Precalculus 2: Polynomials and rational functions Course
Mathematics from high school to university
Precalculus 2: Polynomials and rational functions Mathematics from high school to university. 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.
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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.
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