ALG-TRIG Notes - GCC links

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Oct 10, 2013 (3 years and 9 months ago)

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MAT220 Class Notes.

Algebra and Trigonometry Review

Homework for the Algebra and Trigonometry Review can be found by clicking on the
“UNIT HOMEWORK” link on your class webpage.

This extremely short review is only meant to cover a few of the topics from

Algebra and
Trigonometry that seem to come up quite frequently in MAT220. It is expected that YOU are
well versed in
all

of the necessary
topics from College Algebra and Trigonometry as they are
prerequisites for this course.


There are some web sites th
at you can go spend
SIGNIFICANT

time on if you are lacking in
some of these areas. These web sites are available on your class web page but I have listed a few
of them here below.


1.


If you need a really good review for Trigonometry click
here


(
http://aleph0.clarku.edu/~djoyce/java/trig/
)

2.


Need to read some lessons and practice FACTORING?


click
here
.



(
http://regentsprep.org/regents/Math/math
-
topic.cfm?TopicCode=factor
)

3.


Need some help remembering how to solve polynomial equations?


click
here


(
http://oakroadsystems.com/math/polysol.
htm
)

4.


Factoring quadratic trinomials


click


here



(
http://web.gccaz.edu/~apmckint/Bottoms%20up%20word.doc
)

5.


Trigonometic Identities


click
here



(
http://web.gccaz.edu/~apmckint/MAT220/Trigonometric%20Identities.doc
)



6.


A lot of different Precalculus Review Topics can be found
here


(
http://www.analyzemath.com/
)

I. Factoring (and using factoring to simplify fractions)

Here is a general Factoring Strategy that you should use to factor polynomials.

1.

Always factor out the GCF(Greatest Common Factor) first.

2.

Next check the number of terms i
n your polynomial.

A.

Two terms



i. Factor the difference of two squares





ii. Factor the difference of two cubes







iii. Factor the sum of two cubes








B. Three terms
----

try reverse f
oil

(although sometimes a three term polynomial will factor into the product of two
trinomials)


C. Four terms
----

try factor by grouping


D.

If none of these work you could try to use the rational roots theorem from


College Algebra / Precalculus t
o find a zero of the polynomial (which in turn


will give you one of the factors) and you may be able to go from there.


Note: We will review the Rational Roots Theorem during our review.




3. Repeat step 2. until all factors are pri
me.

Using factoring to simplify a fraction

or an expression containing fractions
.


Examples:


1.

=




2.


=



3.


=





II. The Rational Roots Theorem.


If is a polynomial with
in
teger

coefficients. If the polynomial
has any rational zeros (roots), p/q, then p must be an integer factor of a
0

and q must be a factor
of a
n
.


Example: List the possible rational zeros for .






Other important p
olynomial theorems for College Algebra / Precalculus.

A)

Conjugate Pairs Theorems.


i. If your polynomial has
rational

coefficients and is a zero then so is it’s


conjugate .



ii. If your polynomial has
re
al

coeffic
ients and is a complex zero
then so is it’s


conjugate .


B)

The Remainder Theorem.


If you wish to evaluate a polynomial at a number “c” just do synthetic division using


“c” and whatever remainder y
ou get will be f (c). Note: This works for ANY number,


integer, irrational or imaginary.


C)

The Factor Theorem.




If doing synthetic division with “c” yields a remainder of zero then we say that “c” is


a zero (or root) of f
(x) AND it means that ( x


c ) is a factor of f (x).


D)

The Intermediate Value Theorem.


For any polynomial P(x), with real coefficients, if a is not equal to b and if P(a) and


P(b) have opposite sings (one negative and one positive
) then P(x) MUST have a
t least


one

zero in the interval (a , b).


Note: This Theorem holds for any CONTINUOUS function.


We will study the idea of continuity in MAT220.

Example: Use your “list” of possible rational
zeros of to find ALL


of the zeros for the polynomial. Write the polynomial in factored form.













Let’s T
ry…










III. Using conj
ugation (multiplying by 1)


Fact: Multiplying any expression by the number 1 does NOT change it’s value (although it may


change it’s appearance)


Simplify:


1. =






2. =





3.

=










IV. Trigonometry and the Unit Circle.


1.

http://www.analyzemath.com/unitcircle/unitcircle.html

Look at how the unit circle can be
used to graph the standard trigonometric func
tions Sine, Cosine and Tangent.


Cos A = the “x” coordinate. Sin A = the “y” coordinate. Tan A =
“y” / “x”


Sec A = 1 / “x” Csc A = 1 / “y” Cot A =
“x” / “y”


2.

http://www.libraryofmath.com/animations/unit_circle.gif

Look at the animated GIF to


see how the multiples of pi/6 th’s are labeled on the unit circle.

3.

ht
tp://www.libraryofmath.com/unit
-
circle.html

Scroll down and pick up the multiples of


pi/4 th’s for the unit circle.


Here is a blank copy of the unit circle for us to fill out.
You should practice duplicating this
yourself!!!!




























Note: The equation of the unit circle is





pick any point and notice how the coordinates make the equation true!!!

Examples: Evaluate each of the following using only your unit circle!!!!!







V. Simplifying Tri
gonometric expressions using identities.


Note: Remember a numbered list of trig identities can be found on your class web page


(
http://web.gccaz.edu/~apmckint/
MAT220/Trigonometric%20Identities.doc
)


You can print yourself out the page with all of the identities on them BUT the ones you should
know WITHOUT any doubt are as follows….

1.


2.


3.



4.


5.


6.



7.


8.

Note: These first 8 are the most basic identities. The first three are the reciprocal identities.
Nu
mbers 4 and 6 tell you that the Sine and Tangent functions are “odd” functions and number 5
tells you that the Cosine function is an “even” function. Do you remember what an “odd” and
“even” function is from your College Algeb
ra / Precalculus class? Numb
er

7 defines the
Tangent function in terms of Sine and Cosine.

Number 8 is the result of 7 and 3 (do you see it?).








You should know the “Pythagorean” Identities…


9.


10.


11.

(do you know how to obtain #10 and #11 FROM #9 ?)








You should know the “double angle” identities for Sine and Cosine…

20.

21.

(#21 has two other versions that can be ob
tained by utilizing #9…do you recall how to obtain
them?)







You should also know the sum and difference formulas for Sine and Cosine…

12.



13.



14.


15.


These are the ones that come up most frequently (although on occasion some of the others may
be used).


Examples:
Simplify the following…






1.








2.







3.