Math 73
§6.1
1.
Lecture Notes
Date
Properties of Exponents
XZ
3
.
×
bn · bm = bn+m
÷E±j¥f
.it#atxiiax=x.X6.x9=X4.y6=x4
-
2.
bn
= bn
m
b
m
"
¥=±xxY
3.
←←x?E?xtxn
¥=7±
÷
x÷ox÷eox"
49
e÷,a*x÷
/
(bn )m = bnm
(x4)6=x4
1
x4.x6=
"
(2×3)
xlo
.
-
x6
.IT#t3xxD=i6x'2I
;
8×7
Hit
8×7
-
}
'
=
1
J.M. Villalobos c 2015
2
DX
#
⇒
8×7
"
⇒
'
4.
b
n
÷==
=
1
bn
=
st
.
=
|
8
.
81-6 )
,÷s÷tx
.
2×53=1
(
Fixer
.
×
2×33×8
'I
=×÷
(
=
5.
1-
-6
x
it
3=÷
I
0¥
1=5÷
8
-
.
,E=E
-2
1
=
853
=
x
I
b0 = 1
iii
e=aA÷=a"
20=1
p°=
"
'
1
¥=
=a°
,
÷
¥iH?I¥!|ti¥¥t**
144×14
Y2°
-
=
-
@
⇒
3
(2×755)
3
¥µ⇒s;÷Txx÷an
⇒
Eni
'
'
x÷ii
'
2
J.M. Villalobos c 2015
8×2
'
'
y
's
.
Scientific Notation
←
1,200,000,000.0
q
1.2*10
=
2=7.2*10-7
dodge
q
.
6,320.0*10
Sn
✓
SN
✓
SN
X
?x'9=x
w
-
Sri
6.32*103*109
6.32*1012
=
0.000047*10
who
4.7*10-5*1016=4.7*10
÷
"
0*106 ) ( 45.0*10-3
135.0*103
1.35*102*103
1.351€
)
3
J.M. Villalobos c 2015
×
0122=1
yo9=
106=1
trillion
Billion
Million
"
§6.2
Multiplication of Polynomials
Polynomials
P (x) = an xn + an 1 xn
1
+ ... + a1 x + a0
DESI
{ y
2nd
1/2+5×+1
=
Linear
2×+5
:
zrd
3
4×2-5×+7
4rd
fixity
Ex: Multiply the following
3x(x2
3x + 5)
=
3×3.9×2+1511
Ex: Multiply the following
(4x
a+3×-5
÷
1)(x2
3x + 5)
12×2+20×-112
4×3
-
.
4×3
-
-
13×2+23×-5
4
J.M. Villalobos c 2015
trinomial
Binomial
Monomial
Polynomial
Ex: Multiply the following
F
3)(4x2 + 6x + 9)
(2x
3+14×2+184
x
-
¥7
12×2
18×-27
-
Ex: Multiply the following
(2x
5)(2x + 5)
"
=
4
X
+1¥
x¥
4×2-254
.
X
8
+
Ex: Multiply the following
(2x + 5)2
2×+5>(2×+5)×+5
"y"a÷lYxxt2In|
=
-⇒gµY÷B
(
=
×+su×+
(
×2
,
+8¥
+40
X
-
5
J.M. Villalobos c 2015
13
,
÷
X
+40
←
area
§6.3
GCF and Grouping
¥
};
¥
213146
"
;
11214,81
Xxii;
't
'
GcF=4
lb
××
:{;
'
ac+=x
/
acF=x2y:
5x3y2(4y2
Ex: Factor the following
20x3 y 4
c3y4
15x6 y 2
-
5×372
3×3 )
.
.
p
-
=4y2
15×642
Ey
ACF
=
-
3X3y°
Ex: Factor the following
y(a + b)
[email protected]
x(a + b)
(
)
m
-
5)
(
Ex: Factor the following
x3
3x2 + 4x
(
X
(
x
←
+41×-3
3) (
x2t4
3)
-
.
