ELEMENTARY & INTERMEDIATE 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-books GOLDen Mathematics: Elementary & Intermediate Algebra. This section of material only is downloadable for a nominal fee at www.lulu.com/content/431056. See "Tell me more about Keely's GM book".
(If you are having trouble with exponents, polynomials, and/or factoring, the http://www.lulu.com/content/431056 download covers these two chapters in your text and is probably the best buy of all of the GM's sections available as single downloads and may be able to alleviate any troubles you are having with this material.)
Exponents
GOLDen Mathematics - Elementary Algebra: Section 6.1
Supplemental Sites: MathOL Links - Alg 6.1
Topics of Importance
Integer Exponents (positive, negative, and zero)
Rules of Exponents - product, power, and quotient
Scientific Notation - decimal form
scientific notation,
calculator evaluation
Comments and Cautions
Today we will investigate exponents and hopefully our algebraic expertise
will increase, well, exponent-ially! We have
been working with positive exponents all along, but now we will formalize the
processes and rules. We will also add to our repertoire negative exponents (like
2-4 and x-3) and powers of zero (like 30
and -y0). Negative powers mean to take a reciprocal (flip), so for
example:
x-3 means 1/x3, (2x)-4=1/(2x)4=1/(16x4), 2x-4=2/x4, and (2/x)-4=(x/2)4=x4/16.
Caution: Notice how important the parentheses are? They indicate exactly what is
to the power and what isn't.
We will learn scientific notation which allows us to easily deal with very large (and very small) numbers without having to write out and keep track of dozens of zeros. Read the brief "scientific notation" section of my Calculator Guide: Basics of Arithmetic for examples of scientific notation on various TI calculators. Be sure you can convert between scientific notation and decimal form by hand as well as use a calculator to evaluate expressions containing numbers in scientific notation. Scientific notation opens up a world of applications especially in the physical sciences!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 5.7.)
 | ch 5.1 covers the basic rules of positive exponents; 5.5 covers a zero exponent; 5.7 covers negative exponents. |
| ch 5.7 The many rules of exponents displayed in blue boxes throughout this section are worth memorizing. |
| ch 5.7 includes scientific notation which is important to be able to compute manually and electronically. |
Polynomial Operations
GOLDen Mathematics - Elementary Algebra: Section 6.2
Supplemental Sites: MathOL Links - Alg 6.2
Definition of polynomial & related terminology
Polynomial ops: evaluate, add, subtract, multiply
The FOIL method
Special products: difference of squares, binomial squared, perfect square trinomial
Multiplying polynomials by the vertical method
Using Pascal's triangle to simplify binomials to higher powers
Following order of operations to perform poly ops
Dividing polynomials including long division
Comments and Cautions
Our goal today is to learn to
manipulate polynomials. Remember when you were in elementary school and you
were just learning about numbers? You learned to add and subtract, later
multiply, and then the dreaded long divide. Well, that is what we are going to
do today, only with polynomial expressions rather than with plain numbers. So
put your algebra hats on and get your pencils warmed up, because we are going to
embark on a "polynomic" adventure! There is a ton of information here:
terminology like "degree of a polynomial", operations, the very important FOIL
method to multiply two binomials, formulas like difference of squares, and long
division using polynomials. Take your time as you work through the material and
practice lots of problems. As you are doing the FOIL problems think about how
you might work this process backwards (going from the answer trinomial back to
the product of two binomials) because that ("factoring") is exactly what we
start out with in math 095!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 5.1-5.6.)
