Download Algebra and Trig Review notes (fillable) F15

Algebra / Trigonometry Review (Notes for MAT220) NOTE: For more review on any of these topics just navigate to my
MAT187 Precalculus page and check in the Help section for the topic you wish to review!
I. Factoring (A fundamental skill that YOU must have)
Ex. 1. Factor out the GCF leaving no fractions “inside” and no negative exponents in your final answer.
x2 
2
1
1

 2 x  1 3  2   2 x  1 3  2 x
3
Ex. 2. Factor out the GCF
x3  2e 2 x  e 2 x  3x 2
For each of the following, factor completely.
Ex. 3. 3x5  48 x
Ex. 4. x3  27
Ex. 5. x 3  8
Ex. 6. x 2  5 x  6
Ex. 7. x 2  5 x  6
Ex. 8. x 2  4 x  4
Ex. 9. x 2  4 x  12
Ex. 10. 6 x 2  x  6
Ex. 11. 8 x 2  2 x  15
Ex. 12. 2 x 3  6 x 2  8 x  24
II. Completing the square (adding zero)
Rewrite the following quadratic function in “graphing form” f  x   a  x  h   k
2
f  x   3x2  12x  2
III. Working with fractions (be sure that you know what the various “properties of fractions” are)
PROPERTIES we will discuss.
1.
2.
3.
4.
5.
6.
a c ac
 
b d bd
a a c ac
  
c  0
b b c bc
a c a d
  
 Do YOU know why this is? 
b d b c
a c ad  bc
 
b d
bd
ab a b
 
c
c c
a b a
b
 b  a 
c
c
c
DO NOT INVENT YOUR OWN “PROPERTIES”! (I will discuss some “common” mistakes shortly)
A. “Multiplying by 1”- something you will do A LOT in Calculus.
Here are some examples of where YOU have previously “multiplied by 1” (Property 2)
Ex. Adding or subtracting fractions
5 3
 
6 4
Ex. Rewriting complex fractions in standard form
Ex. “Rationalizing the denominator”
x5
2 x9
2  3i

3  4i
B. Multiplying or Dividing fractions (Properties 1 and 3)
x 2  7 x  12 x 2  x  12


x 3
x2  9
C. Adding and Subtracting fractions
x
5
 2

x  11x  30 x  9 x  20
2
D. Simplifying fractions. (Cancel FACTORS NOT TERMS) i.e “using property 2 backwards”
Ex.
26

4
26

4
26
or

4
Ex.
x2  4

x2
x2  4

x2
We will sometimes use properties 5 and 6 to “split up” fractions to help us with a calculus problem.
Ex.
sin x 1  cos x 

x
sin x 1  cos x 

x
or
sin x 1  cos x 

x
IV. Evaluating polynomials.
f  x   5x4  2 x3  5x2  9 x 11
Find
f  2   5  2   2  2   5  2   9  2   11
4
3
2
=
=
=
OR you can use “synthetic division” and the “remainder theorem”!
2
5
2
5
9
 11
OR you can use your calculator (TI….Table Set….Indep Var….Ask)
g  x   x6  4x5  3x4  2x2  5x  3 . Use synthetic division to find g 1 . Note that g  x  is “missing” the x3 term.
Shortcut for evaluating a “polynomial” at x = 1…….
Since g 1  0 (from the “remainder theorem”) we say that “1 is a zero” and that  x  1 is a factor of g  x  (from the
“factor theorem”)
Synthetic division can often be helpful in factoring when traditional methods fail.
x3  5 x  4

x3  4 x 2  7 x  4
In College Algebra / Precalculus you should have studied “The Rational Roots Theorem” and used it to help factor or find
zeros for polynomials. (See the last page of this packet for more review on this)
f  x   3x4 11x3  10 x  4
V. Graphs of relations
1. Be able to “interpret” a graph. Consider the graph of y  f  x  given below.
A. f  3 
B. f 1 
C. Y-intercept =
D. How many zeros? ______
E. Is y  f  x  a function ? Explain
______________________________
______________________________
F. State the Domain and Range using
interval notation.
Domain:
Range:
y  f  x
2. You need to KNOW what the graphs of some “basic” functions look like from your College Algebra + Trig classes (or
Precalculus class). In particular you should know WHEN (if ever) these functions are zero. Look these up yourself if you
do not already know them and draw them in your notes!
yx
y  x2
y  x3
y x
y
1
x
y  ln x
y  ex
y  sin 
3. Be sure that you “understand” the definition of Absolute value
Official Mathematical Definition:
Examples:
5
3 
 w if w  0
w 
 w if w  0
0
x2  1 
Write f  x   4 x  2 as a piecewise defined function and graph the result.
y  cos 
y  tan 
4. Be able to graph (and read) a piecewise defined function.
Graph:
  x  2 2  1 x  1

g  x    x  3  1  x  2
 x 1
x2

Note: Sometimes the pieces matchup and the function is “continuous” at that x value and other times the pieces do NOT matchup and
the function is discontinuous at that value of x. We will study the concept of continuity in Calculus.
VI. Trigonometry
1. You should know some basic trigonometric identities and be able to use them to simplify trigonometric expressions.
(see a later page in these notes for a list of trigonometric identities).
Examples: Simplify
A.
cos 2 