12
)
)
6
J.M. Villalobos c 2015
htt
)
FOG
¥msahq¥%g€y
3-
terms
not
Ex: Factor the following
2xy + 6xp
÷y
( y
t
3y
3 P)
(
p )
+3
3
-
2
x
9p
( Y
+
3
)
-
3
P
)
Ex: Factor the following
x4
5x2 + 2x2
* 2(
X
2-
5)
5) (
+
×
2
(
10
X
2-
5
)
2+2 )
7
J.M. Villalobos c 2015
÷
-
§6.3
Trinomials
4
12=4.380
Ex: Factor the following
=
x
2
x
12
(
=
X
4
-
+3
)(
x
(
X
)
±
X2
4×+3×-12
-
*
4)
-
(
+3
X
4)
-
⇒
)
411×+3
-
Ex: Factor the following
5)
45=9.5
=
x
3
4x
X(X2
ti
2
45x
)
4×-45
-
×
⇒
(
X
9
-
15
)(xt
F
Ex: Factor the following
24
=
×
4-6
×
12.2
12x2
12×2
11x + 2
8×-3×+2
-
(3×-2) =
(3×-2)
8
(
-
4×-1
1
(
3×-2
)
)
8
J.M. Villalobos c 2015
-
3
-
-
3
2
#
§6.5
Factoring using formulas
FIX
3
L=
,
x=9=cx)2
←
F2
L2 = (F
L)(F + L)
F3
L3 = (F
L)(F 2 + F L + L2)
F 3 + L3 = (F + L)(F 2
F2tL2
3
)(xt3
-
4x
2
25
L=
F=
2×32
-
(5)
-
(2×-5)/2×+5
=
)
5
9m2
Ex : Factor
(
=
F=2X
100y 2
=
=
3M
L=
x2(x
p)
(
X-P
)(
( 3m)2
:
(
3M
4(x
)(x2
x-p
x
.
.
2
4
-
p)
)
)(
Optimus
1/2+25
←
9
J.M. Villalobos c 2015
-
⇐
F L + L2 )
Prime
Ex : Factor
Ex : Factor
x
9x=x(x-I
Xt
←
(
=
( 3,2
.
prime
.
)2
Cloy
IOY )(
3Mt
IOY )
)
3tL3=
F
8x3 + 27
Ex : Factor
F=2x
L=
,
3
-52=4×2
=
(
=
(
2x)3+
( 3
FTL
)(
)3
4×2-6×+9
)(
2×+3
(
)
6=9
,
FL=6X
y3
Ex : Factor
F2=y2
1,4=1
F=Y
L=
=( y )3
1
,
( y
=
)
a
.
HP
-
( y2+y
)
+1
FL=y
2x3
Ex : Factor
50x
2×1×2-25
=
=2X
x6 + 64
Ex : Factor
F=X2
L=4
F.
L=
Ft
,
,
'
x
'
=
4=16
4×2
10
J.M. Villalobos c 2015
(X
=
(
'
(
×
-
5)
)
(
Xt5
)
)3+( 4 )3
x2t4)(x4
-
4×2+16
)
FIFLTE
)
§6.6
Solving equations by factoring
Multiplication Property of zero
A
.b=
0
a=o
⇒
4x(x
Ex : Solve
#
b=
or
3)(2x
0
5) = 0
d
I
X
or
411=0
F
+3
⇒
(
X
(
X
x2(x
2)
9(x
)(
xkq
)=o
-2
-
2)
(
t
4
t
11=3
X=
x
(1/+371×-4)=0
X=
t
-3
X=4
11
J.M. Villalobos c 2015
2) = 0
-
÷
12=0
.
2×-5=0
X=@
-3>(1/+3)--0
×
11=2
a
+3
XD
@o
Ex : Solve
Or
3=0
-
3
+5
2×=5
+5
€
x2
Ex : Solve
9x = 10
1/2-9×-10=0
(
(x
.
Xt 1)
)=O
( o
t
d
11=-1
11=10
,
2x3
2
Ex : Solve
10x
2(x2
21
2€
X
(
112+3×+3×+9=25
1/2+6×+9=25
⇒
=y
.
1/+31=25
Xt3)(
(
X
+
(
×
)=o-2
⇒
12
J.M. Villalobos c 2015
€
gz
)2
→
(
xt
}
-
25=0
-
5)
Xt3t5)=°
(
1×+3
|
(
(
xzta
⇒
0
=
X
-
×t8)=0
2) (
-
¥61205
8)
X=2=
'
0
Xtl
or
⇒
⇒
=
or
(x + 3)2 = 25
⇒
)
xti
6=0
0
=
@
Ex : Solve
µ=¥g4
6) (
-
.