| We will spend several days working through ch 5.1-5.6. |
| ch 5.1 begins with basic rules of positive exponents. |
| ch 5.1 contains much important new terminology plus basic operations like combining like terms. |
| ch 5.1 covers adding and subtracting polynomials as well as introducing the important term "the opposite of a polynomial". |
| ch 5.2 covers polynomial multiplication including the distributive law (for multiplying a monomial through a polynomial), the horizontal method (for multiplying two polynomials that are each binomials or larger), and the vertical method (similar to the horizontal method but written in a different configuration). |
| ch 5.3 covers the very important FOIL method which is a shortcut for multiplying two binomials. |
| ch 5.3 has formulas that can be used instead of FOIL for multiplying a difference of squares or for squaring a binomial. I strongly recommend that you memorize these formulas prior to math 095 when we will take the FOILed out polynomial and backtrack to the ( )( ) form. |
| ch 5.3 The text fails to cover using Pascal's triangle here as I believe it should. For an explanation, see page 971, explanatory posts on the main classroom board in BB, and the supplemental sites linked above. |
| ch 5.4 expands the processes in 5.1-5.3 by performing them with multi-variate polynomials (polys that contain several variables like x, y, and z's not just x's). |
| ch 5.5 covers dividing polynomials by monomials and also discusses a "zero exponent". |
| ch 5.5 pg 345-246 Pay attention to the two "cautions" in the yellow boxes. They describe very common errors! |
| ch 5.6 covers simplifying rational expressions (fractions with a polynomial in the numerator and a polynomial in the denominator) via long division. |
| ch 5.6 SKIP "synthetic division" as shown on pg 356-358 and example 5. |
Factoring Polynomials
GOLDen Mathematics - Intermediate Algebra: Section 6.3
Supplemental Sites: MathOL Links - Alg 6.3
Topics of Importance
Factor polynomials by the following methods:
Greatest common factor, grouping, trial 'n check, difference of squares
Perfect square trinomials
Sum/Difference of cubes
Comments and Cautions
Welcome to math 095! Remember "prime factoring" from an
arithmetic class? Like 12=2*2*3? Well, that is what we are going to do in
chapter 6, except with polynomials. Factoring a polynomial is basically "unFOILing",
so this chapter is basically doing the previous chapter backwards. Instead of starting with
(x-2)(x+5) and FOILing to get the polynomial we will start with x2+3x-10
and factor down to the two binomials. There are several methods of factoring
covered in this chapter. A few things to keep in mind as you work through the
content:
| Always factor the GCF out first - no matter what other methods of factoring might be involved, start with the GCF. |
| Grouping method only works when the poly has an even number of terms (usually 4). |
| The "trial 'n check method" or the "ab method" can be used to factor a trinomial. This is the most common sized polynomial to factor, so get this process down well! Notice the differences between factoring a poly of the form 1x2+bx+c and ax2+bx+c where the coefficient of the x isn't 1. The latter is much trickier and will take some practice! |
| Caution: a binomial squared like (x+3)2 must be FOILed out to get the "perfect square trinomial" x2+6x+9 - never take the power across the addition to get x2+32! Keeping this in mind will help you recognize PSTs when working backwards to factor into a binomial squared. |
| If the original poly has only two terms then it factors as either a difference of squares, difference of cubes, or sum of cubes. These formulas are worth memorizing. |
| There is no way to factor a sum of squares like x2+9 - well not until we cover imaginary numbers in chapter 9 ;} |
| Factoring polynomials enables us to solve polynomial equations via the zero product rule. |
| Caution: Expressions can only factored and not solved. The original problem must contain an equals sign (thus be an equation) to be able to be solved for x. This is a very important difference! The "zero products rule" is only applicable to equations not expressions. |
Factoring is used extensively in the remainder of the course so really spend some quality time this week practicing problems. As you do so look for patterns that may help you to identify which factoring method works on which polynomials.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 6.1-6.5.)
| We will spend several days working through ch 6.1-6.5. |
| ch 6.1 covers the first two types of factoring: factoring out the GCF (greatest common factor) which is basically "undistributing" and grouping method. |
| ch 6.2 and 6.3 cover factoring trinomials into two factors by trial 'n error approach (alternatively using the "ac method"). |
| ch 6.4 includes factoring special forms: difference of squares that factor into the form (_+_)(_-_), perfect square trinomials that factor into a binomial square ( )2, and sum or difference of cubes - the formulas on pg 411 are worth putting in your notes. |
| 6.5 It is very important to practice some exercises in this section since they combine all the factoring methods and mix the processes so you must determine which method of factoring is to be used when and in which order. Enjoy! |
Originally written: 2006-006-15
Last revision:
2008-01-05 03:40 AM