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Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A1 Glencoe Algebra 1
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
3
Gle
ncoe A
lgeb
ra 1
An
tici
pati
on
Gu
ide
Facto
rin
g
B
efo
re y
ou
beg
in C
ha
pte
r 8
•
R
ead
each
sta
tem
en
t.
•
D
eci
de w
heth
er
you
Agre
e (
A)
or
Dis
agre
e (
D)
wit
h t
he s
tate
men
t.
•
W
rite
A o
r D
in
th
e f
irst
colu
mn
OR
if
you
are
not
sure
wh
eth
er
you
agre
e o
r d
isagre
e,
wri
te N
S (
Not
Su
re).
Aft
er y
ou
com
ple
te C
ha
pte
r 8
•
R
ere
ad
each
sta
tem
en
t an
d c
om
ple
te t
he l
ast
colu
mn
by e
nte
rin
g a
n A
or
a D
.
•
D
id a
ny o
f you
r op
inio
ns
abou
t th
e s
tate
men
ts c
han
ge f
rom
th
e f
irst
colu
mn
?
•
F
or
those
sta
tem
en
ts t
hat
you
mark
wit
h a
D,
use
a p
iece
of
pap
er
to w
rite
an
exam
ple
of
wh
y y
ou
dis
agre
e.
8 Ste
p 1
Ste
p 2
ST
EP
1A
, D
, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1.
A m
on
om
ial
is i
n f
act
ore
d f
orm
wh
en
it
is e
xp
ress
ed
as
the
pro
du
ct o
f p
rim
e n
um
bers
an
d v
ari
able
s, a
nd
no v
ari
able
has
an
exp
on
en
t gre
ate
r th
an
1.
A
2.
Th
e g
reate
st c
om
mon
fact
or
(GC
F)
of
two o
r m
ore
mon
om
ials
is
th
e p
rod
uct
of
their
un
iqu
e f
act
ors
wh
en
each
mon
om
ial
is
wri
tten
in
fact
ore
d f
orm
.A
3.
An
y t
wo n
um
bers
th
at
have a
gre
ate
st c
om
mon
fact
or
of
1 a
re
said
to b
e r
ela
tively
pri
me.
A
4.
If t
he p
rod
uct
of
an
y t
wo f
act
ors
is
0,
then
at
least
on
e o
f th
e
fact
ors
mu
st e
qu
al
0.
A
5.
A q
uad
rati
c tr
inom
ial
has
a d
egre
e o
f 4.
D
6.
To s
olv
e a
n e
qu
ati
on
su
ch a
s x
2 =
8 +
2x,
tak
e t
he s
qu
are
root
of
both
sid
es.
D
7.
Th
e p
oly
nom
ial
3r2
- r
- 2
can
not
be f
act
ore
d b
eca
use
th
e
coeff
icie
nt
of
r2 i
s n
ot
1.
D
8.
Th
e p
oly
nom
ial
t2 +
16 i
s n
ot
fact
ora
ble
. A
9.
Th
e n
um
bers
16,
64,
an
d 1
21 a
re p
erf
ect
squ
are
s.A
Answers (Anticipation Guide and Lesson 8-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-1
Ch
ap
ter
8
5
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Mo
no
mia
ls a
nd
Facto
rin
g
8-1
Fact
or
Mo
no
mia
ls A
mon
om
ial
is i
n f
act
ore
d f
orm
wh
en
it
is e
xp
ress
ed
as
the p
rod
uct
of
pri
me n
um
bers
an
d v
ari
able
s, a
nd
no v
ari
able
has
an
exp
on
en
t gre
ate
r th
an
1.
F
acto
r e
ach
mo
no
mia
l co
mp
lete
ly.
a.
42a
3
42a
3 =
2
21
a
a
a
42 =
21
2,
and a
3 =
a
a
a
= 2
3
7
a
a
a
21 =
3
7
Th
us,
42a
3 i
n f
act
ore
d f
orm
is
2
3
7
a
a
a.
b. -
40
x2y
3
-40x
2y
3 =
-1
40 x
2 y
3
Expre
ss -
40 a
s -
1
40
.
= -
1
2
20
x
x
y
y
y
40 =
20
2,
x2 =
x
x,
and y
3 =
y
y
y
= -
1
2
2
10
x
x
y
y
y
20 =
10
2
=
-1
2
2
2
5
x
x
y
y
y
10 =
5
2
Th
us,
-40
x2y
3 i
n f
act
ore
d f
orm
is
-1
2
2
2
5
x
x
y
y
y.
Exerc
ises
Fa
cto
r e
ach
mo
no
mia
l co
mp
lete
ly.
1. 32x
2
2. 18m
2n
3. 49
a3b
2
2
2
2
2
2
x
x
2
3
3
m
m
n
7
7
a
a
a
b
b
4. 18y
3
5. -
9h
3jk
2
6. -
8d
2
2
3
3
y
y
y
-
1
3
3
h
h
h
j
k
k
-
1
2
2
2
d
d
7. 66q
3r3
8. 140x
2y
4z
9. –
ab
2f
2
2 3
1
1 q
q
q
r r r
2 2
5
7
x
x
y
y
y
y
z
-
1 a
b
b
f f
10. -
17t
11. 625jk
2
12. 47w
x3yz2
-
1 1
7 t
5 5
5
5
j k
k
47 w
x
x
x
y
z
z
13. 12r4
14. -
38a
2b
15. 4
mp
2
2
3
r
r
r
r
-
1
2
19
a
a
b
2
2
m
p
Exam
ple
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A2 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
6
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Mo
no
mia
ls a
nd
Facto
rin
g
Greatest C
om
mo
n F
acto
r
Th
e p
rod
uct
of
the c
om
mon
pri
me f
act
ors
is
call
ed
th
e
gre
ate
st c
om
mon
fact
or
(GC
F)
of
the n
um
bers
. T
he g
reate
st c
om
mon
fact
or
is t
he g
reate
st
nu
mber
that
is a
fact
or
of
both
ori
gin
al
nu
mbers
.
If t
wo o
r m
ore
in
tegers
or
mon
om
ials
have n
o c
om
mon
pri
me f
act
ors
, th
eir
GC
F i
s 1 a
nd
th
e
inte
gers
or
mon
om
ials
are
said
to b
e r
ela
tiv
ely
prim
e.
F
ind
th
e G
CF
of
16xy
2z
2 a
nd
72xyz
3.
16
xy
2z2
= 2 !
2 !
2 !
2 !
x !
y !
y !
z !
z
72xyz3
= 2 !
2 !
2 !
3 !
3 !
x !
y !
z !
z !
z
Th
e G
CF
of
16
xy
2z2
an
d 7
2xyz3
is
2 !
2 !
2 !
x !
y !
z !
z o
r 8
xyz2
.
Exerc
ises
Fin
d t
he G
CF
of
ea
ch
set
of
mo
no
mia
ls.
1. 49x,
343
x2
2. 4
a7b
, 28a
b49
x
4ab
3. 96y,
12
x,
8y
4. 12a
, 18a
bc
4
6a
5. 28y
2,
35xy,
49x
2yz
6. 2m
2p
, 12
mp
2,
18m
p7
y
2m
p
7. 12x
2,
32x
2yz,
60xy
2
8. 18a
3b
2,
36a
3b
2
4x
18
a3b
2
9. 15m
n2,
30
m3n
2,
90m
3
10. 2x
2y,
9x
2y
3,
18xy
2
15
m
xy
11. a
4b,
8a
3b
2
12. a
b2,
5a
4b
2,
10
b3
a3b
b
2
13. 2
x2y
2,
8xy
4,
12x
2y
14. 6
a2b
5,
15
a3b
4
2xy
3a
2b
4
15. 21a
4b
7f2
, 84a
3b
4,
28a
5b
2f3
16. 13x
2y
5,
5xy
3,
x4y
7
a3b
2
xy
8-1 Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-1
Ch
ap
ter
8
7
Gle
ncoe A
lgeb
ra 1
Sk
ills
Pra
ctic
e
Mo
no
mia
ls a
nd
Facto
rin
g
Fa
cto
r e
ach
mo
no
mia
l co
mp
lete
ly.
1. 10a
4 2 �
5 �
a �
a �
a �
a
2.
-27x
3y
2
-1 �
3 �
3 �
3 ·
x �
x �
x �
y �
y
3. 28p
r2 2 �
2 �
7 �
p �
r �
r
4. 44m
2n
p3 2
� 2
� 1
1 �
m �
m �
n �
p �
p �
p
5. 9
x3y
2 3 �
3 �
x �
x �
x �
y �
y
6.
-17
ab
2f
-1 �
17 �
a �
b �
b �
f
7. 42g
2 2 �
3 �
7 �
g �
g
8. 36
tu2 2 �
2 �
3 �
3 �
t �
u �
u
9.
-4a
-
1 �
2 �
2 �
a
10.
-10x
4yz2
-
1 �
2 �
5 �
x �
x �
x �
x �
y �
z �
z
Fin
d t
he G
CF
of
ea
ch
set
of
mo
no
mia
ls.
11. 16f,
21a
b2 1
12. 18t,
48t4
6
t
13. 32xyz,
48xy
4 16
xy
14. 12m
3p
2,
44m
p3 4m
p2
15. 4
q2r2
t2,
9q
3r3
t3 q
2r2
t2
16. 14
ab
5,
7a
2b
3c
7ab
3
17. 51xyz2
, 68x
2yz2
17xyz
2
18. 12
t7u
3,
18t3
u7 6t3
u3
19. 11
a4b
3,
44a
2b
5 11
a2b
3
20. 18r3
t, 2
6qr2
t4 2
r2t
8-1
Answers (Lesson 8-1)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A3 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
8
Gle
ncoe A
lgeb
ra 1
Practi
ce
Mo
no
mia
ls a
nd
Facto
rin
g
Fa
cto
r e
ach
mo
no
mia
l co
mp
lete
ly.
1. 30d
5
2. -
72m
p
2
� 3
� 5
� d
� d
� d
� d
� d
-
1 �
2 �
2 �
2 �
3 �
3 �
m �
p
3. 81b
2c3
4. 145
abc3
3
� 3
� 3
� 3
� b
� b
� c
� c
� c
5
� 2
9 �
a �
b �
c �
c �
c
5. 168n
q2r
6. -
121
x2yz2
2
� 2
� 2
� 3
� 7
� n
� q
� q
� r
-
1 �
11 �
11 �
x �
x �
y �
z �
z
7. -
14f
2g
2
8. -
77
w4
-
1 �
2 �
7 �
f �
f �
g �
g
-1 �
7 �
11 �
w �
w �
w �
w
Fin
d t
he G
CF
of
ea
ch
set
of
mo
no
mia
ls.
9. 24fg
5,
56
f 3g 8
fg
10. 72r2
t2,
36
rt3 36
rt2
11. 15a
2b
, 35
ab
2 5
ab
12. 28k
3n
2,
45p
r2 1
13. 40xy
2,
56x
3y
2,
124x
2y
3 4
xy
2
14. 88a
3d
, 40a
2d
2,
32
a2d
8a
2d
15. G
EO
METR
Y T
he a
rea o
f a r
ect
an
gle
is
84 s
qu
are
in
ches.
Its
len
gth
an
d w
idth
are
both
wh
ole
nu
mbers
.
a.
Wh
at
is t
he m
inim
um
peri
mete
r of
the r
ect
an
gle
? 38 i
n.
b.
Wh
at
is t
he m
axim
um
peri
mete
r of
the r
ect
an
gle
? 170 i
n.
16. R
EN
OV
ATIO
N M
s. B
axte
r w
an
ts t
o t
ile a
wall
to s
erv
e a
s a s
pla
shgu
ard
above a
basi
n
in t
he b
ase
men
t. S
he p
lan
s to
use
equ
al-
sized
til
es
to c
over
an
are
a t
hat
measu
res
48 i
nch
es
by 3
6 i
nch
es.
a.
Wh
at
is t
he m
axim
um
-siz
e s
qu
are
til
e M
s. B
axte
r ca
n u
se a
nd
not
have t
o c
ut
an
y o
f
the t
iles?
12-i
n.
sq
uare
b.
How
man
y t
iles
of
this
siz
e w
ill
she n
eed
? 12
8-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-1
Ch
ap
ter
8
9
Gle
ncoe A
lgeb
ra 1
Wo
rd
Pro
ble
m P
racti
ce
Mo
no
mia
ls a
nd
Facto
rin
g
1. M
ATH
GA
MES
M
rs.
Jen
son
’s c
lass
is
pla
yin
g “
Gu
ess
th
e M
on
om
ial.
” O
ne
stu
den
t d
isp
lays
fact
ors
of
the s
ecr
et
mon
om
ial,
an
d t
he t
eam
tri
es
to g
uess
the m
on
om
ial.
Wh
en
it
is J
am
es’
tu
rn,
he
sees
that
the s
ecr
et
mon
om
ial
is 2
10x
2y
2.
Wh
ich
of
the f
oll
ow
ing c
ard
s sh
ou
ld h
e
dis
pla
y s
o h
is t
eam
gu
ess
es
the c
orr
ect
mon
om
ial?
3x
5x
7y
11y
13z
17z
19z
O
rder
may v
ary
:
2
� 3
� 5
� 7
�x �
x�
y�
y
2. PA
RTY
FA
VO
RS
B
all
oon
s co
me i
n
pack
ages
of
18 a
nd
part
y h
ats
com
e i
n
pack
ages
of
8.
Jeff
wan
ts t
o h
ave t
he
sam
e n
um
ber
of
ball
oon
s an
d h
ats
. W
hat
is t
he f
ew
est
pack
ages
of
ball
oon
s an
d
hats
th
at
he n
eed
s to
bu
y s
o h
e h
as
no
hats
or
ball
oon
s le
ft o
ver?
4 p
ackag
es
of
ballo
on
s a
nd
9 p
ackag
es o
f h
ats
3. PA
CK
AG
ING
C
olo
r W
heel
pri
nte
r in
k
com
pan
y w
an
ts t
o d
esi
gn
a n
ew
cart
on
in
wh
ich
to p
ack
pri
nte
r in
k c
art
rid
ges
for
ship
men
t to
sto
res.
Cart
rid
ge b
oxes
are
7 i
nch
es
lon
g a
nd
3 i
nch
es
wid
e.
Wh
at
are
th
e d
imen
sion
s of
the s
mall
est
squ
are
-bott
om
cart
on
th
at
wil
l h
old
th
e
cart
rid
ge b
oxes
wit
hou
t extr
a s
pace
?
21 i
n.
by 2
1 i
n.