)
5×-6
-
X
12 = 0
xtsyx
.
stzx
"
"
Missile
hlt
)
=
16+2+64++80
-
DM
Ground
:
'
'
0
ground
-
hlt
)=
16t2t64t
-
( t2
16
( t
seconds
hit
to
the
5) (
-
5)
=
)
=
ttl
ground
.
to
hlt
#•
to
-
4t
-
¥T€=T⇐x
5
0¥
16
-
#
)=o
16+2+64++80=0
-
got
hit
⇒
)=o
16+2+64
⇒
-
⇒
t=4
t=0
i±H±=°
to
t=4
0
0
Ex : Solve
x2
Ex : Solve
2x3
Ex : Solve
(x + 3)2 = 25
9x = 10
10x
12
J.M. Villalobos c 2015
12 = 0
1 Function
Expressions
Rational
7.1
§
Rlx
Equations
1
#
)=
Qlx )
Thomasin
ALL
:
values
X
4
3X
-
Rix
Domain
)=
.
X
5
-
QCX
'
{
-
X
)=×
-5
5=0
⇒
-
that
such
)±
QIX
0
}
xlx±s
/
X=@
#
7
RCX
)=
-
XZ
XZ
25
-
( X
Domain
EX
=
RC
x
't
X
:
)
5
XF
,
-
11=5
-5
25=0
-
1×+53=0
5)
X=
,
-
7×-3
=
'
X
(
X
(
X
(
-
x
-
'
5) (
x
(
X
5)
-
- 91×-5
5)
-
9)
.
Rcx
D
!
X=5
XF5
,
X=3
,
Xt
}
,
,
-
xtn
X2t49
)
2/2+49=10
D:
=0
⇒
⇒
9
9-
)=
1×+31=0
3)
.
5
X=
Xt
D
-3
-
}
X
?49=
( X
-
7)
CD
:
1×+7
)
,
is
)
ReducingRationalExpressio=
"
"
"
Past
÷= :*
pm
x3
-
"
xt#)
#⇒
=X¥@
)
F3-L3j#4cF2tFL+L2
112+3×+9
27
×
3×+9
-
xE÷x+÷i¥Yx¥n
's
,
→
#
"
now
-
3) (
1/-3
=
¥iex÷="x¥±÷=I¥Q
§
FI.$
*$
g-
=±
7
14
7
(F)
•#-)
#
lX#)
Xtz
(
)
EE
IIIx
X
-
7)
xxky.sn
(
X
Division
5
II
.
(
E¥
1
Multiplication
7.2
x
-
extant
-
X
(
X
-
41=4
1- #
-
(4#
-
X
.
)
z
2)
I÷t=o
€-4±x¥=t
*
1/+4=4+11
X
5
(#l=1
Cttx
)
)
(
X
XZ
)
1×+2
3)
-
×
.
( ×
X2
-
1/2+2×-8
(
4) (
+
x
×
(
2)
.
X
2)
-
ifxxz
6
-
3) (
-
(
XI
#
X
5×+6
.
2)
-
⇒
(
)
xtz
.
4) (
(
)
1×+1
)(
)
Xtl
x
-
2)
=
C
(
)(
Xtz
X
4) (
t
)
xti
x
z
.
)
Addinykationaxpns
$7.3
t÷÷÷
÷
+
LlD=2°3
2-
,+÷,÷i¥
÷
¥
×3zg
+
( X
l
-
)
5X
-
¥4
#
(
X
-
2) (
XTZ
(11×+5-2
#
.
(
Xtz
L
c
D=
c
)
3YIt=8x+
×÷z
+
+
×÷z
)
+3×+6+4×-8
a-
LCD
LcD=cX
i¥xIitt÷iYI÷tx÷i¥z
X
)
3×+6
5
51¥
⇒
3
⇒a
8×-2
.
2)
(
xtz
)
xtz
)(
x.
i
)
L
+3+2+5
÷
+
(
D=
4
X
(
X
+3
)
4x÷+Ix
4×(11+3)
+4×37+5
,
@*(xt=
#
't
axis
,
x÷;¥x+@÷i2zYx¥
16Xt2xt6t3_
.iq#@xxIjI
⇒
LCD
Rational
§±4_
Equations
¥4=x÷z+×÷z
(
X
*
Dong
;×÷Io
Di
Xto
,
xtz
xt
.
I
'It4×
III
@
's
@
,=a÷i¥¥ta÷it¥d|¥T@EE
⇒
6x
-
¥
=
⇒
2
-
¥3III.
+
(X
7×-28
+2
X
9×-28=3×+9
⇒
⇒
6 X
=
37
⇒
=
Xtz
.÷=
x=
÷
*
,
)
it=÷i×x÷
#
×÷six¥n+
⇒
4) (
-
=
-
2
,
¥o"÷Ygi÷⇒
XTZ )
2) (
-
fat
3×+9
×=3@
✓
×F4
→