1  sin 
B.
cos x

sin 2 x
C.
1  tan 4 

sec 2 
2. You should know your “unit circle” and how to use it to find trig function values AND inverse trig function values. See
a page later in these notes for more information on this.
Examples: Draw “partial unit circles” to show where the value for each of the following comes from…
 
1. sin   
 3
 2 
2. tan 

 3 
 1
3. arcsin   
 2

3
4. tan 1  

 3 
Here are 28 trigonometric identities that you studied in your Trigonometry or Precalculus class. You do NOT need to memorize all 28
of them BUT there are several that you should know because they come up frequently in Calculus.
Trigonometric Identities
1. Sin x 
1
Csc x
2. Cos x 
4. Sin x   Sin x
7. Tan x 
Sin x
Cos x
9. Cos 2 
1
Sec x
3. Tan x 
5. Cos x  Cos x
8. Cot x 
 Sin2   1
1
Cot x
6. Tan x  Tan x
Cos x
Sin x
10. Cot 2 
 1  Csc 2 
11. 1  Tan 2   Sec 2 
12. Cos A  B  CosACosB  SinASinB
13. Cos A  B  CosACosB  SinASinB
14. Sin A  B  SinACosB  CosASinB
15. Sin A  B  SinACosB  CosASinB
16. Tan A  B  
TanA  TanB
1  TanATanB
17. Tan A  B  


 x   Cos x
2

TanA  TanB
1  TanATanB


 x   Sin x
2

18. Sin 
19. Cos
note: #’s 18 and 19 also hold for the function pairs….tan x, cot x AND sec x, csc x
Cos 2 x  Cos 2 x  Sin 2 x
20. Sin 2 x  2 SinxCosx
21.
 2Cos 2 x  1
22. Tan 2 x 
=1  2Sin 2 x
23. Sin 2 x 
1  Cos2 x
2
24. Cos 2 x 
1  Cos2 x
2
26. Sin
x
1  Cosx

2
2
28. Tan
x
1  Cosx
Sinx
1  Cosx

=
=
1  Cosx
Sinx
2
1  Cosx
27. Cos
25. Tan 2 x 
2Tan x
1  Tan 2 x
1  Cos2 x
1  Cos2 x
x
1  Cosx

2
2
The identities that come up often in calculus are #’s 1 – 3, 7, 9 – 11, 12 – 15, 20, 21.
Note: #’s 4 – 6 state that our trigonometric functions are either “odd” or “even” functions!
A function is “even” if f   x   f  x  for all x in the domain of the function.
A function is “odd” if f   x    f  x  for all x in the domain of the function.
Those of you going on to take Calc II will also need to know the power reducing identities #’s 23 and 34
Here is a copy of The Unit Circle…YOU should know this completely!!!!!
As YOU know from your Trigonometry background, the “x” coordinate of a point on the unit circle is the cosine of the given angle
and the “y” coordinate of a point on the unit circle is the sine of the given angle. In terms of the coordinates on the unit circle we
know….
cos   x
sec  
sin   y
1
x
csc  
1
y
x
y
y
cot  
x
tan  
Note: the equation of the unit circle is
x 2  y 2  1 and as is the case with every graph, if a point lies on the graph then the
coordinates of the point must make the equation true. So, if you take any of the coordinates shown on the graph and substitute them
into the equation
x 2  y 2  1 you will get a true statement.
When you studied Trigonometry you restricted the domain on your trigonometric functions so they became 1-to-1 and consequently
would have an inverse that was also a function.
f  x   sin x 

2
x

f 1  x   sin 1 x  1  x  1

2
1  y  1
f  x   cos x 0  x  



2
x
2
 y

2
f 1  x   cos 1 x  1  x  1
0 y 
1  y  1
f  x   tan x 


2
  y  

f 1  x   tan 1 x    y  


2
 y

2
Knowing the RANGE for the inverse trigonometric functions will be very important later in this class.
REMEMBER THAT IF YOU NEED TO REVIEW MORE TRIGONOMETRY THEN YOU CAN ALWAYS GO TO MY MAT187
PRECACLULUS PAGE AND CHECK IN THE HELP SECTION TOWARRDS THE BOTTOM FOR VARIOUS TRIGONOMETRY
TOPICS.
Rational Roots Theorem:
Let f  x   an x  an1 x
n
n 1
 an2 xn2  .....  a2 x 2  a1x  a0 be a polynomial with integer 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 an.
Example: List the possible rational zeros for
f  x   3x4 11x3  10 x  4
.
p : 1,  2,  4
q : 1,  3
p
1
2
4
: 1,  , 2,  ,  4, 
q
3
3
3
Other important polynomial theorems for College Algebra / Precalculus.
Conjugate Pairs Theorems.
i. If your polynomial has rational coefficients and
ii. If your polynomial has real coefficients and
a  b c is a zero then so is it’s conjugate a  b c
a  bi
is a complex zero then so is it’s conjugate
A) 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.
B) 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).
C) The Upper Bound Theorem
If doing synthetic division with a positive number yields a whole row of non-negatives
then there is no zero greater than the one that you just tried.
D) The Lower Bound Theorem
If doing synthetic division with a negative number yields a whole row of alternating signs
then there is no zero smaller than the one that you just tried.
E)
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: The Intermediate Value Theorem holds for any CONTINUOUS function.
We will study the idea of continuity in MAT220.
a  bi