4. M
ATH
EM
ATIC
IAN
S A
Gre
ek
math
em
ati
cian
an
d a
stro
nom
er
nam
ed
Era
tost
hen
es
create
d a
way t
o s
ep
ara
te
pri
me n
um
bers
fro
m c
om
posi
te n
um
bers
.
His
meth
od
is
kn
ow
n a
s th
e S
ieve o
f
Era
tost
hen
es.
It
pro
ceed
s as
foll
ow
s.
So
urc
e:
Math
Foru
m
Recr
eate
th
e S
ieve o
f E
rato
sth
en
es
to
fin
d t
he f
irst
11 p
rim
e n
um
bers
. 2,
3,
5,
7,
11,
13,
17,
19,
23,
29,
31
5. R
EPA
IRS
Heid
i w
an
ts t
o r
ep
lace
th
e
floor
in h
er
16-f
oot
by 1
8-f
oot
rect
an
gu
lar
dan
ce s
tud
io.
Sh
e w
an
ts t
o u
se s
qu
are
wood
til
es,
an
d s
he d
oes
not
wan
t to
have
to c
ut
an
y o
f th
e t
iles
nor
leave a
ny g
ap
s.
a.
Su
pp
ose
th
e f
loori
ng c
om
pan
y c
an
use
an
y s
ize t
ile.
Wh
at
is t
he l
arg
est
squ
are
til
e t
hat
Heid
i ca
n u
se f
or
the
new
flo
or?
2 f
eet
by 2
feet
b.
If H
eid
i fi
rst
kn
ock
s ou
t a w
all
an
d
incr
ease
s th
e s
tud
io t
o 2
4 f
eet
by
18 f
eet,
wh
at
is t
he l
arg
est
squ
are
til
e
she c
an
use
for
the n
ew
flo
or?
6 f
oo
t b
y 6
fo
ot
8-1
Write
num
bers
1 t
o 5
0.
Sin
ce 1
is n
either
prim
e n
or
com
posite,
ignore
1.
Circle
the n
um
ber
2,
and t
hen c
ross o
ff e
very
num
ber
that
is d
ivis
ible
by 2
.
Circle
the n
ext
num
ber
that
is n
ot
cro
ssed o
ff,
3,
and c
ross o
ff a
ll m
ultip
les o
f 3.
Circle
the n
ext
num
ber
that
is n
ot
cro
ssed o
ff,
5,
and c
ross o
ff t
he
multip
les o
f 5,
etc
…
Answers (Lesson 8-1)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A4 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
10
Gle
ncoe A
lgeb
ra 1
En
rich
men
t
Fin
din
g t
he G
CF b
y E
uclid
’s A
lgo
rith
mF
ind
ing t
he g
reate
st c
om
mon
fact
or
of
two l
arg
e n
um
bers
can
tak
e a
lo
ng t
ime u
sin
g p
rim
e f
act
ori
zati
on
s. T
his
meth
od
can
be a
void
ed
by
usi
ng E
ucl
id’s
Alg
ori
thm
as
show
n i
n t
he f
oll
ow
ing e
xam
ple
.
F
ind
th
e G
CF
of
209 a
nd
532.
Div
ide t
he g
reate
r n
um
ber,
532,
by t
he l
ess
er,
209.
Th
e d
ivis
or,
19,
is t
he G
CF
of
209 a
nd
532.
Su
pp
ose
th
e G
CF
of
two n
um
bers
is
fou
nd
to b
e 1
. T
hen
th
e n
um
bers
are
said
to b
e r
ela
tiv
ely
prim
e.
Fin
d t
he G
CF
of
ea
ch
gro
up
of
nu
mb
ers b
y u
sin
g E
ucli
d’s
Alg
orit
hm
.
1. 187;
578 17
2. 1802;
106 106
3. 161;
943 23
4. 215;
1849 43
5. 1325;
3498 53
6. 3484;
5963 67
7. 33,5
83;
4257 473
8. 453;
484 1 (
rela
tively
pri
me)
9. 95;
209;
589 19
10. 518;
407;
851 37
11. 17a
2x
2z;
1615
axz2
17axz
12. 752
af 3
; 893
a3f 3
47af 3
13. 979
r2t2
; 495
rt3,
154
r3t3
11rt
2
14. 360
x5y
7;
328
xy;
568
x3y
3 8
xy
8-1 Exam
ple
Div
ide t
he r
em
ain
der
into
th
e d
ivis
or
above.
Rep
eat
this
pro
cess
u
nti
l th
e r
em
ain
der
is z
ero
. T
he l
ast
n
on
zero
rem
ain
der
is
the G
CF
.
2
209 � ����
����������
���� 532
__
__
418 1
114 � �
����������
�������
209
__
__
114 1
95 � �
����������
�������
114
__
_ 95 5
19 � �
����������
�� 95
__
_ 95
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-2
Ch
ap
ter
8
11
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Usin
g t
he D
istr
ibu
tive P
rop
ert
y
Use
th
e D
istr
ibu
tive P
rop
ert
y t
o F
act
or
Th
e D
istr
ibu
tive P
rop
ert
y h
as
been
use
d
to m
ult
iply
a p
oly
nom
ial
by a
mon
om
ial.
It
can
als
o b
e u
sed
to e
xp
ress
a p
oly
nom
ial
in
fact
ore
d f
orm
. C
om
pare
th
e t
wo c
olu
mn
s in
th
e t
able
belo
w.
U
se t
he D
istr
ibu
tiv
e
Pro
perty
to
fa
cto
r 1
2mp
+ 8
0m
2.
Fin
d t
he G
CF
of
12m
p a
nd
80m
2.
12m
p =
2 �
2 �
3 �
m �
p
80
m2 =
2 �
2 �
2 �
2 �
5 �
m �
m
GC
F =
2 �
2 �
m o
r 4m
Wri
te e
ach
term
as
the p
rod
uct
of
the G
CF
an
d i
ts r
em
ain
ing f
act
ors
.
12
mp
+ 8
0m
2 =
4m
(3 �
p)
+ 4
m(2
� 2
� 5
� m
)
= 4
m(3
p)
+ 4
m(2
0m
)
= 4
m(3
p +
20m
)
Th
us
12
mp
+ 8
0m
2 =
4m
(3p
+ 2
0m
).
F
acto
r
6ax +
3ay +
2bx
+ by b
y g
ro
up
ing
.
6a
x +
3a
y +
2bx +
by
= (
6a
x +
3a
y)
+ (
2bx +
by)
= 3
a(2
x +
y)
+ b
(2x +
y)
= (
3a
+ b
)(2x +
y)
Ch
eck
usi
ng t
he F
OIL
meth
od
.
(3a
+ b
)(2
x +
y)
= 3
a(2
x)
+ (
3a
)(y)
+ (
b)(
2x)
+ (
b)(
y)
= 6
ax +
3a
y +
2bx +
by ✓
Exerc
ises
Fa
cto
r e
ach
po
lyn
om
ial.
1. 24x +
48y
2. 30
mp
2 +
m2p
- 6
p
3. q
4 -
18q
3 +
22
q
2
4(x
+ 2
y)
p(3
0m
p +
m2 -
6)
q(q
3 -
18q
2 +
22)
4. 9
x2 -
3x
5. 4
m +
6p
- 8
mp
6. 45
r3 -
15r2
3
x(3
x -
1)
2(2
m +
3p
- 4
mp
) 1
5r2
(3r -
1)
7. 14t3
- 4
2t5
- 4
9t4
8. 55p
2 -
11p
4 +
44
p5
9. 14y
3 -
28
y2 +
y
7
t3(2
- 6
t2 -
7t)
1
1p
2(5
- p
2 +
4p
3)
y(1
4y
2 -
28y +
1)
10. 4
x +
12x
2 +
16x
3
11. 4a
2b
+ 2
8a
b2 +
7a
b
12. 6y +
12x -
8z
4
x(1
+ 3
x +
4x
2)
ab
(4a +
28b
+ 7
) 2
(3y +
6x -
4z)
13. x
2 +
2x +
x +
2
14. 6
y2 -
4y +
3y -
2
15. 4
m2 +
4m
p +
3m
p +
3p
2
(
x +
1)(
x +
2)
(2
y +
1)(
3y -
2)
(4
m +
3p
)(m
+ p
)
16. 12a
x +
3xz
+ 4
ay +
yz
17. 12
a2 +
3a
- 8
a -
2
18. xa
+ y
a +
x +
y
(
3x +
y)(
4a
+ z
) (
4a +
1)(
3a
- 2
) (
x +
y)(
a +
1)
8-2
Exam
ple
1Exam
ple
2
Mu
ltip
lyin
gF
acto
rin
g
3(a
+ b
) =
3a
+ 3
b3
a +
3b
= 3
(a +
b)
x(y
- z
) =
xy -
xz
xy -
xz =
x(y
- z
)
6y(2
x +
1)
= 6
y(2
x)
+ 6
y(1
)
=
12
xy +
6y
12
xy +
6y =
6y(2
x)
+ 6
y(1
)
=
6y(2
x +
1)
Answers (Lesson 8-1 and Lesson 8-2)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A5 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
12
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Usin
g t
he D
istr
ibu
tive P
rop
ert
y
So
lve E
qu
ati
on
s b
y F
act
ori
ng
T
he f
oll
ow
ing p
rop
ert
y,
alo
ng w
ith
fact
ori
ng,
can
be
use
d t
o s
olv
e c
ert
ain
equ
ati
on
s.
S
olv
e 9x
2 +
x =
0.
Th
en
ch
eck
th
e s
olu
tio
ns.
Wri
te t
he e
qu
ati
on
so t
hat
it i
s of
the f
orm
ab
= 0
.
9x
2 +
x =
0
Origin
al equation
x(9
x +
1) =
0
Facto
r th
e G
CF
of
9x
2 +
x,
whic
h is x
.
x =
0 or
9x +
1 =
0
Zero
Pro
duct
Pro
pert
y
x =
0 x
= -
1 −
9
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {0
, -
1 −
9 } .
Ch
eck
S
ubst
itu
te 0
an
d -
1 −
9 f
or
x i
n t
he o
rigin
al
equ
ati
on
.
9x
2 +
x =
0
9x
2 +
x =
0
9(0
)2 +
0 "
0
9 (- 1
−
9 ) 2
+ (-
1 −
9 ) "
0
0 =
0 ✓
1
−
9 +
(- 1
−
9 ) "
0
0 =
0 ✓
So
lve e
ach
eq
ua
tio
n.
Ch
eck
yo
ur s
olu
tio
ns.
1. x(x
+ 3
) =
0
2. 3m
(m -
4) =
0
3. (r
- 3
)(r +
2) =
0
{
0, -
3}
{0,
4}
{-
2,
3}
4. 3
x(2
x -
1) =
0
5. (4
m +
8)(
m -
3) =
0
6. 5t2
= 2
5t
{
0,
1
−
2 }
{-
2,
3}
{0,
5}
7. (4
c +
2)(
2c -
7) =
0
8. 5p
- 1
5p
2 =
0
9. 4y
2 =
28y
{- 1
−
2 ,
7
−
2 }
{0,
1
−
3 }
{0,
7}
10. 12x
2 =
-6
x
11. (4
a +
3)(
8a
+ 7
) =
0
12. 8y =
12y
2
{- 1
−
2 ,
0}
{- 7
−
8 , - 3
−
4 }
{0,
2
−
3 }
13. x
2 =
-2
x
14. (6
y -
4)(
y +
3) =
0
15. 4m
2 =
4m
{-
2,
0}
{-
3,
2
−
3 }
{0,
1}
16. 12x =
3x
2
17. 12a
2 =
-3
a
18. (1
2a
+ 4
)(3a
- 1
) =
0
{
0,
4}
{- 1
−
4 ,
0}
{- 1
−
3 ,
1
−
3 }
8-2 Exam
ple
Exerc
ises
Zero
Pro
du
ct
Pro
pert
yF
or
any r
eal num
bers
a a
nd b
, if
ab
= 0
, th
en e
ither
a =
0,
b =
0,
or
both
a a
nd b
equal 0.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-2
Ch
ap
ter
8
13
Gle
ncoe A
lgeb
ra 1
Sk
ills
Pra
ctic
e
Usin
g t
he D
istr
ibu
tive P
rop
ert
y
Fa
cto
r e
ach
po
lyn
om
ial.
1. 7
x +
49
2. 8
m -
6
7
(x +
7)
2(4
m -
3)
3. 5
a2 -
15
4. 10
q -
25q
2
5
(a2 -
3)
5q
(2 -
5q
)
5. 8
ax -
56a
6. 81r +
48
rt
8
a(x
- 7
) 3
r(2
7 +
16
t)
7. t2
h +
3t
8. a
2b
2 +
a
t
(th
+ 3
) a
(ab
2 +
1)
9. x +
x2y +
x3y
2
10. 3
p2r2
+ 6
pr +
p
x
(1 +
xy +
x2y
2)
p(3
pr2
+ 6
r +
1)
11. 4a
2b
2 +
16a
b +
12a
12. 10
h3n
3 -
2h
n2 +
14h
n
4
a(a
b2 +
4b
+ 3
) 2
hn
(5h
2n
2 -
n +
7)
13. x
2 +
3x +
x +
3
14. b
2 -
2b
+ 3
b -
6
(
x +
1)(
x +
3)
(b
+ 3
)(b
- 2
)
15. 2
j 2 +
2j +
3j +
3
16. 2a
2 -
4a
+ a
- 2
(
2j +
3)(
j +
1)
(2a
+ 1
)(a -
2)
17. 6
t2 -
4t -
3t +
2
18. 9x
2 -
3xy +
6x -
2y
(
2t -
1)(
3t -
2)
(3x +
2)(
3x -
y)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
yo
ur s
olu
tio
ns.
19. x(x
- 8
) =
0 {0
, 8}
20. b(b
+ 1
2) =
0 {-
12,
0}
21. (m
- 3
)(m
+ 5
) =
0 {-
5,
3}
22. (a
- 9
)(2
a +
1) =
0 {- 1
−
2 ,
9}
23. x
2 -
5x =
0 {0
, 5}
24. y
2 +
3y =
0 {-
3,
0}
25. 3a
2 =
6a
{0
, 2}
26. 2
x2 =
3x {0
, 3
−
2 }
8-2
Answers (Lesson 8-2)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A6 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
14
Gle
ncoe A
lgeb
ra 1
Practi
ce
Usin
g t
he D
istr
ibu
tive P
rop
ert
y
Fa
cto
r e
ach
po
lyn
om
ial.
