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Complex Numbers
Pre-Calc
Name
~,....
__
(' =
Simplify the following powers ofi.
l.
iSO
,
2.
".-I
i2S
".
,
3. i67
-- - (
l
:0-1
4. il22
Write the following complex numbers in standard form.
5.
,;( .!: )L
6. 2-,/-27
7.
-I -&'L
-6/ +i'
G
8. -4/' + 2/
Perform the specified operation and write the result in standard form.
F;
3
,J::6,.J2
9..
,
'.r-
L'.
L;Z;-'Vi).;
""
"1
10. n',)-10 .
J5L'vW~
,1"7.
v5'OL
II.
(,)-10)'
?-
lOlL-
co
3 -;;l.i
.
foOL
3+-(
13.
-;;)~
_"-
t-2
.
-lei{.
(1-3iX3-41~:='
5-'I(
_ qL
'/
(f:tiX:f::' 2/)
12.
ClFOiJ
.")..
.;>..
t t;;Jt
-.
-3 --,;-_.-''),
- ,:;I
15. (4+5i?'
,,-)
Iv, -t '-jo'- :;).("[1
16. (2+31)' + (2-3i)'
"'- '-j -L/ -\- 1;X
t~, ~
10 doJ
4
17.
4-5/
(lj-fSi. )
_
t
B
-L
,I
1/ "+L )
'-j
<I.•..
t' (
.2
-c
.,,(pf}.O[,
<-11
(;1)'('-
/8-7i \ <1"';/')
19.
(
q.L
:.4 t 4'-
2+1 )(J..f-i)
18-: -,2-1
(Ljtrc:)
!G~')-O~
''''
it- tz:=
L
/~,,,,,'t
<
(1-2/ )u-r;U )
'i tl(,~_1;"!,/;L
I _'-j~2.,
~
i..~'
'Yt'-i +- (.,. 3i -,;Lh -;;( f
I,
;f.-5?S" i) ,;as,) : ;;110 +- 5~()~t ~~
(21-71)(4 + 3/)
20.----2 - 5i
+
6
-(~-:si)(~ri)
[~'B
(lOS'
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( '?5:' t '')q~i~1
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Section 4 - 1 Polynomial Functions
">egree of a polvnomial in one variable: the greatest exponent of its variable,
tells how,many possible solutions (complex ,solutions) the function has ..
.
5,><.,0
l-
'3>c;;LO t 'f...
.
liP} .
Leading Coefficient: The coefficient of the variable with the greatest
Exponent.
':).
Zeros (roots or solution): The place where the graph crosses the x- axis (or xintercept; to find, set y = 0)
. Number of turns: The greatest exponent - 1 = the maximum number of turns.
Example 1: State the degree and leading coefficient of the 'polynomial function f{i) = 6x5
8)(2 - 8x. Then determine whether-3and
0 are zeros of the f{x).
.
d C'4v-ee-=' S-' Lt6£,~
cere-f-h ClM ; ~
o
f
s
oj' -lGvO
I
.
bu--+
-)
1'$ {L01
Example 2: Write a polynomial equation of least degree with roots 0, .fii and -.fii.
(Y.-D'/><1
)( (x.9-_
-Vii..
V.:2,i)(x-fii)",
;;\c.,:>')
)~~(-l)=
~--:-:..-'
.
.
,
)<.3 y:
-J~ 4.
~\
---
...
Example 3: State the number of co~
roots of the equation
then find the roots.
~ ,e.cd S{)Lu-f,(JYls'
.- ....••
3x2 +
C*- ',Yx + Lf)
3><- - l =- 0
"3' Ie -;. I
l'- 0 '(3
X
~
11x- 4 = 0
0
-rLj::..O
X. __ - L(
+
Name
Difference Quotient
Evaluate each of the following functions using the difference quotient.
[Cx
+ h)
SHOW ALL STEPS!
- [Cx)
h
+2
2. [Cx) = -8x
- '6(q~)T~-
t8~i-~)
.h
- ~-
3. [Cx)
I"h +-i
= 2x2
3
')--,3 -
~(~;-11
¥' -,,;
'3
-~0
f:-
E])
S.
(~J(;l-~
~(,."-I- ;!~h t ~'') _;2>c ~ ~
10+Lh h t.;l.h",.3'
1
h
-
Ix (Lf,( t;JV
K
x2 + Sx ,; 6
L 1(1'h) ~+-'5 {t1h} - (" - (/'
6. [Cx) =
_2_
3x-4
/'~
J~~th2.t4xt5h-;{-~-s:/P&_
h
.'
.h ?
