Download MAT220 Class Notes

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 that 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 in your polynomial.
A. Two terms
i. Factor the difference of two squares
a 2  b 2   a  b  a  b 
ii. Factor the difference of two cubes
a 3  b3   a  b   a 2  ab  b 2 
iii. Factor the sum of two cubes
a 3  b3   a  b   a 2  ab  b 2 
B. Three terms ---- try reverse foil
(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 to 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 prime.
Using factoring to simplify a fraction or an expression containing fractions.
Examples:
1.
4 x3  16 x 2
2 x3  6 x 2  8 x
2.
x 2  7 x  12 x 2  x  12

x 3
x2  9
3.
=
=
x
5

x 2  11x  30 x 2  9 x  20
=
II. The Rational Roots Theorem.
If F  x   an x n  an1 x n1  ....  a1 x  a0 is a polynomial with integer coefficients. If the polynomial
has any rational zeros (roots), p/q, then p must be an integer factor of a0 and q must be a factor
of an.
Example: List the possible rational zeros for
F  x   3x 4  11x3  10 x  4
.
Other important polynomial theorems for College Algebra / Precalculus.
A) Conjugate Pairs Theorems.
i. If your polynomial has rational coefficients and a  b c is a zero then so is it’s
conjugate a  b c .
ii. If your polynomial has real coefficients and a  bi is a complex zero then so is it’s
conjugate a  bi .
B) The Remainder Theorem.
If you wish to evaluate a polynomial at a number “c” just do synthetic division using
“c” and whatever remainder you 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 at 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 F  x   3x 4  11x3  10 x  4 to find ALL
of the zeros for the polynomial. Write the polynomial in factored form.
F  x   6 x5  19 x 4  23x3  82 x 2  4 x  24
Let’s Try…
III. Using conjugation (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:
2  5i
1. 3  2i =
2.
3.
x4
3 5 x
=
x  27
5 x 2
=
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 functions 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. http://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
x2  y 2  1
pick any point and notice how the coordinates make the equation true!!!
Examples: Evaluate each of the following using only your unit circle!!!!!
A)sin 2100
 5 
B) cos  300  C ) tan    D)sec5
 6 
E ) csc 7200
F ) cot
2
3
V. Simplifying Trigonometric 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
1
1
1. Sin x 
2. Cos x 
3. Tan x 
Sec x
Cot x
Csc x
4. Sin x  Sin x
5. Cos x  Cos x
6. Tan x  Tan x
Sin x
Cos x
8. Cot x 
Sin x
Cos x
Note: These first 8 are the most basic identities. The first three are the reciprocal identities.
Numbers 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 Algebra / Precalculus class? Number 7 defines the
Tangent function in terms of Sine and Cosine. Number 8 is the result of 7 and 3 (do you see it?).
7. Tan x 
You should know the “Pythagorean” Identities…
9. Sin 2   Cos 2   1
10. Cot 2   1  Csc 2 
(do you know how to obtain #10 and #11 FROM #9 ?)
11. 1  Tan 2   Sec 2 
You should know the “double angle” identities for Sine and Cosine…
20. Sin 2 x  2SinxCosx
21. Cos 2 x  Cos 2 x  Sin 2 x
(#21 has two other versions that can be obtained by utilizing #9…do you recall how to obtain
them?)
You should also know the sum and difference formulas for Sine and Cosine…
12. Cos A  B  CosACosB  SinASinB
13. Cos A  B  CosACosB  SinASinB
14. Sin A  B  SinACosB  CosASinB
15. Sin A  B  SinACosB  CosASinB
These are the ones that come up most frequently (although on occasion some of the others may
be used).
Examples: Simplify the following…
1.
1.
2.
3.
sin 
sin 2
2.
1  sin 
cos2 
3.
cos 2
sin   cos 