1. 64 -
40ab
2. 4d
2 +
16
3. 6r2t
- 3rt
2
8(8
- 5
ab
) 4
(d2 +
4)
3rt
(2r -
t)
4. 15ad
+ 3
0a
2d
2
5. 32a
2 +
24b
2
6. 36xy
2 -
48x
2y
15
ad
(1 +
2ad
) 8
(4a
2 +
3b
2)
12
xy(3
y -
4x)
7. 30x
3y +
35x
2y
2
8. 9a
3d
2 -
6ad
3
9. 75b
2g
3 +
60bg
3
5x
2y(6
x +
7y)
3ad
2(3
a2 -
2d
) 1
5b
g3(5
b +
4)
10. 8p
2r2
- 2
4pr3
+ 1
6pr
11. 5x
3y
2 +
10x
2y +
25x
12. 9ax
3 +
18bx
2 +
24cx
8p
r(p
r -
3r2
+ 2
) 5
x(x
2y
2 +
2xy +
5)
3x(3
ax
2 +
6b
x +
8c)
13. x
2 +
4x +
2x +
8
14. 2a
2 +
3a
+ 6a
+ 9
15. 4b
2 -
12b +
2b
- 6
(x +
2)(
x +
4)
(a +
3)(
2a +
3)
(4b
+ 2
)(b
- 3
)
16. 6xy -
8x +
15y -
20
17.
-6mp
+ 4m
+ 1
8p
- 1
2
18. 12a
2 -
15ab
- 1
6a
+ 2
0b
(2x +
5)(
3y -
4)
(-
2m
+ 6
)(3
p -
2)
(3a
- 4
)(4a
- 5
b)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
yo
ur s
olu
tio
ns.
19. x(x
- 3
2)
= 0
20. 4b
(b +
4)
= 0
21. (y
- 3
)(y +
2)
= 0
{
0,
32}
{-
4,
0}
{-
2,
3}
22. (a
+ 6
)(3a
- 7
) =
0
23. (2y +
5)(y -
4)
= 0
24. (4y +
8)(
3y -
4)
= 0
{-
6,
7
−
3 }
{- 5
−
2 ,
4}
{-2,
4
−
3 }
25. 2z2
+ 2
0z
= 0
26. 8p
2 -
4p
= 0
27. 9x
2 =
27x
{-
10,
0}
{ 0,
1
−
2 }
{0,
3}
28. 18x
2 =
15x
29. 14x
2 =
-21x
30. 8x
2 =
-26x
{0
, 5
−
6 }
{- 3
−
2 ,
0}
{- 1
3
−
4 ,
0}
31. LA
ND
SC
APIN
G A
lan
dsc
ap
ing c
om
pan
y h
as
been
com
mis
sion
ed
to d
esi
gn
a t
rian
gu
lar
flow
er
bed
for
a m
all
en
tran
ce.
Th
e f
inal
dim
en
sion
s of
the f
low
er
bed
have n
ot
been
d
ete
rmin
ed
, bu
t th
e c
om
pan
y k
now
s th
at
the h
eig
ht
wil
l be t
wo f
eet
less
th
an
th
e b
ase
.
Th
e a
rea o
f th
e f
low
er
bed
can
be r
ep
rese
nte
d b
y t
he e
qu
ati
on
A =
1
−
2 b
2 -
b.
a.
Wri
te t
his
equ
ati
on
in
fact
ore
d f
orm
. A
= b
( 1 −
2 b
- 1)
b.
Su
pp
ose
th
e b
ase
of
the f
low
er
bed
is
16 f
eet.
Wh
at
wil
l be i
ts a
rea?
112 f
t2
32. PH
YSIC
AL S
CIE
NC
E M
r. A
lim
’s s
cien
ce c
lass
lau
nch
ed
a t
oy r
ock
et
from
gro
un
d l
evel
wit
h a
n i
nit
ial
up
ward
velo
city
of
60 f
eet
per
seco
nd
. T
he h
eig
ht h
of
the r
ock
et
in f
eet
above t
he g
rou
nd
aft
er t
seco
nd
s is
mod
ele
d b
y t
he e
qu
ati
on
h =
60t
- 1
6t2
. H
ow
lon
g
was
the r
ock
et
in t
he a
ir b
efo
re i
t re
turn
ed
to t
he g
rou
nd
? 3.7
5 s
8-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-2
Ch
ap
ter
8
15
Gle
ncoe A
lgeb
ra 1
Wo
rd
Pro
ble
m P
racti
ce
Usin
g t
he D
istr
ibu
tive P
rop
ert
y
1. PH
YSIC
S A
ccord
ing t
o l
egen
d,
Gali
leo
dro
pp
ed
obje
cts
of
dif
fere
nt
weig
hts
fro
m
the s
o-c
all
ed
“le
an
ing t
ow
er”
of
Pis
a
wh
ile d
evelo
pin
g h
is f
orm
ula
for
free
fall
ing o
bje
cts.
Th
e r
ela
tion
ship
th
at
he
dis
covere
d w
as
that
the d
ista
nce
d a
n
obje
ct f
all
s aft
er t
seco
nd
s is
giv
en
by
d=
16t2
(ig
nori
ng a
ir r
esi
stan
ce).
Th
is
rela
tion
ship
can
be f
ou
nd
in
th
e e
qu
ati
on
h
= 4t
- 1
6t2
, w
here
h i
s th
e h
eig
ht
of
an
obje
ct t
hro
wn
up
ward
fro
m g
rou
nd
level
at
a r
ate
of
32 f
eet
per
seco
nd
. S
olv
e t
he
equ
ati
on
for h
= 0
. t =
0.2
5 a
nd
0
2. SW
IMM
ING
PO
OL T
he a
rea A
of
a
rect
an
gu
lar
swim
min
g p
ool
is g
iven
by
the e
qu
ati
on
A=
12w
-w
2,
wh
ere
w i
s th
e w
idth
of
on
e s
ide.
Wri
te a
n
exp
ress
ion
for
the o
ther
sid
e o
f th
e
pool.
12 -
w
3. C
ON
STR
UC
TIO
N U
niq
ue B
uil
din
g
Com
pan
y i
s co
nst
ruct
ing a
tri
an
gu
lar
roof
tru
ss f
or
a b
uil
din
g.
Th
e w
ork
ers
ass
em
ble
th
e t
russ
wit
h t
he d
imen
sion
s sh
ow
n o
n t
he d
iagra
m b
elo
w.
Usi
ng t
he
Pyth
agore
an
Th
eore
m,
fin
d t
he l
en
gth
of
the s
ides
of
the t
russ
. 3 y
d,
4 y
d,
5 y
d
4. V
ER
TIC
AL J
UM
P Y
ou
r vert
ical
jum
p
heig
ht
is m
easu
red
by s
ubtr
act
ing
you
r st
an
din
g r
each
heig
ht
from
th
e
heig
ht
of
the h
igh
est
poin
t you
can
reach
by j
um
pin
g w
ith
ou
t ta
kin
g a
ru
nn
ing
start
. T
yp
icall
y,
NB
A p
layers
have
vert
ical
jum
p h
eig
hts
of
up
to 3
4 i
nch
es.
If
an
NB
A p
layer
jum
ps
this
hig
h,
his
h
eig
ht h
in
in
ches
above h
is s
tan
din
g
reach
heig
ht
aft
er t
seco
nd
s ca
n b
e
mod
ele
d b
y t
he e
qu
ati
on
h
= 1
62t
- 1
92t2
. S
olv
e t
he e
qu
ati
on
for
h=
0 a
nd
in
terp
ret
the s
olu
tion
. R
ou
nd
you
r an
swer
to t
he n
eare
st h
un
dre
dth
.
t=
0 a
nd
t≈
0.8
44;
Th
e p
layer
lan
ds a
fter
ab
ou
t 0.8
4 s
eco
nd
s.
5. PETS
C
on
ner
toss
es
a d
og t
reat
up
ward
w
ith
an
in
itia
l velo
city
of
13.7
mete
rs p
er
seco
nd
. T
he h
eig
ht
of
the t
reat
above t
he
dog’s
mou
th h
in
mete
rs a
fter t
seco
nd
s is
giv
en
by t
he e
qu
ati
on
.h
= 1
3.7t
- 4
.9t2
a
. A
ssu
min
g t
he d
og d
oesn
’t j
um
p,
aft
er
how
man
y s
eco
nd
s d
oes
the d
og c
atc
h
the t
reat?
2.7
95
b. T
he d
og t
reat
reach
es
its
maxim
um
h
eig
ht
half
way b
etw
een
wh
en
it
was
thro
wn
an
d w
hen
it
was
cau
gh
t. W
hat
is i
ts m
axim
um
heig
ht?
9.6
m
c. H
ow
fast
wou
ld C
on
nor
have t
o t
hro
w
the d
og t
reat
in o
rder
to m
ak
e i
t fl
y
thro
ugh
th
e a
ir f
or
6 s
eco
nd
s?
at
29.4
m/s
2x -
1 y
d
x +
1 y
d
x y
d
8-2
Answers (Lesson 8-2)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A7 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
16
Gle
ncoe A
lgeb
ra 1
Lin
ear
Co
mb
inati
on
sT
he g
reate
st c
om
mon
fact
or,
GC
F,
of
two n
um
bers
can
be w
ritt
en
as
a l
inear
com
bin
ati
on
of
the t
wo n
um
bers
. A
lin
ear
com
bin
ati
on
is
an
exp
ress
ion
of
the f
orm
Ax +
By.
W
rit
e t
he g
rea
test
co
mm
on
fa
cto
r o
f 52 a
nd
36 a
s a
lin
ea
r
co
mb
ina
tio
n.
Fir
st,
use
th
e E
ucl
idean
Alg
ori
thm
to f
ind
th
e g
reate
st c
om
mon
fact
or
of
the t
wo n
um
bers
.
1
36 """""
""""""""
52
D
ivid
e t
he g
reate
r num
ber
by t
he lesser
num
ber.
36
2
16 """"
"""""""""
36
ori
gin
al
div
isor;
T
hen d
ivid
e u
sin
g t
he r
em
ain
der
as t
he n
ew
div
isor.
32
4
4 """"
"""""""""
16
se
con
d d
ivis
or;
D
ivid
e a
gain
.
16
0
Sto
p d
ivid
ing.
Last
div
isor
use
d i
s th
e G
CF
. In
th
is c
ase
, 4 i
s th
e G
CF
for
36 a
nd
52.
To w
rite
4 a
s a l
inear
com
bin
ati
on
of
36 a
nd
52,
it n
eed
s to
be w
ritt
en
as:
4 =
36
x +
52y,
wh
ere
x a
nd
y a
re s
om
e i
nte
gers
.
Use
tri
al
an
d e
rror
to d
ete
rmin
e t
he t
wo i
nte
gers
.
Th
e t
wo i
nte
gers
th
at
work
are
x =
3 a
nd
y =
-2.
So,
the l
inear
com
bin
ati
on
for
the g
reate
st
com
mon
fact
or
of
52 a
nd
36 i
s:
4 =
36(3
) +
52(-
2)
Exerc
ises
Writ
e t
he g
rea
test
co
mm
on
fa
cto
r f
or e
ach
pa
ir o
f n
um
bers a
s a
lin
ea
r
co
mb
ina
tio
n.
1. 16,
64
2. 21,
28
16 =
16(1
) +
64(0
)
7 =
21(-
1) +
28(1
)
3. 3,
18
4. 15,
36
3 =
3(1
) +
18(0
) 3
= 1
5(5
) -
36(2
)
5. 6,
8
6. 18,
42
2 =
6(-
1) +
8(1
)
6 =
18(-
2) +
42(1
)
En
rich
men
t8-2 Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-3
Ch
ap
ter
8
17
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Qu
ad
rati
c E
qu
ati
on
s:
x2 +
bx +
c =
0
Facto
r x
2 +
bx +
c
To f
act
or
a t
rin
om
ial
of
the f
orm
x2 +
bx +
c,
fin
d t
wo i
nte
gers
, m
an
d p
, w
hose
su
m i
s equ
al
to b
an
d w
hose
pro
du
ct i
s equ
al
to c
.
F
acto
r e
ach
po
lyn
om
ial.
a. x
2 +
7x
+ 1
0
In
th
is t
rin
om
ial,
b =
7 a
nd
c =
10.
Facto
rs o
f 10
Su
m o
f F
acto
rs
1,
10
11
2,
57
S
ince
2 +
5 =
7 a
nd
2 #
5 =
10,
let
m =
2
an
d p
= 5
.
x
2 +
7x +
10 =
(x +
5)(
x +
2)
b. x
2 -
8x
+ 7
In t
his
tri
nom
ial,
b =
-8 a
nd
c =
7.
Noti
ce t
hat
m +
p i
s n
egati
ve a
nd
mp
is
posi
tive,
so m
an
d p
are
both
negati
ve.
Sin
ce -
7 +
(-
1)
= -
8 a
nd
(-
7)(
-1)
= 7
, m
= -
7 a
nd
p =
-1.
x2 -
8x +
7 =
(x -
7)(
x -
1)
F
acto
r x
2 +
6x
- 1
6.
In t
his
tri
nom
ial,
b =
6 a
nd
c =
-16.
Th
is
mean
s m
+ p
is
posi
tive a
nd
mp
is
negati
ve.
Mak
e a
lis
t of
the f
act
ors
of
-16,
wh
ere
on
e
fact
or
of
each
pair
is
posi
tive.
Facto
rs o
f -
16
Su
m o
f F
acto
rs
1, -
16
-15
-1,
16
15
2, -
8-
6
-2,
86
Th
ere
fore
, m
= -
2 a
nd
p =
8.
x2 +
6x -
16 =
(x -
2)(
x +
8)
Exerc
ises
Fa
cto
r e
ach
po
lyn
om
ial.
1. x
2 +
4x +
3
2. m
2 +
12m
+ 3
2
3. r2
- 3
r +
2
(
x +
3)(
x +
1)
(m
+ 4
)(m
+ 8
) (
r -
2)(
r -
1)
4. x
2 -
x -
6
5. x
2 -
4x -
21
6. x
2 -
22
x +
121
(
x -
3)(
x +
2)
(x -
7)(
x +
3)
(x -
11)(
x -
11)
7. t2
- 4
t -
12
8. p
2 -
16p
+ 6
4
9. 9 -
10
x +
x2
(
t +
2)(
t -
6)
(p
- 8
)(p
- 8
) (
9 -
x)(
1 -
x)
10. x
2 +
6x +
5
11. a
2 +
8a
- 9
12. y
2 -
7y -
8
(
x +
5)(
x +
1)
(a
- 1
)(a +
9)
(y -
8)(
y +
1)
13. x
2 -
2x -
3
14. y
2 +
14
y +
13
15. m
2 +
9m
+ 2
0
(
x -
3)(
x +
1)
(y +
1)(
y +
13)
(m
+ 4
)(m
+ 5
)
16. x
2 +
12
x +
20
17. a
2 -
14a
+ 2
4
18. 18 +
11
y +
y2
(
x +
10)(
x +
2)
(a
- 2
)(a -
12)
(9 +
y)(
2 +
y)
19. x
2 +
2xy +
y2
20. a
2 -
4a
b +
4b
2
21. x
2 +
6xy -
7y
2
(
x +
y)(
x +
y)
(a
- 2
b)(
a -
2b
) (
x +
7y)(
x -
y)
8-3
Exam
ple
1Exam
ple
2
Facto
rin
g x
2 +
bx +
cx
2 +
bx +
c =
(x +
m)(
x +
p),
where
m +
p =
b a
nd m
p =
c.
