COLLEGE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: this material is extensively elaborated upon in my optional e-book GOLDen Mathematics: College Algebra. This section of material only is downloadable for a nominal fee at www.lulu.com/content/449300. See "Tell me more about Keely's GM book".
Polynomial Functions and Graphs
GOLDen Mathematics - College Algebra: Section 3.1
Supplemental Sites:
MathOL Links - Calg
3.1
Topics of Importance
Definitions: polynomial function, polynomial degree
Finding y-intercept point and x-intercept points (zeros, roots)
End behavior via the leading coefficient test
Odd vs. even powers on factors, effect on graph, multiplicity
Turning point theorem
Synthetic division
Remainder theorem; Factor theorem
Factor in the rational, real, or complex realm
Find poly fn that has given roots
Complex zeros (conjugate pairs)
Fundamental theorem of algebra
Rational Roots theorem
Descartes rule of signs; Descartes root chart
Upper/Lower bounds theorem
Comments and Cautions
These sections cover polynomial functions and their
graphs including several important theorems. These theorems will enable us to
factor, solve, and graph large polynomials. As you learn a new theorem, practice
using just that single process, but in the long run you need to pull all the
separate theorems together in one big multi-step problem. For example,
completely algebraically without the aid of a calculator factor and graph a
polynomial like f(x) = 4x5 + 12x4 - 41x3 - 99x2
+ 10x + 24.
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 3.1-3.4.)
 | ch 3.1 should be almost all review from an intermediate algebra class. All the stuff about parabolas should be review. The only topic that may be new is applications of quadratic functions pgs 319-323. |
| ch 3.1 Concentrate on finding the vertex of a parabola both algebraically and graphically. Be able to use a grapher to find the local extrema points of a parabola (see my online Calculator Guide: Extrema Points). Use information obtained from the parabola's vertex to solve quadratic optimization application problems. |
| ch 3.2-3.4 are crucial material for this course. They cover theory of polynomial functions including lots of new terminology and many theorems, processes, and rule. By the end of these sections you should be able to factor and solve any polynomial! |
Rational Functions and Graphs
GOLDen Mathematics - College Algebra: Section 3.2
Supplemental Sites:
MathOL Links - Calg 3.2
Topics of Importance
Definition: rational function, asymptote line
Find domain, missing points, and intercept points of rational functions
Find vertical, horizontal, and oblique asymptote lines
Find a rational function given its graph
Comments and Cautions
Today we study the graphs of rational functions. Rational functions are
fractions with a polynomial in each of the numerator and the denominator. The
graph of a rational functions have special features such as asymptote lines and
holes (missing points). Our goal is to
algebraically find these features from the function and then put the information
together to produce the graph without the aid of a calculator.
I have uploaded two guides to rational functions and asymptotes. A former colleagues, Prof. Louise Hoover, offers this "note sheet" with examples:
web.clark.edu/skeely/FILES/PDF/111/guide_ratasy_hoover.pdf
and I have a "template" style guide sheet that walks you thru all the things you must look for when faced with a rational function:
web.clark.edu/skeely/FILES/PDF/111/guide_ratasy.pdf.
I hope one or both are helpful.
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 3.5-3.7.)
| ch 3.5 is meant to be covered over two days as it is contains several theorems and processes to learn and practice. Be sure to try the problems both algebraically and graphically. Can you find all the needed information using either approach? |
| ch 3.6 was covered earlier in the course. |
| ch 3.7 SKIP |
Partial Fractions
GOLDen Mathematics - College Algebra: Section 3.3
Supplemental Sites:
MathOL Links - Calg 3.3
Topics of Importance
Decompose proper rational functions into partial fractions
(including linear factors, quadratic factors, repeated factors)
Decompose improper rational functions into partial fractions
Comments and Cautions
The goal today is to rewrite a large rational function
as a sum of smaller fractions. This is the act of "decomposing" rational
functions into "partial fractions" and it is really useful in Integral Calculus!
It is FAR easier to integrate the partial fractions than to integrate the
original big old fraction.
A caution on wording: To "expand" the fraction means to equate it to a sum of fractions with unknown numerators (involving A, B, C, etc.) but to "decompose" the fraction means to complete the full decomposition including figuring out the A, B, C, etc.
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 5.3.)
| ch 5.3 is meant to be covered over two days as it is packed with theorems and processes to learn and practice. Be sure to try problems that include linear factors, quadratic factors, and repeated factors as each type expands in a slightly different way. |
Originally written: 2006-09-04
Last revision:
2009-10-13 08:30 AM