## Topics in Algebra (Math 5900)

Instructor: Aaron Bertram
Spring 2011
Time: Mon, Wed 6:30-8:00 PM
Location: UofU Murray Site

### Texts:

(1) Ira J. Papick, Algebra Connections (Mathematics for Middle School Teachers)

(2) Algebra I (any middle school textbook).

(3) Algebra II (any high school textbook).

### Our Algebra Matrix (work in progress):

 Topic\Level Algebra 0 Algebra I Algebra II Algebra N Solving Linear Equations One Variable, One Equation Two Variables, Two Equations Three Variables, Three Equations Many Variables, Many Equations Fractions Rational Numbers and their Arithmetic Rational Numbers as Coefficients in Polynomials Rational Functions Fields of Fractions (Localization) Exponents and Logarithms Place Value and Different Bases (especially binary) Arithmetic Rules for Exponents Arithmetic Rules for Logarithms The Natural Log as an Integral Induction Find the Next Term in a Sequence Find the nth Term in a Sequence Proofs by Induction Factoring Prime Factorization of Natural Numbers Factoring Quadratic Polynomials Factoring Special Polynomials (e.g. Differences and Sums of Cubes) Techniques for Factoring, Proofs of Unique Factorization The Distributive Law Distributive Law of Arithmetic, Multiplying Numbers with more than One Digit, Distributing the Minus Sign Multiplying Linear Polynomials ("FOIL") Multiplying Polynomials in General Multiplying Power Series

### For Discussion:

(a) What is algebra?

(b) What is the meaning of 0.999... and is it true that 0.999... = 1?

(c) Why are we "not allowed" to divide by zero?

(d) Should 1 be considered to be a prime number?

(e) Could there be a biggest prime? Why or why not?

(f) What is the greatest common divisor and how should you go about finding it?

(g) Why is it necessarily the case that the product of two negative numbers is a positive number?

(h) What, precisely, does it mean to say that the square root of 2 is irrational, and why is it irrational?

(i) What is wrong with this? -1 = (√-1)(√-1) = √(-1)(-1) = √1 = 1

(j) What is a root of a polynomial and why does a polynomial of degree d have at most d roots?

(k) How is the number "e" defined and what does it have to do with algebra?

(l) Why do the coefficients of (x+y)^n match the numbers in the corresponding row of Pascal's triangle?

### Required Presentations (following the template) on the Following Topics:

(1) Due 1/24. Find the roots of a quadratic polynomial by factoring.

(2) Due 1/31. Complete the square in a quadratic expression to reveal the maximum or minimum value.

(3) Due 2/7. Solve a pair of linear equations in two variables, either by substitution, elimination or graphically.

(4) Due 2/14. Explain how to graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

(5) Due 2/23. Explain how to divide polynomials with long division. Compare and contrast this with long division of natural numbers.

(6) Due 2/28-3/2. Explain how to multiply powers a^n and b^n or a^n and a^m, and how to raise a^n to the mth power. Or, explain the corresponding properties of the "inverse" function log_a(x) to the power function a^x. Give some examples involving compound interest or half-life.

(7) Due 3/7. Explain the difference between permutations and combinations and prove Pascal's identity.

### Lecture Notes (with Exercises)

Mod p Arithmetic and Algebra (with 4 Exercises). Due 4/11.

The Chinese Remainder Theorem (Impress your friends!)

### Final Two Projects

Frequently Asked (Algebra) Questions Write concise (one or two paragraph) answers to these questions, which consolidate the discussion items (a)-(l) above.

Find a word problem that illustrates each of the presentation topics (1)-(7) above.

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148