Answers (Lesson 8-2 and Lesson 8-3)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A8 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
18
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Qu
ad
rati
c E
qu
ati
on
s:
x2 +
bx +
c =
0
So
lve E
qu
ati
on
s b
y F
act
ori
ng
F
act
ori
ng a
nd
th
e Z
ero
Pro
du
ct P
rop
ert
y c
an
be u
sed
to
solv
e m
an
y e
qu
ati
on
s of
the f
orm
x2 +
bx +
c =
0.
S
olv
e x
2 +
6x
= 7
. C
heck
yo
ur s
olu
tio
ns.
x
2 +
6x =
7
Origin
al equation
x
2 +
6x -
7 =
0
Rew
rite
equation s
o t
hat
one s
ide e
quals
0.
(x
- 1
)(x +
7) =
0
Facto
r.
x -
1 =
0 or
x +
7 =
0
Zero
Pro
duct
Pro
pert
y
x =
1
x =
-7
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {
1, -
7}.
Sin
ce 1
2 +
6 =
7 a
nd
(-
7)2
+ 6
(-7) =
7,
the s
olu
tion
s ch
eck
.
R
OC
KET L
AU
NC
H A
ro
ck
et
is f
ired
wit
h a
n i
nit
ial
velo
cit
y o
f 2288
feet
per s
eco
nd
. H
ow
ma
ny
seco
nd
s w
ill
it t
ak
e f
or t
he r
ock
et
to h
it t
he g
ro
un
d?
Th
e f
orm
ula
h =
vt -
16t2
giv
es
the h
eig
ht
h o
f th
e r
ock
et
aft
er
t se
con
ds
wh
en
th
e i
nit
ial
velo
city
v i
s giv
en
in
feet
per
seco
nd
.
h
= v
t -
16t2
F
orm
ula
0 =
2288
t -
16
t2
Substitu
te.
0 =
16
t(143 -
t)
Facto
r.
16t =
0 or
143 -
t =
0
Zero
Pro
duct
Pro
pert
y
t =
0
t =
143
Solv
e e
ach e
quation.
Th
e v
alu
e t
= 0
rep
rese
nts
th
e t
ime a
t la
un
ch.
Th
e r
ock
et
retu
rns
to t
he g
rou
nd
in
143
seco
nd
s, o
r a l
ittl
e l
ess
th
an
2.5
min
ute
s aft
er
lau
nch
.
Exerc
ises
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
1. x
2 -
4x +
3 =
0 {1
, 3}
2. y
2 -
5y +
4 =
0 {1
, 4}
3. m
2 +
10m
+ 9
= 0
4. x
2 =
x +
2 {-
1,
2}
5. x
2 -
4x =
5 {-
1,
5}
6. x
2 -
12
x +
36 =
0 {6}
7. t2
- 8
= -
7t {-
8, 1}
8. p
2 =
9p -
14 {2
, 7}
9. -
9 -
8x +
x2 =
0 {-
1, 9}
10. x
2 +
6 =
5x {2,
3}
11. a
2 =
11a
- 1
8 {2,
9}
12. y
2 -
8y +
15
= 0
{3,
5}
13. x
2 =
24 -
10
x {-
12,
2}
14. a
2 -
18a
= -
72 {6,
12}
15. b
2 =
10b
- 1
6 {2,
8}
Use t
he f
orm
ula
h =
vt -
16t
2 t
o s
olv
e e
ach
pro
ble
m.
16. FO
OTB
ALL A
pu
nte
r ca
n k
ick
a f
ootb
all
wit
h a
n i
nit
ial
velo
city
of
48 f
eet
per
seco
nd
. H
ow
man
y s
eco
nd
s w
ill
it t
ak
e f
or
the b
all
to r
etu
rn t
o t
he g
rou
nd
? 3 s
eco
nd
s
17. B
ASEB
ALL A
ball
is
thro
wn
up
wit
h a
n i
nit
ial
velo
city
of
32 f
eet
per
seco
nd
. H
ow
man
y
seco
nd
s w
ill
it t
ak
e f
or
the b
all
to r
etu
rn t
o t
he g
rou
nd
? 2 s
eco
nd
s
18. R
OC
KET L
AU
NC
H If
a r
ock
et
is l
au
nch
ed
wit
h a
n i
nit
ial
velo
city
of
1600 f
eet
per
seco
nd
, w
hen
wil
l th
e r
ock
et
be 1
4,4
00 f
eet
hig
h?
at
10 s
eco
nd
s a
nd
at
90 s
eco
nd
s
8-3
Exam
ple
1
Exam
ple
2
{-1, -
9}
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Sk
ills
Pra
ctic
e
Qu
ad
rati
c E
qu
ati
on
s:
x2 +
bx +
c =
0
Fa
cto
r e
ach
po
lyn
om
ial.
1. t2
+ 8
t +
12
2. n
2 +
7n
+ 1
2
(
t +
2)(
t +
6)
(n
+ 3
)(n
+ 4
)
3. p
2 +
9p
+ 2
0
4. h
2 +
9h
+ 1
8
(
p +
5)(
p +
4)
(h
+ 6
)(h
+ 3
)
5. n
2 +
3n
- 1
8
6. x
2 +
2x -
8
(
n +
6)(
n -
3)
(x +
4)(
x -
2)
7. y
2 -
5y -
6
8. g
2 +
3g -
10
(
y +
1)(
y -
6)
(g
+ 5
)(g
- 2
)
9. r2
+ 4
r -
12
10. x
2 -
x -
12
(
r -
2)(
r +
6)
(x -
4)(
x +
3)
11. w
2 -
w -
6
12. y
2 -
6y +
8
(
w -
3)(
w +
2)
(y -
2)(
y -
4)
13. x
2 -
8x +
15
14. b
2 -
9b
+ 8
(
x -
5)(
x -
3)
(b
- 1
)(b
- 8
)
15. t2
- 1
5t +
56
16. -
4 -
3m
+ m
2
(
t -
7)(
t -
8)
(m
- 4
)(m
+ 1
)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
17. x
2 -
6x +
8 =
0 {2,
4}
18. b
2 -
7b
+ 1
2 =
0 {3,
4}
19. m
2 +
5m
+ 6
= 0
{-
3, -
2}
20. d
2 +
7d
+ 1
0 =
0 {-
5, -
2}
21. y
2 -
2y -
24
= 0
{-
4,
6}
22. p
2 -
3p
= 1
8 {-
3,
6}
23. h
2 +
2h
= 3
5 {-
7,
5}
24. a
2 +
14a
= -
45 {-
9, -
5}
25. n
2 -
36 =
5n
{-
4,
9}
26. w
2 +
30
= 1
1w
{5,
6}
8-3
Ch
ap
ter
8
19
Gle
ncoe A
lgeb
ra 1
Answers (Lesson 8-3)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A9 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
20
Gle
ncoe A
lgeb
ra 1
Practi
ce
Qu
ad
rati
c E
qu
ati
on
s:
x2 +
bx +
c =
0
Fa
cto
r e
ach
po
lyn
om
ial.
1. a
2 +
10a
+ 2
4
2. h
2 +
12h
+ 2
7
3. x
2 +
14
x +
33
(a +
4)(
a +
6)
(h
+ 3
)(h
+ 9
) (
x +
11)(
x +
3)
4. g
2 -
2g -
63
5. w
2 +
w -
56
6. y
2 +
4y -
60
(g +
7)(
g -
9)
(w
+ 8
)(w
- 7
) (
y +
10)(
y -
6)
7. b
2 +
4b
- 3
2
8. n
2 -
3n
- 2
8
9. t2
+ 4
t -
45
(b -
4)(
b +
8)
(n
- 7
)(n
+ 4
) (
t -
5)(
t +
9)
10. z2
- 1
1z +
30
11. d
2 -
16d
+ 6
3
12. x
2 -
11
x +
24
(z -
6)(
z -
5)
(d
- 9
)(d
- 7
) (
x -
3)(
x -
8)
13. q
2 -
q -
56
14. x
2 -
6x -
55
15. 32 +
18r +
r2
(q -
8)(
q +
7)
(x +
5)(
x -
11)
(r +
16)(
r +
2)
16. 48 -
16g +
g2
17. j 2
- 9
jk -
10
k2
18. m
2 -
mv -
56v
2
(g -
12)(
g -
4)
( j -
10
k)(
j +
k)
(m
- 8
v)(
m +
7v)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
19. x
2 +
17x +
42
= 0
20. p
2 +
5p
- 8
4 =
0
21. k
2 +
3k -
54 =
0{-
14, -
3}
{-
12,
7}
{-
9,
6}
22. b
2 -
12
b -
64 =
0
23. n
2 +
4n
= 3
2
24. h
2 -
17h
= -
60
{-
4,
16}
{-
8,
4}
{5,
12}
25. t2
- 2
6t =
56
26. z2
- 1
4z =
72
27. y
2 -
84
= 5
y
{-
2,
28}
{-
4,
18}
{-
7,
12}
28. 80 +
a2 =
18a
29. u
2 =
16u
+ 3
6
30. 17
r +
r2 =
-52
{8,
10}
{-
2,
18}
{-
13, -
4}
31. F
ind
all
valu
es
of
k s
o t
hat
the t
rin
om
ial
x2 +
kx -
35 c
an
be f
act
ore
d u
sin
g i
nte
gers
. -
34, -
2,
2,
34
32. C
ON
STR
UC
TIO
N A
con
stru
ctio
n c
om
pan
y i
s p
lan
nin
g t
o p
ou
r co
ncr
ete
for
a d
rivew
ay.
Th
e l
en
gth
of
the d
rivew
ay i
s 16 f
eet
lon
ger
than
its
wid
th w
.
a.
Wri
te a
n e
xp
ress
ion
for
the a
rea o
f th
e d
rivew
ay.
w(w
+ 1
6)
ft 2
b.
Fin
d t
he d
imen
sion
s of
the d
rivew
ay i
f it
has
an
are
a o
f 260 s
qu
are
feet.
10 f
t b
y 2
6 f
t
32. W
EB
DESIG
N Jan
eel
has
a 1
0-i
nch
by 1
2-i
nch
ph
oto
gra
ph
. S
he w
an
ts t
o s
can
th
e
ph
oto
gra
ph
, th
en
red
uce
th
e r
esu
lt b
y t
he s
am
e a
mou
nt
in e
ach
dim
en
sion
to p
ost
on
her
Web s
ite.
Jan
eel
wan
ts t
he a
rea o
f th
e i
mage t
o b
e o
ne e
igh
th t
hat
of
the o
rigin
al
ph
oto
gra
ph
.
a.
Wri
te a
n e
qu
ati
on
to r
ep
rese
nt
the a
rea o
f th
e r
ed
uce
d i
mage.
(10 -
x)(
12 -
x) =
15,
or
x2 -
22x +
105 =
0
b.
Fin
d t
he d
imen
sion
s of
the r
ed
uce
d i
mage.
3 i
n.
by 5
in
.
8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-3
Ch
ap
ter
8
21
Gle
ncoe A
lgeb
ra 1
1. C
OM
PA
CT D
ISC
S A
sta
nd
ard
jew
el
case
fo
r a c
om
pact
dis
c h
as
a w
idth
2 c
m
gre
ate
r th
an
its
len
gth
. T
he a
rea f
or
the
fron
t co
ver
is 1
68 s
qu
are
cen
tim
ete
rs.
Th
e f
irst
tw
o s
tep
s to
fin
din
g t
he v
alu
e o
f x a
re s
how
n b
elo
w.
Solv
e t
he e
qu
ati
on
an
d f
ind
th
e l
en
gth
of
the c
ase
.
Len
gth
×w
idth = a
rea
x(x + 2
) = 1
68
x2 + 2
x - 1
68 = 0
-14 o
r 12;
12 c
m
2. M
ATH
GA
MES
F
ion
a a
nd
Gre
g p
lay a
n
um
ber
gu
ess
ing g
am
e.
Gre
g g
ives
Fio
na
this
hin
t abou
t h
is t
wo s
ecr
et
nu
mbers
, “T
he p
rod
uct
of
the t
wo c
on
secu
tive
posi
tive i
nte
gers
th
at
I am
th
ink
ing o
f is
11 m
ore
th
an
th
eir
su
m.”
Wh
at
are
G
reg’s
nu
mbers
? 4 a
nd
5
3. B
RID
GE E
NG
INEER
ING
A
car
dri
vin
g
over
a s
usp
en
sion
bri
dge i
s su
pp
ort
ed
by
a c
able
han
gin
g b
etw
een
th
e e
nd
s of
the
bri
dge.
Sin
ce i
ts s
hap
e i
s p
ara
boli
c, i
t ca
n
be m
od
ele
d b
y a
qu
ad
rati
c equ
ati
on
. T
he
heig
ht
above t
he r
oad
bed
of
a b
rid
ge’s
ca
ble
h (
in i
nch
es)
measu
red
at
dis
tan
ce
d (
in y
ard
s) f
rom
th
e f
irst
tow
er
is g
iven
by t
he e
qu
ati
on
h=
d2-
36d+
324.
If t
he d
river
of
a c
ar
look
s ou
t at
a h
eig
ht
of
49 i
nch
es
above t
he r
oad
bed
, at
wh
at
dis
tan
ce(s
) fr
om
th
e t
ow
er
wil
l th
e
dri
ver’
s eyes
be a
t th
e s
am
e h
eig
ht
as
the
cable
? at
11 a
nd
25 y
ds f
rom
th
e
firs
t to
wer
4. PH
YSIC
AL S
CIE
NC
E T
he b
oil
ing p
oin
t of
wate
r d
ep
en
ds
on
alt
itu
de.
