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.

Icon Precalculus 2

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

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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

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How to solve problems concerning polynomials or rational functions (illustrated with 160 solved problems) and why these methods work.
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Definition and basic terminology for polynomials: variable, coefficient, degree; a brief repetition about powers with rational exponents, and main power rules.
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Arithmetical operations (addition, subtraction, multiplication) on polynomials; the polynomial ring R[x].
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Completing the square for solving second degree equations and plotting parabolas; derivation of the quadratic formula.
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Polynomial division: quotient and remainder; three methods for performing the division: factoring out the dividend, long division, undetermined coefficients.
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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.
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The Remainder Theorem and The Factor Theorem with many applications; the proofs, based on the Division Theorem (proven in an article), are presented.
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Ruffini-Horner Scheme for division by monic binomials of degree one, with many examples of applications; the derivation of the method is presented.
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Factoring polynomials, its applications for solving polynomial equations and inequalities, and its importance for Calculus.
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Polynomials as functions: their domain, range, zeros, intervals of monotonicity, and graphs (just rough sketches).
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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.
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Rational functions: their definition, domain, zeros, (y-intercept), intervals of monotonicity, asymptotes (infinite limits), and graphs (just rough sketches).
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Application of factoring polynomials for solving *rational* equations and inequalities, and its importance for Calculus.
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Partial fraction decomposition and its importance for Integral Calculus; some simple examples of integration.
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Derivatives and antiderivatives of polynomials are polynomials; a brief introduction to derivatives.
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Derivatives of rational functions are rational functions; antiderivatives can also involve inverse tangent (arctan) and logarithm.
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.
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Udemy student
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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
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