1.b n bm bn m .it#atxiiax=x.x6.x9=x4.y6=x4 ÷e±j¥f...j.m. villalobos c 2015 §6.3 gcf and grouping...
TRANSCRIPT
J.M. Villalobos
c
� 2015
Math 73 Lecture Notes Date
§6.1 Properties of Exponents
1. b
n· b
m= b
n+m
2.
b
n
b
m= b
n�m
3. (b
n)
m= b
nm
1
XZ . ×3
÷E±j¥f.it#atxiiax=x.X6.x9=X4.y6=x4
-
¥=±xxY"
←←x?E?xtxn ¥=7± 49
÷x÷ox÷eox" / e÷,a*x÷ '
(2×3)"
.x6
(x4)6=x4 1 x4.x6= xlo -
8×7
Hit;.IT#t3xxD=i6x'2I8×7
-
⇒ #} = 2
'
8×7
⇒DX "
J.M. Villalobos
c
� 2015
4. b
�n=
1
b
n
5. b
0
= 1
2
- 8 1- 0¥×
. 8.
81-6 )
1=5 Fixerx
-6 ÷I 3=÷ it 'I.
|= ,÷s÷tx.
= st( 2×53=1 =×÷ . ,E=E
(
2×33×8=
1 = x-2
853 = I
iii÷==20=1 ¥= ,
e=aA÷=a"' "
=a°
p°= 1
÷¥iH?I¥!|ti¥¥t**- -
@ 3= 144×14Y2°
⇒ (2×755)�3�
⇒ 8×2'
y' 's
¥µ⇒s;÷Txx÷an'Eni
' x÷ii '
J.M. Villalobos
c
� 2015
.
Scientific Notation
.
3
q
1,200,000,000.0 = 1.2*10SN
✓
←dodge2=7.2*10-7 SN ✓
q
6,320.0*10 Sn X
-w
× ?x'9=x"
6.32*103*109 = 6.32*1012
Sri0122=1 trillion
0.000047*10 yo9= Billionwho
4.7*10-5*1016=4.7*10"
106=1 Million
÷0*106 ) ( 45.0*10-3 )135.0*1031.35*102*103
1.351€
J.M. Villalobos
c
� 2015
§6.2 Multiplication of Polynomials
Polynomials
P (x) = anxn+ an�1
x
n�1
+ ...+ a
1
x+ a
0
Ex: Multiply the following
3x(x
2
� 3x+ 5)
Ex: Multiply the following
(4x� 1)(x
2
� 3x+ 5)
4
{ yDESI trinomial
= 1/2+5×+1 2nd
2×+5Linear
Binomial
3 zrdMonomial
: 4×2-5×+7 4rd Polynomial
fixity-
= 3×3.9×2+1511
a÷4×3 -
12×2+20×-112+3×-5.
4×3 - 13×2+23×-5-
J.M. Villalobos
c
� 2015
Ex: Multiply the following
(2x� 3)(4x
2
+ 6x+ 9)
Ex: Multiply the following
(2x� 5)(2x+ 5)
Ex: Multiply the following
(2x+ 5)
2
5
Fx3+14×2+184
- 12×2 - 18×-27¥7-
= 4 X
"
+1¥x¥4×2-254
.X + 8
= (2×+5>(2×+5)×+5-⇒gµY÷B= "y"a÷lYxxt2In|
( ×+su×+ , ,←
area
×2 +8¥X +40
÷- 13 X +40
J.M. Villalobos
c
� 2015
§6.3 GCF and Grouping
Ex: Factor the following
20x
3
y
4
� 15x
6
y
2
Ex: Factor the following
x(a+ b)� y(a+ b)
Ex: Factor the following
x
3
� 3x
2
+ 4x� 12
6
¥ ;} ;" 213146 '
't
GcF=411214,81 lb
¥Xxii; ac+=x
'
/ ××;:{
acF=x2y:c3y4- =4y25x3y2(4y2.