Th
e f
oll
ow
ing
equ
ati
on
ap
pro
xim
ate
s th
e n
um
ber
of
degre
es
D b
elo
w 2
12ºF
at
wh
ich
wate
r w
ill
boil
at
alt
itu
de h
.
D2+
520
D=
H
In D
en
ver,
Colo
rad
o,
the a
ltit
ud
e i
s ap
pro
xim
ate
ly 5
300 f
eet
above s
ea l
evel.
A
t ap
pro
xim
ate
ly w
hat
tem
pera
ture
does
wate
r boil
in
Den
ver?
D=
10°
dro
p T
he b
oilin
g p
oin
t is
ab
ou
t 202°F
.
5. M
ON
UM
EN
TS
Su
san
is
desi
gn
ing a
p
yra
mid
al
ston
e m
on
um
en
t fo
r a l
oca
l p
ark
. T
he d
esi
gn
sp
eci
fica
tion
s te
ll h
er
that
the h
eig
ht
need
s to
be 9
feet,
th
e
wid
th o
f th
e b
ase
mu
st b
e 5
feet
less
th
an
th
e l
en
gth
, an
d t
he v
olu
me s
hou
ld
be 1
50 c
ubic
feet.
Reca
ll t
hat
the
volu
me o
f a p
yra
mid
is
giv
en
by V
= 1 − 3
Bh
,
wh
ere
B i
s th
e a
rea o
f th
e b
ase
an
d h
is
the h
eig
ht.
a.
Wri
te a
nd
solv
e a
n e
qu
ati
on
to f
ind
th
e
wid
th o
f th
e b
ase
of
the m
on
um
en
t.
150 =
1 − 3w
(w+
5) ·
9 o
r
3w
2+
15
w-
150 =
0;
w=
{5, -
10}
b.
Inte
rpre
t each
an
swer
in t
erm
s of
the
situ
ati
on
. w=
5:
the w
idth
of
the
pyra
mid
is 5
feet;
w=-
10:
neg
ati
ve l
en
gth
do
esn
’t m
ake
sen
se i
n t
he s
itu
ati
on
.
Wo
rd
Pro
ble
m P
racti
ce
Qu
ad
rati
c E
qu
ati
on
s:
x2 +
bx +
c =
0
d
h
8-3
Answers (Lesson 8-3)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A10 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
22
Gle
ncoe A
lgeb
ra 1
En
rich
men
t
Pu
zzlin
g P
rim
es
A p
rim
e n
um
ber
has
on
ly t
wo f
act
ors
, it
self
an
d 1
. T
he n
um
ber
6 i
s n
ot
pri
me b
eca
use
it
has
2 a
nd
3 a
s fa
ctors
; 5 a
nd
7 a
re p
rim
e.
Th
e n
um
ber
1 i
s n
ot
con
sid
ere
d t
o b
e p
rim
e.
1. U
se a
calc
ula
tor
to h
elp
you
fin
d t
he 2
5 p
rim
e n
um
bers
less
th
an
100.
2
, 3,
5,
7,
11,
13,
17,
19,
23,
29,
31,
37 4
1,
43,
47,
53,
59,
61,
67,
71,
73,
79,
83,
89,
97
Pri
me n
um
bers
have i
nte
rest
ed
math
em
ati
cian
s fo
r ce
ntu
ries.
Th
ey h
ave t
ried
to f
ind
exp
ress
ion
s th
at
wil
l giv
e a
ll t
he p
rim
e n
um
bers
, or
on
ly p
rim
e n
um
bers
. In
th
e 1
700s,
E
ule
r d
isco
vere
d t
hat
the t
rin
om
ial
x2 +
x +
41 w
ill
yie
ld p
rim
e n
um
bers
for
valu
es
of
x
from
0 t
hro
ugh
39.
2. F
ind
th
e p
rim
e n
um
bers
gen
era
ted
by E
ule
r’s
form
ula
for
x f
rom
0 t
hro
ugh
7.
4
1,
43,
47,
53,
61,
71,
83,
97
3. S
how
th
at
the t
rin
om
ial
x2 +
x +
31 w
ill
not
giv
e p
rim
e n
um
bers
for
very
man
y v
alu
es
of
x.
I
t w
ork
s f
or
x =
0,
2,
3,
5,
an
d 6
bu
t n
ot
for
x =
1,
4,
an
d 7
.
4. F
ind
th
e l
arg
est
pri
me n
um
ber
gen
era
ted
by E
ule
r’s
form
ula
.
1601
Gold
ba
ch’s
Con
ject
ure
is
that
every
non
zero
even
nu
mber
gre
ate
r th
an
2 c
an
be w
ritt
en
as
the s
um
of
two p
rim
es.
No o
ne h
as
ever
pro
ved
th
at
this
is
alw
ays
tru
e,
bu
t n
o o
ne h
as
fou
nd
a c
ou
nte
rexam
ple
, eit
her.
5. S
how
th
at
Gold
bach
’s C
on
ject
ure
is
tru
e f
or
the f
irst
5 e
ven
nu
mbers
gre
ate
r th
an
2.
4
= 2
+ 2
, 6 =
3 +
3,
8 =
3 +
5,
10
= 3
+ 7
, 12 =
5 +
7
6. G
ive a
way t
hat
som
eon
e c
ou
ld d
isp
rove G
old
bach
’s C
on
ject
ure
.
F
ind
an
even
nu
mb
er
that
can
no
t b
e w
ritt
en
as t
he s
um
of
two
pri
mes.
8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-4
Ch
ap
ter
8
23
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Qu
ad
rati
c E
qu
ati
on
s:
ax
2 +
bx +
c =
0
Facto
r ax
2 +
bx
+ c
T
o f
act
or
a t
rin
om
ial
of
the f
orm
ax
2 +
bx +
c,
fin
d t
wo i
nte
gers
, m
an
d p
wh
ose
pro
du
ct i
s equ
al
to a
c an
d w
hose
su
m i
s equ
al
to b
. If
th
ere
are
no i
nte
gers
th
at
sati
sfy t
hese
requ
irem
en
ts,
the p
oly
nom
ial
is c
all
ed
a p
rim
e p
oly
no
mia
l.
F
acto
r 2x
2 +
15x +
18.
In t
his
exam
ple
, a
= 2
, b
= 1
5,
an
d c
= 1
8.
You
need
to f
ind
tw
o n
um
bers
th
at
have a
su
m o
f 15 a
nd
a p
rod
uct
of
2
18 o
r 36.
Mak
e a
lis
t of
the f
act
ors
of
36 a
nd
look
for
the p
air
of
fact
ors
wit
h a
su
m o
f 15.
Facto
rs o
f 36
Su
m o
f F
acto
rs
1,
36
37
2,
18
20
3,
12
15
U
se t
he p
att
ern
ax
2 +
mx +
px +
c,
wit
h
a
= 2
, m
= 3
, p
= 1
2,
an
d c
= 1
8.
2
x2 +
15x +
18 =
2x
2 +
3x +
12x +
18
= (
2x
2 +
3x) +
(1
2x +
18)
= x
(2x +
3) +
6(2
x +
3)
= (
x +
6)(
2x +
3)
Th
ere
fore
, 2x
2 +
15x +
18 =
(x +
6)(
2x +
3).
F
acto
r 3x
2 -
3x -
18.
Note
th
at
the G
CF
of
the t
erm
s 3
x2,
3x,
an
d 1
8 i
s 3.
Fir
st f
act
or
ou
t th
is G
CF
.
3x
2 -
3x -
18 =
3(x
2 -
x -
6).
Now
fact
or
x2 -
x -
6.
Sin
ce a
= 1
, fi
nd
th
e
two f
act
ors
of -
6 w
ith
a s
um
of -
1.
Facto
rs o
f -
6S
um
of
Facto
rs
1, -
6-
5
-1,
65
-2,
31
2, -
3-
1
Now
use
th
e p
att
ern
(x +
m)(
x +
p)
wit
h
m =
2 a
nd
p =
-3.
x2 -
x -
6 =
(x +
2)(
x -
3)
Th
e c
om
ple
te f
act
ori
zati
on
is
3x
2 -
3x -
18 =
3(x
+ 2
)(x -
3).
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
usin
g
inte
gers,
writ
e prim
e.
1. 2
x2 -
3x -
2
2. 3
m2 -
8m
- 3
3. 16
r2 -
8r
+ 1
(
2x +
1)(
x -
2)
(3
m +
1)(
m -
3)
(4
r -
1)(
4r -
1)
4. 6
x2 +
5x -
6
5. 3
x2 +
2x -
8
6. 18
x2 -
27
x -
5
(
2x +
3)(
3x -
2)
(3
x -
4)(
x +
2)
(3
x -
5)(
6x +
1)
7. 2
a2 +
5a
+ 3
8. 18
y2 +
9y -
5
9. -
4t2
+ 1
9t -
21
(
2a
+ 3
)(a
+ 1
) (
6y +
5)(
3y -
1)
(4t -
7)(
3 -
t)
10. 8
x2 -
4x -
24
11. 28
p2 +
60p
- 2
5
12. 48x
2 +
22
x -
15
(
4x -
8)(
2x +
3)
(2
p +
5)(
14p
- 5
) (
6x +
5)(
8x -
3)
13. 3y
2 -
6y -
24
14. 4
x2 +
26x -
48
15. 8
m2 -
44
m +
48
3
(y +
2)(
y -
4)
2(x
+ 8
)(2x -
3)
4(2
m -
3)(
m -
4)
16. 6x
2 -
7x +
18
17. 2
a2 -
14
a +
18
18. 18 +
11y +
2y
2
p
rim
e
2(a
2 -
7a +
9)
pri
me
8-4
Exam
ple
1Exam
ple
2
Exerc
ises
Answers (Lesson 8-3 and Lesson 8-4)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A11 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
24
Gle
ncoe A
lgeb
ra 1
So
lve E
qu
ati
on
s b
y F
act
ori
ng
F
act
ori
ng a
nd
th
e Z
ero
Pro
du
ct P
rop
ert
y c
an
be u
sed
to
solv
e s
om
e e
qu
ati
on
s of
the f
orm
ax
2 +
bx +
c =
0.
S
olv
e 1
2x
2 +
3x =
2 -
2x
. C
heck
yo
ur s
olu
tio
ns.
12
x2 +
3x =
2 -
2x
Origin
al equation
12x
2 +
5x -
2 =
0
Rew
rite
equation s
o t
hat
one s
ide e
quals
0.
(3
x +
2)(
4x -
1)
= 0
F
acto
r th
e left s
ide.
3
x +
2 =
0 o
r 4
x -
1 =
0
Zero
Pro
duct
Pro
pert
y
x =
- 2
−
3
x =
1
−
4
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {-
2
−
3 ,
1
−
4 } .
Sin
ce 1
2 (-
2
−
3 ) 2
+ 3
(- 2
−
3 ) =
2 -
2 (-
2
−
3 ) a
nd
12 ( 1
−
4 ) 2
+ 3
( 1
−
4 ) =
2 -
2 ( 1
−
4 ) ,
the s
olu
tion
s ch
eck
.
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
1. 8x
2 +
2x -
3 =
0
2. 3n
2 -
2n
- 5
= 0
3. 2
d2 -
13
d -
7 =
0
{
1
−
2 , -
3
−
4 }
{-
1,
5
−
3 }
{-
1
−
2 ,
7}
4. 4
x2 =
x +
3
5. 3x
2 -
13x =
10
6. 6
x2 -
11x -
10 =
0
{
1, -
3
−
4 }
{-
2
−
3 ,
5}
{-
2
−
3 ,
5
−
2 }
7. 2
k2 -
40 =
-11k
8. 2p
2 =
-21p
- 4
0
9.
-7 -
18x +
9x
2 =
0
{-
8,
5
−
2 }
{-
5
−
2 , -
8}
{ 7
−
3 , -
1
−
3 }
10. 12x
2 -
15 =
-8
x
11. 7a
2 =
-65a
- 1
8
12. 16
y2 -
2y -
3 =
0
{- 3
−
2 ,
5
−
6 }
{-
2
−
7 , -
9 }
{ 1
−
2 , -
3
−
8 }
13. 8
x2 +
5x =
3 +
7x
14. 4a
2 -
18
a +
5 =
15
15. 3
b2 -
18
b =
10b
- 4
9
{ 3
−
4 , -
1
−
2 }
{-
1
−
2 ,
5}
{ 7
−
3 ,
7}
16. T
he d
iffe
ren
ce o
f th
e s
qu
are
s of
two c
on
secu
tive o
dd
in
tegers
is
24.
Fin
d t
he i
nte
gers
. -
5, -
7 a
nd
5,
7
17. G
EO
METR
Y T
he l
en
gth
of
a C
harl
ott
e,
Nort
h C
aro
lin
a,
con
serv
ato
ry g
ard
en
is
20 y
ard
s gre
ate
r th
an
its
wid
th.
Th
e a
rea i
s 300 s
qu
are
yard
s. W
hat
are
th
e d
imen
sion
s?
30 y
d b
y 1
0 y
d
18. G
EO
METR
Y A
rect
an
gle
wit
h a
n a
rea o
f 24 s
qu
are
in
ches
is
fo
rmed
by c
utt
ing s
trip
s of
equ
al
wid
th f
rom
a r
ect
an
gu
lar
pie
ce
of
pap
er.
Fin
d t
he d
imen
sion
s of
the n
ew
rect
an
gle
if
the o
rigin
al
rect
an
gle
measu
res
8 i
nch
es
by 6
in
ches.
6 i
n.
by 4
in
.
8 in
.
x
6 in
.
x
x x
8-4 Exam
ple
Exerc
ises
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Qu
ad
rati
c E
qu
ati
on
s:
ax
2 +
bx +
c =
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-4
Ch
ap
ter
8
25
Gle
ncoe A
lgeb
ra 1
Sk
ills
Pra
ctic
e
Qu
ad
rati
c E
qu
ati
on
s:
ax
2 +
bx +
c =
0
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
usin
g
inte
gers,
writ
e prim
e.
