Precalculus 2: Polynomials and rational functions

The Algebra 2 topics of polynomials and rational functions, preparing for studies of Calculus

Created by: Hania Uscka-Wehlou
University teacher in mathematics, PhD

Created by: Martin Wehlou
Editor at MITM AB

What Will I 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.

Target audience

• 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.