Bernard X.
Udemy student
J'ai pris beaucoup de plaisir à suivre ce cours.
merciMachine translation: I really enjoyed taking this course.
Thank you
merciMachine translation: I really enjoyed taking this course.
Thank you
Maciej K.
Udemy student
Great course, as usually. Thank you, Hania!
Maja K.
Udemy student
This course is amazing. I really appreciate the part about partial fraction decomposition.
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 !
Tetyana M.
Lecturer
A next great course! Useful for students. A very good example for teachers.
Wanda W.
Udemy student
Great course, very detailed. Hania is a very skilled lecturer and exercises are really helpful. Everything from the computation rules to applications of polynomials is explained. I also like that the theory (and practice) of rational functions is clearly connected to the theory (and practice) for polynomials; it makes it easier to study rational functions. This course makes it easy to understand some more advanced topics in Calculus.
Richard B.
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.
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!
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!
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.
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 instructor explains everything in an easy to follow manner.