1. 2
x2 +
5x +
2
2. 3n
2 +
5n
+ 2
(
x +
2)(
2x +
1)
(3
n +
2)(
n +
1)
3. 2
t2 +
9t
- 5
4. 3g
2 -
7g +
2
(
t +
5)(
2t -
1)
(3
g -
1)(
g -
2)
5. 2
t2 -
11t
+ 1
5
6. 2x
2 +
3x -
6
(
t -
3)(
2t -
5)
pri
me
7. 2
y2 +
y -
1
8. 4
h2 +
8h
- 5
(
y +
1)(
2y -
1)
(2
h +
5)(
2h
- 1
)
9. 4
x2 -
3x -
3
10. 4
b2 +
15b
- 4
p
rim
e
(4
b -
1)(
b +
4)
11. 9
p2 +
6p
- 8
12. 6
q2 -
13q
+ 6
(
3p
- 2
)(3p
+ 4
) (
3q
- 2
)(2q
- 3
)
13. 3
a2 +
30a
+ 6
3
14. 10w
2 -
19w
- 1
5
3
(a +
7)(
a +
3)
(2w
- 5
)(5
w +
3)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
15. 2x
2 +
7x +
3 =
0 {-
3, - 1
−
2 }
16. 3w
2 +
14w
+ 8
= 0
{-
4, -
2
−
3 }
17. 3
n2 -
7n
+ 2
= 0
{ 1
−
3 ,
2}
18. 5d
2 -
22
d +
8 =
0 { 2
−
5 ,
4}
19. 6
h2 +
8h
+ 2
= 0
{-
1, -
1
−
3 }
20. 8
p2 -
16
p =
10 {-
1
−
2 ,
5
−
2 }
21. 9
y2 +
18y -
12 =
6y {-
2,
2
−
3 }
22. 4
a2 -
16
a =
-15 { 3
−
2 ,
5
−
2 }
23. 10
b2 -
15
b =
8b -
12 { 4
−
5 ,
3
−
2 }
24. 6
d2 +
21d
= 1
0d
+ 3
5 {-
7
−
2 ,
5
−
3 }
8-4
Answers (Lesson 8-4)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A12 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
26
Gle
ncoe A
lgeb
ra 1
Practi
ce
Qu
ad
rati
c E
qu
ati
on
s:
ax
2 +
bx +
c =
0
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
usin
g
inte
gers,
writ
e prim
e.
1. 2b
2 +
10b
+ 1
2
2. 3g
2 +
8g +
4
3. 4x
2 +
4x -
3
2
(b +
2)(
b +
3)
(3
g +
2)(
g +
2)
(2x +
3)(
2x -
1)
4. 8b
2 -
5b -
10
5. 6m
2 +
7m
- 3
6. 10d
2 +
17d
- 2
0
p
rim
e
(3m
- 1
)(2m
+ 3
) (
5d
- 4
)(2
d +
5)
7. 6a
2 -
17a
+ 1
2
8. 8w
2 -
18w
+ 9
9. 10x
2 -
9x +
6
(
3a
- 4
)(2
a -
3)
(4
w -
3)(
2w
- 3
) p
rim
e
10. 15n
2 -
n -
28
11. 10x
2 +
21x -
10
12. 9r2
+ 1
5r
+ 6
(
5n
- 7
)(3
n +
4)
(2x +
5)(
5x -
2)
3(3
r +
2)(
r +
1)
13. 12y
2 -
4y -
5
14. 14k
2 -
9k
- 1
8
15. 8z2
+ 2
0z
- 4
8
(
2y +
1)(
6y -
5)
(2
k -
3)(
7k
+ 6
) 4
(z +
4)(
2z -
3)
16. 12q
2 +
34q
- 2
8
17. 18h
2 +
15h
- 1
8
18. 12p
2 -
22p
- 2
0
2
(3q
- 2
)(2
q +
7)
3(2
h +
3)(
3h
- 2
) 2
(3p
+ 2
)(2
p -
5)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
19. 3h
2 +
2h
- 1
6 =
0
20. 15n
2 -
n =
2
21. 8q
2 -
10q +
3 =
0
{-
8
−
3 ,
2}
{-
1
−
3 ,
2
−
5 }
{ 1
−
2 ,
3
−
4 }
22. 6b
2 -
5b =
4
23. 10r2
- 2
1r
= -
4r
+ 6
24. 10g
2 +
10 =
29g
{-
1
−
2 ,
4
−
3 }
{-
3
−
10 ,
2}
{ 2
−
5 ,
5
−
2 }
25. 6y
2 =
-7y -
2
26. 9z2
= -
6z
+ 1
5
27. 12k
2 +
15k
= 1
6k +
20
{-
2
−
3 , -
1
−
2 }
{-
5
−
3 ,
1}
{-
5
−
4 ,
4
−
3 }
28. 12x
2 -
1 =
-x
29. 8a
2 -
16a
= 6a
- 1
2
30. 18a
2 +
10a
= -
11a
+ 4
{-
1
−
3 ,
1
−
4 }
{ 3
−
4 ,
2}
{-
4
−
3 ,
1
−
6 }
31. D
IVIN
G L
au
ren
dove i
nto
a s
wim
min
g p
ool
from
a 1
5-f
oot-
hig
h d
ivin
g b
oard
wit
h a
n
init
ial
up
ward
velo
city
of
8 f
eet
per
seco
nd
. F
ind
th
e t
ime t
in
seco
nd
s it
took
Lau
ren
to
en
ter
the w
ate
r. U
se t
he m
od
el
for
vert
ical
moti
on
giv
en
by t
he e
qu
ati
on
h
= -
16t2
+ vt
+ s
, w
here
h i
s h
eig
ht
in f
eet,
t i
s ti
me i
n s
eco
nd
s, v
is
the i
nit
ial
up
ward
velo
city
in
feet
per
seco
nd
, an
d s
is
the i
nit
ial
heig
ht
in f
eet.
(Hint:
Let h
= 0
rep
rese
nt
the s
urf
ace
of
the p
ool.
) 1.2
5 s
32. B
ASEB
ALL B
rad
toss
ed
a b
ase
ball
in
th
e a
ir f
rom
a h
eig
ht
of
6 f
eet
wit
h a
n i
nit
ial
up
ward
velo
city
of
14 f
eet
per
seco
nd
. E
nri
qu
e c
au
gh
t th
e b
all
on
its
way d
ow
n a
t a p
oin
t 4 f
eet
above t
he g
rou
nd
. H
ow
lon
g w
as
the b
all
in
th
e a
ir b
efo
re E
nri
qu
e c
au
gh
t it
? U
se
the m
od
el
of
vert
ical
moti
on
fro
m E
xerc
ise 3
1.
1 s
8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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TE
PE
RIO
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Lesson 8-4
Ch
ap
ter
8
27
Gle
ncoe A
lgeb
ra 1
Wo
rd
Pro
ble
m P
racti
ce
Qu
ad
rati
c E
qu
ati
on
s:
ax
2 +
bx +
c =
0
1. B
REA
K E
VEN
B
reak
ing e
ven
occ
urs
w
hen
th
e r
even
ues
for
a b
usi
ness
equ
al
the c
ost
. A
loca
l ch
ild
ren
’s m
use
um
st
ud
ied
th
eir
cost
s (w
ages,
ele
ctri
city
, etc
.) a
nd
reven
ues
from
paid
ad
mis
sion
. T
hey f
ou
nd
th
at
their
bre
ak
-even
tim
e i
s giv
en
by t
he e
qu
ati
on
2h
2-
2h
- 2
4 =
0,
wh
ere
h i
s th
e n
um
ber
of
hou
rs t
he
mu
seu
m i
s op
en
per
day.
How
man
y
hou
rs m
ust
th
e m
use
um
be o
pen
per
day
to r
each
th
e b
reak
even
poin
t?
4 h
ou
rs
2. C
AR
PEN
TR
Y M
iko w
an
ts t
o b
uil
d a
toy
box f
or
her
sist
er.
It
is t
o b
e 2
feet
hig
h,
an
d t
he w
idth
is
to b
e 3
feet
less
th
an
its
le
ngth
. If
it
need
s to
hold
a v
olu
me o
f 80 c
ubic
feet,
fin
d t
he l
en
gth
an
d w
idth
of
the b
ox.
len
gth
= 8
ft;
wid
th =
5 f
t
3. FU
RN
ITU
RE
T
he s
tud
en
t co
un
cil
wan
ts
to p
urc
hase
a t
able
for
the s
chool
lobby.
Th
e t
able
com
es
in a
vari
ety
of
dim
en
sion
s, b
ut
for
every
table
, th
e
len
gth
is
1 m
ete
r gre
ate
r th
an
tw
ice t
he
wid
th.
Th
e s
tud
en
t co
un
cil
has
bu
dgete
d
for
a t
able
top
wit
h a
n a
rea o
f exact
ly
3 s
qu
are
mete
rs.
Fin
d t
he w
idth
an
d l
en
gth
of
the t
able
th
ey c
an
pu
rch
ase
. w
idth
= 1
m;
len
gth
= 3
m
4. LA
DD
ER
S A
lad
der
is r
est
ing a
gain
st a
w
all
. T
he t
op
of
the l
ad
der
tou
ches
the
wall
at
a h
eig
ht
of
15 f
eet,
an
d t
he l
en
gth
of
the l
ad
der
is o
ne f
oot
more
th
an
tw
ice
its
dis
tan
ce f
rom
th
e w
all
. F
ind
th
e
dis
tan
ce f
rom
th
e w
all
to t
he b
ott
om
of
the l
ad
der.
(Hint:
Use
th
e P
yth
agore
an
T
heore
m t
o s
olv
e t
he p
roble
m.)
8 f
t
15 ft.
Wall
Ladder
5. FA
RM
ING
M
r. H
en
sley h
as
a t
ota
l of
480 s
qu
are
feet
of
sheet
meta
l w
ith
w
hic
h h
e w
ou
ld l
ike t
o c
on
stru
ct a
cy
lin
dri
cal
tan
k f
or
stori
ng g
rain
. T
he
loca
l zon
ing l
aw
lim
its
the h
eig
ht
of
the
tan
k t
o 1
3.5
feet.
Reca
ll t
hat
a f
orm
ula
fo
r th
e s
urf
ace
are
a o
f a b
ott
om
less
cy
lin
der
wit
h r
ad
ius r
an
d h
eig
ht h
is
A =
πr2
+ 2
πrh
.
a.
Wri
te a
qu
ad
rati
c equ
ati
on
(se
t equ
al
to z
ero
) to
rep
rese
nt
the i
nfo
rmati
on
.
0 =
πr2
+ 2
7π
r -
480
b.
Usi
ng 3
as
an
ap
pro
xim
ati
on
for
π,
solv
e t
he e
qu
ati
on
for r.
{5
, -
32}
c.
Wh
at
rad
ius
shou
ld M
r. H
en
sley u
se
for
his
tan
k?
5 f
t
8-4
2w
+1
w
Answers (Lesson 8-4)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A13 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
28
Gle
ncoe A
lgeb
ra 1
En
rich
men
t
Are
a M
od
els
fo
r Q
uad
rati
c T
rin
om
ials
Aft
er
you
have f
act
ore
d a
qu
ad
rati
c tr
inom
ial,
you
can
use
th
e f
act
ors
to
dra
w g
eom
etr
ic m
od
els
of
the t
rin
om
ial.
x
2 +
5x -
6 =
(x -
1)(
x +
6)
To d
raw
a r
ect
an
gu
lar
mod
el,
th
e v
alu
e 2
w
as
use
d f
or
x s
o t
hat
the s
hort
er
sid
e w
ou
ld
have a
len
gth
of
1.
Th
en
th
e d
raw
ing w
as
don
e i
n c
en
tim
ete
rs.
So,
the a
rea o
f th
e
rect
an
gle
is
x2 +
5x -
6.
To d
raw
a r
igh
t tr
ian
gle
mod
el,
reca
ll t
hat
th
e a
rea o
f a t
rian
gle
is
on
e-h
alf
th
e b
ase
ti
mes
the h
eig
ht.
So,
on
e o
f th
e s
ides
mu
st
be t
wic
e a
s lo
ng a
s th
e s
hort
er
sid
e o
f th
e
rect
an
gu
lar
mod
el.
x2 +
5x -
6 =
(x -
1)(
x +
6)
= 1
−
2 (
2x -
2)(
x +
6)
Th
e a
rea o
f th
e r
igh
t tr
ian
gle
is
als
o x
2 +
5x -
6.
Fa
cto
r e
ach
trin
om
ial.
Th
en
fo
llo
w t
he d
irecti
on
s t
o d
ra
w e
ach
mo
del
of
the t
rin
om
ial.
1. x
2 +
2x -
3 U
se x
= 2
. D
raw
a
2. 3
x2 +
5x -
2 U
se x
= 1
. D
raw
a
rect
an
gle
in
cen
tim
ete
rs.
re
ctan
gle
in
cen
tim
ete
rs.
(x
+ 2
)(3x -
1)
(
x +
3)(
x -
1)
3. x
2 -
4x +
3 U
se x
= 4
. D
raw
tw
o d
iffe
ren
t ri
gh
t tr
ian
gle
s in
cen
tim
ete
rs.
(x
- 1
)(x -
3)
4. 9
x2 -
9x +
2 U
se x
= 2
. D
raw
tw
o d
iffe
ren
t ri
gh
t tr
ian
gle
s.
Use
0.5
cen
tim
ete
r fo
r each
un
it.
(3x -
2)(
3x -
1)
3x
- 2
6x
- 2
3x
- 1
6x
- 4
2x
- 6
x-
1
x-
3
2x
- 2
3x
- 1
x+
2
x-
1
x+
3
x+
6
2x
- 2
x+
6
x-
1
8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-4
Ch
ap
ter
8
29
Gle
ncoe A
lgeb
ra 1
Gra
ph
ing C
alc
ula
tor
Act
ivit
y
Usin
g T
ab
les i
n F
acto
rin
g b
y G
rou
pin
g
Th
e T
AB
LE f
eatu
re c
an
be u
sed
to h
elp
fact
or
a p
oly
nom
ial
by f
ind
ing t
he f
act
ors
of
a c
ert
ain
p
rod
uct
, w
hic
h h
ave a
sp
eci
fic
sum
.
F
acto
r 1
0x
2-
43x
+ 2
8 b
y g
ro
up
ing
.
Mak
e a
table
of
the n
egati
ve f
act
ors
of
10
! 2
8 o
r 280.
Look
for
a p
air
of
fact
ors
wh
ose
su
m i
s -
43.
En
ter
the e
qu
ati
on
y =
280
−
x
in
Y1 t
o f
ind
th
e f
act
ors
of
280.
Th
en
,
fin
d t
he s
um
of
the f
act
ors
usi
ng y
= 2
80
−
x
+ x
in
Y2.
Set
up
th
e t
able
to d
isp
lay t
he n
egati
ve f
act
ors
of
280 b
y s
ett
ing ∆
Tb
l =
to
-1.
Exam
ine t
he r
esu
lts.
Th
e l
ast
lin
e o
f th
e t
able
sh
ow
s th
at
-43x m
ay b
e r
ep
lace
d w
ith
-
8x
+ (
-35x).