• .1
+- ;10{
)<:
h
:~~
/
,
"
Review:
_\ -.1
~
2..:
~I
,
1,.'. I
-h
h -l-!>
h
h (ht2~
v - n\'
lj~\y,'.
t5'L -l" )
_
+-5)
Name --------
Pre-Calc
Pedorm the indicated operations. Write the resulting polynomial in standard form.
1. (15x2-6)-(-8x -14x -n)
3
2
1.
ax
'}
f.;("I)("'+11
Find the product. Write the resulting polynomial in standard form.
3.
3. (3x-2Y
4. [(x+l)- y]2
5.
Completely factor the following expressions.
6. 12x2
48
-
7. 16+6x-x2
7.
+ 40
9.
9. 5x3
Simplify. Express answers with positive exponents.
'5/ 'i'LJ
10.
(:)3(~r
10.
l'
:~:
'-'
:-;,
11.
:-:;
( .::.-L
-3 4f
5
IJ- S11.
\1
.,;:1.
<:J
'1
~
W rite the following expression in reduced form.
12+x-x2
19. ----2
2x -9x+4
_I (
?-IX)
19"_~_~7=-
I
'J){
_
-I)
Perform the indicated operation on the following expressions and simplify.
(x+~)( )<-~)
2
20
2x +x-6,
•
2
X
+4x-5
x3 -8
21. ---,'--x2 -4
3
2
x -3x
+2x
2
4x -6x
x2 +2x+4
x3 +8
.
20.
;( L \(-rt))
21.
Simplify the following compl@xfractions.
22.
,'23.
x-2
(~-3)
1 __ 1 )
(
x-I
22.
---------
(J.-1'i ') (X-I)_
23.
'Y,(x-.;;I..)
Name -------
Precalc Review
Exponentials, Logs, and Trig
,
Solve for x.' Show all your work.
2. 4e2x=5
<j
7j
eP.t.: s/"
)1 V1
e~.: jVl >I'f
~ ~ .Q.:S7,!
eX:.
;J
)/"1./--/(\ ~
Cx~
I~q~
)(-:- 0,111)7
x
=6
x
4. 8 _2e
x
= 300
e-'f ~ l@
5. 500e-
'?.l: -~
eX;
or-Lrll
\
X
)t'l t
- 't : 1. t'l f3!E. )
~ ,
~~
cY~o~
6. 20(100-e
70
2)
= 500
-,ro
,em. /1;.:
-} lSi)
-ell/;;.::
X~OS(og
et/)..
'-1~
~
7' '
1Vlt '1~:J.vt
.i-
Evaluate the expression without using a calculator.
7. log216
J~
-/ro
/(.,)( -;.
~
11
if
-)
~(i~
'if, t..~S-
9. 10glO 0.01
8. log'64
~ 1':. I""
J-
7~
'-/
I
'-}~ :;'i
~.
.2.x--'
~
Use the properties oflogs to write the expression as a sum, difference, and/or constant
multiple oflogarithms.
../xy4
11. log-4-
10. log~x2(x+2)
-i!!09
X;'
f-.JO~
L ~~
i7-
Joqx +- -1-
(\t+'~) ,
~ (t+~)
JLXj Cd).)
12.I{X~~l)
'J- V1 (X ':. I)
-
3)1) 'j.
+.
PC:\ l\
_ {-- l-j ()I,l
~d<-\ 0. \
~[) Lf'
,
d-
Jn
11:31-):::.;;1.
.e~::: 3)'.
3
x.~
:::1.%3
1/~,1/1;\YH
.
:( -,)
;)..
(-X-"J.) ("J'c') =. jn
C x=;)Z;;I)( '3');
.
c."-l~
~
= 2lnx
><
)<.,;:L .;L
J J<;;1.- 3)( - 4", +G> =.
-
.3
-
'1-'') (})
&, ~
14. In(x-2)+ln(2x-3)
-
-a-
~ ~i
G
Solve the logarithmic equation algebraically. Show all your wor~
13. 2ln(3x) = 4
I~
X
X;;' - '7~ •..(.,~c
Use trig identities to 0 tain solutions to these equations giving all answers to the nearest
degree.
18. 2cosx = 1,-2;r:O;x:O; 2;r
CD5x;.
1
19.2sin2x=1::0:O;x:O;2;r
{5;n\::~
) ~ I ).J
-,
't)
COS-l(~)o
()(1f
3
. 20. 2coi'-x-smx
(
= 1:: 0:0;x :::;2;r
21.
.J3 sin x
= cosx:: 0:0; x:O; 2;r
.8i,?:"
't
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