3×3 ) 5×372.
p - 15×642ACF Ey
= - 3X3y°
[email protected]) ( m - 5) ( htt )
¥msahq¥%g€yFOG3- terms
not
←( X - 3) +41×-3 )
( x . 3) ( x2t4 )
J.M. Villalobos
c
� 2015
Ex: Factor the following
2xy + 6xp� 3y � 9p
Ex: Factor the following
x
4
� 5x
2
+ 2x
2
� 10
7
÷y t 3 P ) - 3 ( Y + 3 P )
( y +3 p ) ( 2 x - 3 )
* 2- 5) + 2 ( X 2- 5 )
( X 2- 5) ( × 2+2 )
J.M. Villalobos
c
� 2015
§6.3 Trinomials
Ex: Factor the following
x
2
� x� 12
Ex: Factor the following
x
3
� 4x
2
� 45x
Ex: Factor the following
12x
2
� 11x+ 2
8
-
÷4
12=4.3802= #
= ( X - 4 )( x+3 )
±
X2 - 4×+3×-12
* - 4) +3 ( X - 4) ⇒ ( X - 411×+3 )
45=9.5= 15 - 3
X(X2- 4×-45 ) ⇒ × ( X - 9 )(xt
5)tiF
24 = 4-6
×
12.2 ×
8 - 3-
12×2 - 8×-3×+2
=(3×-2) - 1 ( 3×-2 )
(3×-2) ( 4×-1 )
J.M. Villalobos
c
� 2015
§6.5 Factoring using formulas
F
2
� L
2
= (F � L)(F + L)
F
3
� L
3
= (F � L)(F
2
+ FL + L
2
)
F
3
+ L
3
= (F + L)(F
2
� FL + L
2
)
Ex : Factor 4x
2
� 25
Ex : Factor 9m
2
� 100y
2
Ex : Factor x
2
(x� p)� 4(x� p)
9
FIX,
L= 3
←
x=9=cx)2 . (3,2
= ( x - 3 )(xt3 )
⇐9x=x(x-I
F2tL2 ← Prime-
= ( 2×32 - (5)-
F=2X = (2×-5)/2×+5 )
L= 5
= ( 3m)2 . Cloy)2
F= 3M= ( 3M - IOY )( 3Mt IOY )
L= :x-p )(x2 . 4 )
( X-P )( x . 2 )(
XtOptimus
1/2+25 ← prime
J.M. Villalobos
c
� 2015
Ex : Factor 8x
3
+ 27
Ex : Factor y
3
� 1
Ex : Factor 2x
3
� 50x
Ex : Factor x
6
+ 64
10
F 3tL3= ( FTL )( FIFLTE )
= ( 2x)3+( 3 )3
F=2x,
-52=4×2= ( 2×+3 )( 4×2-6×+9 )
L= 3,
6=9
FL=6X
=(y )3 - HP
F=Y,
F2=y2 = ( y . a ) ( y2+y +1 )
L=1,4=1FL=y
= 2×1×2-25 )
=2X ( × - 5) ( Xt5 )
= ( X' )3+( 4 )3
F=X2,
Ft x' '
= ( x2t4)(x4 - 4×2+16 )
L=4,
4=16
F. L= 4×2
J.M. Villalobos
c
� 2015
§6.6 Solving equations by factoring
Multiplication Property of zero
Ex : Solve 4x(x� 3)(2x� 5) = 0
Ex : Solve x
2
(x� 2)� 9(x� 2) = 0
11
A .b= 0 ⇒ a=o or b= 0
#
d +5+5
411=0or X - 3=0 Or 2×-5=0
I F +3 +32×=5
@o XD X=@
⇒
( X -2 )( xkq )=o€
( X - 2) ( × -3>(1/+3)--0
t 4 t
11=2 11=3 X= - 3
÷x . 12=0
(1/+371×-4)=0a t
X= -3 X=4
J.M. Villalobos
c
� 2015
Ex : Solve x
2
� 9x = 10
Ex : Solve 2x
3
� 10x� 12 = 0
Ex : Solve (x + 3)
2
= 25
12
1/2-9×-10=0( Xt 1)( x . ( o )=O
d t
11=-1,
11=10
2
2(x2 - 5×-6 ) = 0
21 X - 6) ( xti ) = 0
X . 6=0 or Xtl = 02€@or €
gz
→
1×+3)2 - 25=0
⇒ ( Xt3)( 1/+31=25 ( xt } - 5) ( Xt3t5)=°
⇒ 112+3×+3×+9=25 ( X - 2) ( ×t8)=0
⇒ 1/2+6×+9=25-
µ=¥g4⇒ xzta¥61205 | ( xtsyx . stzx
=y. ' ⇒ ( X +8) ( × -2)=o
⇒X=2=
"
Missile"
hlt ) = - 16+2+64++80
Ground ⇒ hit )=o
got DM- 16+2+64++80=0:' '
#- 16 ( t2 - 4t - 5) = 0
0ground - 16 ( t - 5) ( ttl ) = 0
¥T€=T⇐x5 seconds to hit the ground .