10x
2 -
43x +
28 =
10x
2 -
8x +
(-
35
x)
+ 2
8
=
2x(5
x -
4)
+ (
-7)(
5x -
4)
=
(5
x -
4)(
2x -
7)
Th
us,
10
x2 -
43
x +
28 =
(5x -
4)(
2x -
7).
F
acto
r 1
2x
2 -
7x -
12.
Look
at
the f
act
ors
of
12
! -
12 o
r -
144 f
or
a p
air
wh
ose
su
m i
s -
7.
En
ter
an
equ
ati
on
to d
ete
rmin
e t
he f
act
ors
in
Y1 a
nd
an
equ
ati
on
to
fin
d t
he s
um
of
fact
ors
in
Y2.
Exam
ine t
he t
able
to f
ind
a s
um
of
-7.
12x
2 -
7x -
12
= 1
2x
2 +
9x +
(-
16x)
- 1
2
= 3
x(4
x +
3)
- 4
(4x +
3)
=
(4
x +
3)(
3x -
4)
Th
us,
12x
2 -
7x -
12 =
(4
x +
3)(
3x -
4).
Fa
cto
r e
ach
qu
ad
ra
tic p
oly
no
mia
l if
po
ssib
le.
1. x
2 +
29
x -
96
2. x
2 -
14
x -
51
3. 3
z2 +
16z
- 3
5
(y +
32)(
y -
3)
(x -
17)(
x +
3)
(3
z -
5)(
z +
7)
4. 4
y2 -
25y +
18
5. 6a
2 -
a -
15
6. 6m
2 +
13m
+ 6
p
rim
e
(3a
- 5
)(2a
+ 3
) (
2m
+ 3
)(3m
+ 2
)
7. 12z2
- z
- 6
8. 16y
2 +
40y +
25
9. 4b
2 +
24b
- 4
93
(
4z -
3)(
3z +
2)
(4y +
5)2
(
2b
+ 2
9)(
2b
- 1
7)
8-4
Exam
ple
1
Exam
ple
2
Exerc
ises
Answers (Lesson 8-4)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A14 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
30
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Qu
ad
rati
c E
qu
ati
on
s:
Dif
fere
nces o
f S
qu
are
s
Fact
or
Dif
fere
nce
s o
f Sq
uare
s T
he b
inom
ial
exp
ress
ion
a2 -
b2 i
s ca
lled
th
e
dif
feren
ce o
f tw
o s
qu
ares.
Th
e f
oll
ow
ing p
att
ern
sh
ow
s h
ow
to f
act
or
the d
iffe
ren
ce o
f sq
uare
s.
F
acto
r e
ach
po
lyn
om
ial.
a. n
2 -
64
n
2 -
64
= n
2 -
82
Write
in t
he f
orm
a2 -
b2.
= (
n +
8)(
n -
8)
Facto
r.
b.
4m
2 -
81n
2
4m
2 -
81n
2
=
(2
m)2
- (
9n
)2
Write
in t
he f
orm
a2 -
b2.
=
(2
m -
9n
)(2
m +
9n
) F
acto
r.
F
acto
r e
ach
po
lyn
om
ial.
a.
50a
2 -
72
50a
2 -
72
=
2(2
5a
2 -
36)
Fin
d t
he G
CF
.
=
2[(
5a
)2 -
62)]
25
a2 =
5a
5a
and 3
6 =
6
6
=
2(5
a +
6)(
5a
- 6
) F
acto
r th
e d
iffe
rence o
f square
s.
b.
4x
4 +
8x
3 -
4x
2 -
8x
4
x4 +
8x
3 -
4x
2 -
8x
Origin
al poly
nom
ial
=
4x(x
3 +
2x
2 -
x -
2)
Fin
d t
he G
CF
.
=
4x[(
x3 +
2x
2) -
(x +
2)]
G
roup t
erm
s.
=
4x[x
2(x
+ 2
) -
1(x
+ 2
)]
Fin
d t
he G
CF
.
=
4x[(
x2 -
1)(
x +
2)]
F
acto
r by g
roupin
g.
=
4x[(
x -
1)(
x +
1)(
x +
2)]
Fa
cto
r th
e d
iffe
rence
of
square
s.
Exerc
ises
Fa
cto
r e
ach
po
lyn
om
ial.
1. x
2 -
81
2. m
2 -
100
3. 16
n2 -
25
(
x +
9)(
x -
9)
(m
+ 1
0)(
m -
10)
(4n
- 5
)(4
n +
5)
4. 36x
2 -
100
y2
5. 49x
2 -
36
6. 16
a2 -
9b
2
(
6x +
10y)(
6x -
10y)
(7x +
6)(
7x -
6)
(4a
- 3
b)(
4a
+ 3
b)
7. 225
b2 -
a2
8. 72p
2 -
50
9. -
2 +
2x
2
(
15b
- a
)(15b
+ a
) 2
(6p
+ 5
)(6p
- 5
) 2
(x -
1)(
x +
1)
10. -
81 +
a4
11. 6 -
54a
2
12. 8y
2 -
200
(
a -
3)(
a +
3)(
a2 +
9)
6(1
+ 3
a)(
1 -
3a
) 8
(y +
5)(
y -
5)
13. 4x
3 -
10
0x
14. 2y
4 -
32y
2
15. 8
m3 -
12
8m
4
x(x
+ 5
)(x -
5)
2y
2(y
+ 4
)(y -
4)
8m
(m +
4)(
m -
4)
16. 4x
2 -
25
17. 2a
3 -
98
ab
2
18. 18
y2 -
72
y4
(
2x +
5)(
2x -
5)
2a
(a -
7b
)(a
+ 7
b)
18
y2(1
- 2
y)(
1 +
2y)
19. 169x
3 -
x
20. 3
a4 -
3a
2
21. 3
x4 +
6x
3 -
3x
2 -
6x
x
(13x +
1)(
13
x -
1)
3a
2(a
+ 1
)(a
- 1
) 3
x(x
- 1
)(x +
1)(
x +
2)
8-5
Exam
ple
1Exam
ple
2
Dif
fere
nce o
f S
qu
are
sa
2 -
b2 =
(a
- b
)(a
+ b
) =
(a +
b)(
a -
b).
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-5
Ch
ap
ter
8
31
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Qu
ad
rati
c E
qu
ati
on
s:
Dif
fere
nces o
f S
qu
are
s
So
lve E
qu
ati
on
s b
y F
act
ori
ng
F
act
ori
ng a
nd
th
e Z
ero
Pro
du
ct P
rop
ert
y c
an
be u
sed
to
solv
e e
qu
ati
on
s th
at
can
be w
ritt
en
as
the p
rod
uct
of
an
y n
um
ber
of
fact
ors
set
equ
al
to 0
.
S
olv
e e
ach
eq
ua
tio
n.
Ch
eck
yo
ur s
olu
tio
ns.
a. x
2 -
1 −
25 =
0
x2 -
1
−
25 =
0
Origin
al equation
x2 -
( 1
−
5 ) 2
= 0
x
2 =
x ·
x a
nd
1
−
25 =
( 1
−
5 ) (
1
−
5 )
(x
+ 1
−
5 ) (x
- 1
−
5 ) =
0
Facto
r th
e d
iffe
rence o
f square
s.
x +
1
−
5 =
0
or
x -
1
−
5 =
0
Zero
Pro
duct
Pro
pert
y
x =
- 1
−
5
x =
1
−
5
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {-
1
−
5 ,
1
−
5 } .
Sin
ce (-
1
−
5 ) 2
-
1
−
25 =
0 a
nd
( 1
−
5 ) 2
-
1
−
25 =
0,
the s
olu
tion
s ch
eck
.
b.
4x
3 =
9x
4
x3 =
9x
Origin
al equation
4x
3 -
9x =
0
Subtr
act
9x f
rom
each s
ide.
x(4
x2 -
9) =
0
Fin
d t
he G
CF
.
x[(
2x)2
- 3
2] =
0
4x
2 =
2x
2x a
nd 9
= 3
3
x[(
2x)2
- 3
2] =
x[(
2x -
3)(
2x +
3)]
F
acto
r th
e d
iffe
rence o
f square
s.
x =
0 or
(2
x -
3) =
0 or
(2
x +
3) =
0
Zero
Pro
duct
Pro
pert
y
x =
0
x =
3
−
2
x =
- 3
−
2
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {0
, 3
−
2 , -
3
−
2 } .
Sin
ce 4
(0)3
= 9
(0),
4 ( 3
−
2 ) 3
= 9
( 3
−
2 ) ,
an
d 4
(- 3
−
2 ) 3
= 9
(- 3
−
2 ) ,
the s
olu
tion
s ch
eck
.
So
lve e
ach
eq
ua
tio
n b
y f
acto
rin
g.
Ch
eck
th
e s
olu
tio
ns.
1. 81
x2 =
49 { 7
−
9 , -
7 −
9 }
2. 36n
2 =
1 {-
1 −
6 ,
1 −
6 }
3. 25
d2 -
100 =
0 {2
, -
2}
4. 1
−
4 x
2 =
25
{1
0, -
10}
5. 36 =
1
−
25 x
2 {-
30,
30}
6. 4
9
−
100 -
x2
= 0
{-
7 −
10 ,
7 −
10 }
7. 9
x3 =
25x {0
, -
5 −
3 ,
5 −
3 }
8. 7
a3 =
175a
{0
, -
5,
5}
9. 2
m3 =
32m
{0
, -
4,
4}
10. 16y
3 =
25y {0
, -
5 −
4 ,
5 −
4 }
11. 1
−
64 x
2 =
49 {-
56,
56}
12. 4
a3 -
64a
= 0
{0
, -
4,
4}
13. 3
b3 -
27b
= 0
{0
, -
3,
3}
14. 9
−
25 m
2 =
121 {-
55 −
3 ,
55 −
3 }
15. 48n
3 =
147
n {0
, -
7 −
4 ,
7 −
4 }
8-5
Exerc
ises
Exam
ple
Answers (Lesson 8-5)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A15 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
32
Gle
ncoe A
lgeb
ra 1
Sk
ills
Practi
ce
Qu
ad
rati
c E
qu
ati
on
s:
Dif
fere
nces o
f S
qu
are
s
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
,
writ
e prim
e.
1. a
2 -
4
2. n
2 -
64
(
a +
2)(
a -
2)
(n
+ 8
)(n
- 8
)
3. 1 -
49
d2
4. -
16 +
p2
(
1 +
7d
)(1 -
7d
) (
p +
4)(
p -
4)
5. k
2 +
25
6. 36 -
100w
2
p
rim
e
(6 -
10
w)(
6 +
10w
)
7. t2
- 8
1u
2
8. 4
h2 -
25
g2
(
t +
9u
)(t -
9u
) (
2h
+ 5
g)(
2h
- 5
g)
9. 64m
2 -
9y
2
10. 4
c2 -
5d
2
(
8m
- 3
y)(
8m
+ 3
y)
pri
me
11. -
49r2
+ 4
t2
12. 8
x2 -
72p
2
(
2t +
7r)
(2t -
7r)
8
(x +
3p
)(x -
3p
)
13. 20q
2 -
5r2
14. 32a
2 -
50b
2
5
(2q
+ r
)(2
q -
r)
2(4
a +
5b
)(4
a -
5b
)
So
lve e
ach
eq
ua
tio
n b
y f
acto
rin
g.
Ch
eck
th
e s
olu
tio
ns.
15. 16x
2 -
9 =
0 {±
3
−
4 }
16. 25p
2 -
16 =
0 {±
4
−
5 }
17. 36q
2 -
49
= 0
{±
7
−
6 }
18. 81 -
4b
2 =
0 {±
9
−
2 }
19. 16d
2 =
4 {±
1
−
2 }
20. 18a
2 =
8 {±
2
−
3 }
21. n
2 -
9
−
25 =
0 {±
3
−
5 }
22. k
2 -
49 −
64 =
0 {±
7
−
8 }
23. 1
−
25 h
2 -
16 =
0 {±
20}
24. 1
−
16 y
2 =
81 {±
36}
8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-5
Ch
ap
ter
8
33
Gle
ncoe A
lgeb
ra 1
Practi
ce
Qu
ad
rati
c E
qu
ati
on
s:
Dif
fere
nces o
f S
qu
are
s
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
, w
rit
e
prim
e.
1. k
2 -
10
0
2. 81 -
r2
3. 16
p2 -
36
(
k +
10)(
k -
10)
(9 +
r)(
9 -
r)
(4
p +
6)(
4p
- 6
)
4. 4x
2 +
25
5. 144 -
9f 2
6. 36
g2 -
49
h2
p
rim
e
(12 +
3f)(
12 -
3f)
(6
g +
7h
)(6g
- 7
h)
7. 121m
2 -
14
4p
2
8. 32 -
8y
2
9. 24
a2 -
54
b2
(
11m
- 1
2p
)(11
m +
12p
) 8
(2 -
y)(
2 +
y)
6(2
a -
3b
)(2a
+ 3
b)
10. 32t2
- 1
8u
2
11. 9d
2 -
32
12. 36z3
- 9
z
2
(4t -
3u
)(4t +
3u
) p
rim
e
9z
(2z +
1)(
2z -
1)
13. 45q
3 -
20q
14. 100
b3 -
36
b
15. 3
t4 -
48t2
5
q(3
q +
2)(
3q
- 2
) 4
b(5
b +
3)(
5b
- 3
) 3
t2(t
+ 4
)(t -
4)
So
lve e
ach
eq
ua
tio
n b
y f
acto
rin
g.
Ch
eck
yo
ur s
olu
tio
ns.
16. 4y
2 =
81
17. 64p
2 =
9
18. 98b
2 -
50 =
0
{± 9
−
2 }
{± 3
−
8 }
{± 5
−
7 }
19. 32 -
162
k2 =
0
20. t2
-
64 −
121 =
0
21. 1
6 −
49 -
v2 =
0
{± 4
−
9 }
{± 8
−
11 }
{± 4
−
7 }
22.
1 −
36 x
2 -
25 =
0
23. 27h
3 =
48h
24. 75
g3 =
14
7g
{±
30}
{± 4
−
3 ,
0}
{± 7
−
5 ,
0}
25. ER
OSIO
N A
rock
bre
ak
s lo
ose
fro
m a
cli
ff a
nd
plu
nges
tow
ard
th
e g
rou
nd
400 f
eet
belo
w.
Th
e d
ista
nce
d t
hat
the r
ock
fall
s in
t s
eco
nd
s is
giv
en
by t
he e
qu
ati
on
d =
16t2
.