0¥ hlt )= -16t2t64t to
hlt )=o
⇒- 16+2+64 t=0#•
⇒ i±H±=°to t=4 to t=4
J.M. Villalobos
c
� 2015
Ex : Solve x
2
� 9x = 10
Ex : Solve 2x
3
� 10x� 12 = 0
Ex : Solve (x + 3)
2
= 25
12
§ 7.1 RationalExpressions 1 Function 1 Equations
Rlx )= #Qlx )
Thomasin: ALL X values such that QIX )± 0
4- 3X
Rix )= - { xlx±s }X - 5
Domain '
.
QCX )=× -5
X - 5=0 ⇒ X=@
#/
7
RCX )= -
XZ - 25 XZ - 25=0
( X - 5) 1×+53=0
Domain : X 't 5, XF -5 11=5 , X= - 5
9EX RC x ) =
7×-3 Rcx )= 9- -
=
- xtnX
'
( x - 5) -91×-5 ) X2t49
( X - 5) ( x'
. 9) =0 D: 2/2+49=10( X - 5) ( X . 3) 1×+31=0
⇒ X=5 , X=3 , X= -3⇒ D : CD , is )
D ! XF5 , Xt }, Xt - } X ?49= ( X - 7) 1×+7 )
ReducingRationalExpressio="
Past"
"
now"
÷= :* 's xE÷x+÷i¥Yx¥n
pm ,
F3-L3j#4cF2tFL+L2
) =X¥@#
→ x3 - 27"#× - 3) (xt#)⇒
- 3×+9112+3×+9
= 1/-3
¥iex÷="x¥±÷=I¥Q
§ 7.2 Multiplication 1 Division
*
$* 5
g- .II =±
14 7I.$F-7
#( Xtz ) ( X - 7)
lX#)EEIIIx• xxky.sn - extantX ( X . z )
X#-)(F)
( x - 2)
E¥ 1/+4=4+11 * 5
(#l=1X - 41=4 - XCttx)
1- #€-4 )I÷t=o±x¥=t- (4#
( X - 3) 1×+2 )( × - 3) ( X - 2)
XZ . × - 6 X2 . 5×+6- ifxxz1/2+2×-8
( x + 4) ( × . 2) ( X - 2) 1×+1 )
#( xtz ).# ( Xtl )
=C Xtz )( xti )⇒ 4) ( # (XI )( x - 2) ( X t 4) ( x . z )
$7.3 Addinykationaxpns
÷ +t÷÷÷ 2- 3
LlD=2°3
÷ ,+÷,÷i¥
¥ (11×+5-2+ ×3zg . # ) L c D= c xtz )( x. i )
( X - l ) ( Xtz )
5X - 5 3×+6
⇒
51¥ +3YIt=8x-LCD
# ¥4 + ×÷z + ×÷z LcD=cX . 2) ( xtz )
( X - 2) ( XTZ )
i¥xIitt÷iYI÷tx÷i¥zX +3×+6+4×-8 8×-2
a- ⇒a
L ( D= 4 X ( X +3 )
÷+3+2+5 +
4x÷+Ix4×(11+3)
@*(xt=#
x÷;¥x+@÷i2zYx¥,
+4×37+5 't axis ,
⇒ 16Xt2xt6t3_ .iq#@xxIjILCD
§±4_ Rational Equations
¥4=x÷z+×÷z fat ;×÷Io Dong
Di Xto ,xtz ,
xt - 2
( X - 2) ( XTZ )
* .
,=a÷i¥¥ta÷it¥d|¥T@EEIIII'It4×'s @
¥o"÷Ygi÷⇒
⇒- 6x = - 2 ⇒ x= .÷= @
¥ ¥3+ III. = ÷,
* →
( X - 4) ( Xtz )×F4
×÷six¥n+ # it=÷i×x÷⇒ 7×-28 +2 X = 3×+9
⇒ 9×-28=3×+9⇒ 6 X = 37 ⇒ ×=3@
✓