How
lon
g d
oes
it t
ak
e t
he r
ock
to h
it t
he g
rou
nd
? 5 s
26. FO
REN
SIC
S M
r. C
oop
er
con
test
ed
a s
peed
ing t
ick
et
giv
en
to h
im a
fter
he a
pp
lied
his
bra
kes
an
d s
kid
ded
to a
halt
to a
void
hit
tin
g a
noth
er
car.
In
tra
ffic
cou
rt,
he a
rgu
ed
th
at
the l
en
gth
of
the s
kid
mark
s on
th
e p
avem
en
t, 1
50 f
eet,
pro
ved
th
at
he w
as
dri
vin
g
un
der
the p
ost
ed
sp
eed
lim
it o
f 65 m
iles
per
hou
r. T
he t
ick
et
cite
d h
is s
peed
at
70 m
iles
per
hou
r. U
se t
he f
orm
ula
1 −
24 s
2 =
d,
wh
ere
s i
s th
e s
peed
of
the c
ar
an
d d
is
the l
en
gth
of
the s
kid
mark
s, t
o d
ete
rmin
e M
r. C
oop
er’
s sp
eed
wh
en
he a
pp
lied
th
e b
rak
es.
Was
Mr.
Coop
er
corr
ect
in
cla
imin
g t
hat
he w
as
not
speed
ing w
hen
he a
pp
lied
th
e b
rak
es?
60 m
ph
; yes
8-5
Answers (Lesson 8-5)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A16 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
34
Gle
ncoe A
lgeb
ra 1
Wo
rd
Pro
ble
m P
racti
ce
Qu
ad
rati
c E
qu
ati
on
s:
Dif
fere
nces o
f S
qu
are
s
1. LO
TTER
Y A
sta
te l
ott
ery
com
mis
sion
an
aly
zes
the t
ick
et
pu
rch
asi
ng p
att
ern
s of
its
citi
zen
s. T
he f
oll
ow
ing e
xp
ress
ion
is
develo
ped
to h
elp
off
icia
ls c
alc
ula
te t
he
lik
ely
nu
mber
of
peop
le w
ho w
ill
bu
y
tick
ets
for
a c
ert
ain
siz
e j
ack
pot.
81
a2-
36
b2
Fact
or
the e
xp
ress
ion
com
ple
tely
.
3(3
a+
2b
) (3
a-
2b
)
2. O
PTIC
S A
refl
ect
or
on
th
e i
nsi
de o
f a
cert
ain
fla
shli
gh
t is
a p
ara
bola
giv
en
by
the e
qu
ati
on
y=
x2-
25.
Fin
d t
he p
oin
ts
wh
ere
th
e r
efl
ect
or
meets
th
e l
en
s by
fin
din
g t
he v
alu
es
of
x w
hen
y=
0.
5, -
5
3. A
RC
HIT
EC
TU
RE
T
he d
raw
ing s
how
s a
tria
ngu
lar
roof
tru
ss w
ith
a b
ase
m
easu
rin
g t
he s
am
e a
s it
s h
eig
ht.
Th
e
are
a o
f th
e t
russ
is
98 s
qu
are
mete
rs.
Fin
d t
he h
eig
ht
of
the t
russ
. 14 m
4. B
ALLO
ON
ING
T
he f
un
ctio
n
f (t)
=-
16t2+
576 r
ep
rese
nts
th
e h
eig
ht
of
a f
reely
fall
ing b
all
ast
bag t
hat
start
s fr
om
rest
on
a b
all
oon
576 f
eet
above t
he
gro
un
d.
Aft
er
how
man
y s
eco
nd
s t
does
the b
all
ast
bag h
it t
he g
rou
nd
?
aft
er
6 s
eco
nd
s
5. D
EC
OR
ATIN
G M
arv
in w
an
ts t
o
pu
rch
ase
a r
ect
an
gu
lar
rug.
It h
as
an
are
a o
f 80 s
qu
are
feet.
He c
an
not
rem
em
ber
the l
en
gth
an
d w
idth
, bu
t h
e
rem
em
bers
th
at
the l
en
gth
was
8 m
ore
th
an
som
e n
um
ber
an
d t
he w
idth
was
8 l
ess
th
an
th
at
sam
e n
um
ber.
a
. Wri
te a
qu
ad
rati
c equ
ati
on
usi
ng t
he
info
rmati
on
giv
en
. x
2-
64 =
80 o
r x
2-
144 =
0
b
. Wh
at
are
th
e l
en
gth
an
d w
idth
of
the
rug?
20 f
t an
d 4
ft
Are
a =
98m
2he
ight
base
x+
8
x-
8
8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-5
Ch
ap
ter
8
35
Gle
ncoe A
lgeb
ra 1
En
ric
hm
en
t
Facto
rin
g T
rin
om
ials
of
Fo
urt
h D
eg
ree
Som
e t
rin
om
ials
of
the f
orm
a4 +
a2b
2 +
b4 c
an
be w
ritt
en
as
the
dif
fere
nce
of
two s
qu
are
s an
d t
hen
fact
ore
d.
F
acto
r 4x
4 -
37x
2y
2 +
9y
4.
Ste
p 1
F
ind
th
e s
qu
are
roots
of
the f
irst
an
d l
ast
term
s.
√ ""
4x
4 =
2x
2
√ ""
9y
4 =
3y
2
Ste
p 2
F
ind
tw
ice t
he p
rod
uct
of
the s
qu
are
roots
.
2(2
x2)(
3y
2) =
12
x2y
2
Ste
p 3
S
ep
ara
te t
he m
idd
le t
erm
in
to t
wo p
art
s. O
ne p
art
is
eit
her
you
r an
swer
to S
tep
2 o
r it
s op
posi
te.
Th
e o
ther
part
sh
ou
ld b
e
the o
pp
osi
te o
f a p
erf
ect
squ
are
.
-
37x
2y
2 =
-12x
2y
2 -
25x
2y
2
Ste
p 4
R
ew
rite
th
e t
rin
om
ial
as
the d
iffe
ren
ce o
f tw
o s
qu
are
s an
d
then
fact
or.
4x
4 -
37x
2y
2 +
9y
4 =
(4
x4 -
12x
2y
2 +
9y
4) -
25x
2y
2
= (
2x
2 -
3y
2)2
- 2
5x
2y
2
= [
(2x
2 -
3y
2) +
5xy][
(2x
2 -
3y
2) -
5xy]
= (
2x
2 +
5xy -
3y
2)(
2x
2 -
5xy -
3y
2)
Fa
cto
r e
ach
po
lyn
om
ial.
1. x
4 +
x2y
2 +
y4
2. x
4 +
x2 +
1
(
x2 + x
y +
y2)(
x2 -
xy +
y2)
(x
2 +
x +
1)(
x2 -
x +
1)
3. 9
a4 -
15a
2 +
1
4. 16a
4 -
17a
2 +
1
(
3a
2 +
3a
- 1
)(3
a2 -
3a -
1)
(4
a -
1)(
a +
1)(
4a
+ 1
)(a
- 1
)
5. 4
a4 -
13a
2 +
1
6. 9
a4 +
26a
2b
2 +
25
b4
(
2a
2 +
3a -
1)(
2a
2 -
3a -
1)
(3
a2 +
2ab
+ 5
b2)(
3a
2 -
2ab
+ 5
b2)
7. 4
x4 -
21x
2y
2 +
9y
4
8. 4
a4 -
29a
2b
2 +
25
b4
(
2x
2 +
3xy -
3y
2)(
2x
2 -
3xy -
3y
2)
(2
a +
5b
)(a
- b
)(2
a -
5b
)(a
+ b
)
8-5 Exam
ple
Answers (Lesson 8-5)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A17 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
36
Gle
ncoe A
lgeb
ra 1
Sp
read
sheet
Act
ivit
y
Dif
fere
nces o
f S
qu
are
s
Th
ere
is
a s
peci
al
patt
ern
you
can
use
to f
act
or
bin
om
ials
of
the f
orm
a2 -
b2.
You
can
use
a s
pre
ad
sheet
to d
isco
ver
this
rela
tion
ship
.
U
se a
sp
rea
dsh
eet
to i
nv
esti
ga
te t
he v
alu
es o
f th
e e
xp
ress
ion
s (a
2 -
b2),
(a
- b
)2, (a
- b
)(a
+ b
), a
nd
(a
+ b
)2. W
ha
t co
nje
ctu
re c
an
yo
u m
ak
e a
bo
ut
the e
xp
ressio
ns?
Ste
p 1
Y
ou
wil
l u
se C
olu
mn
s A
an
d B
to e
nte
r vari
ou
s valu
es
that
you
ch
oose
for a
an
d b
.
Ste
p 2
E
nte
r th
e f
orm
ula
s fo
r (a
2 -
b2),
(a
- b
)2,
(a -
b)(a
+ b
), a
nd
(a
+ b
)2
in C
olu
mn
s C
, D
, E
, an
d F
. T
o e
nte
r an
exp
on
en
t, u
se t
he s
ym
bol
^
foll
ow
ed
by t
he e
xp
on
en
t. F
or
exam
ple
, th
e s
qu
are
of
the v
alu
e i
n
cell
A2 i
s en
tere
d a
s A
2^
2.
1. E
nte
r vari
ou
s valu
es
of a
an
d b
in
Colu
mn
s A
an
d B
. L
ook
for
a p
att
ern
. W
hat
do y
ou
obse
rve a
bou
t th
e e
xp
ress
ion
s? F
or
an
y v
alu
es o
f a
an
d b
, (a
2 -
b2) =
(a -
b)(
a +
b).
2. F
ind
th
e p
rod
uct
s of
(a -
b)2
, (a
- b
)(a
+ b
), a
nd
(a
+ b
)2.
Do t
he r
esu
lts
veri
fy y
ou
r co
nje
ctu
re?
(a -
b)2
= a
2 -
2ab
+ b
2;
(a -
b)(
a +
b) =
a
2 -
b2;
an
d (
a +
b)2
= a
2 +
2ab
+ b
2;
yes
Use t
he p
att
ern
yo
u o
bserv
ed
to
fa
cto
r e
ach
bin
om
ial.
3. m
2 -
t2
4. x
2 -
4
5. y
2 -
16
(
m -
t)(
m +
t)
(x -
2)(
x +
2)
(y -
4)(
y +
4)
6. q
2 -
121
7. r2
- 1
69
8. b
2 -
1
(q
- 1
1)(
q +
11)
(r -
13)(
r +
13)
(b
- 1
)(b
+ 1
)
9. 4x
2 -
1
10. 16t2
- r
2
11. 25a
2 -
81d
2
(
2x -
1)(
2x +
1)
(4
t -
r)(
4t +
r)
(5a
- 9
d)(
5a +
9d
)
A
1 2 3 4 5
BC
DE
Fa^
2 -
b^
2(a
- b
)(a +
b)
(a -
b)^
2(a
+ b
)^2
ab
=(A
2-
B2)^
2=
A2^2-
B2^2
=(A
2-
B2)*
(A2+
B2)
=(A
2+
B2)^
2
=(A
3-
B3)^
2=
A3^2-
B3^2
=(A
3-
B3)*
(A3+
B3)
=(A
3+
B3)^
2
=(A
4-
B4)^
2=
A4^2-
B4^2
=(A
4-
B4)*
(A4+
B4)
=(A
4+
B4)^
2
Sh
eet
1S
heet
2S
heet
3
8-5
Exerc
ises
Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-6
Ch
ap
ter
8
37
Gle
ncoe A
lgeb
ra 1
Fact
or
Perf
ect
Sq
uare
Tri
no
mia
ls
Th
e p
att
ern
s sh
ow
n b
elo
w c
an
be u
sed
to f
act
or
perf
ect
squ
are
tri
nom
ials
.
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Qu
ad
rati
c E
qu
ati
on
s:
Perf
ect
Sq
uare
s
D
ete
rm
ine w
heth
er
16n
2 -
24n
+ 9
is a
perfe
ct
sq
ua
re
trin
om
ial.
If
so
, fa
cto
r i
t.
Sin
ce 1
6n
2 =
(4n
)(4n
), t
he f
irst
term
is
a
perf
ect
squ
are
.
Sin
ce 9
= 3
! 3
, th
e l
ast
term
is
a p
erf
ect
sq
uare
.
Th
e m
idd
le t
erm
is
equ
al
to 2
(4n
)(3).
Th
ere
fore
, 16n
2 -
24n
+ 9
is
a p
erf
ect
sq
uare
tri
nom
ial.
16n
2 -
24n
+ 9
= (
4n
)2 -
2(4n
)(3)
+ 3
2
=
(4n
- 3
)2
F
acto
r 1
6x
2 -
32x +
15.
Sin
ce 1
5 i
s n
ot
a p
erf
ect
squ
are
, u
se a
dif
fere
nt
fact
ori
ng p
att
ern
.
16x
2 -
32x +
15
O
rigin
al tr
inom
ial
=
16x
2 +
mx +
px +
15
Write
the p
attern
.
=
16x
2 -
12x -
20x +
15
m =
-12 a
nd p
= -
20
=
(1
6x
2 -
12x)
- (
20x -
15)
Gro
up t
erm
s.
=
4x(4x -
3)
- 5
(4x -
3)
Fin
d t
he G
CF
.
=
(4x -
5)(
4x -
3)
Facto
r by g
roupin
g.
Th
ere
fore
16x
2 -
32x +
15
= (
4x -
5)(
4x -
3).
Exerc
ises
Dete
rm
ine w
heth
er e
ach
trin
om
ial
is a
perfe
ct
sq
ua
re t
rin
om
ial.
Writ
e yes o
r no
. If
so
, fa
cto
r i
t.
1. x
2 -
16x +
64
2. m
2 +
10m
+ 2
5
3. p
2 +
8p
+ 6
4
y
es;
(x -
8)(
x -
8)
yes;
(m +
5)(
m +
5)
no
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
, w
rit
e
prim
e.
4. 98x
2 -
20
0y
2
5. x
2 +
22x +
121
6. 81 +
18j
+ j
2
2
(7x +
10y)(
7x -
10
y)
(x +
11)2
(
9 +
j)2
7. 25c2
- 1
0c
- 1
8. 169 -
26r
+ r
2
9. 7x
2 -
9x +
2
p
rim
e
(13 -
r)2
(
7x -
2)(
x -
1)
10. 16m
2 +
48m
+ 3
6
11. 16 -
25a
2
12. b
2 -
16b
+ 2
56
4
(2m
+ 3
)2
(4 +
5a
)(4 -
5a
) p
rim
e
13. 36x
2 -
12x +
1
14. 16a
2 -
40ab
+ 2
5b
2
15. 8m
3 -
64m
(
6x -
1)2
(
4a
- 5
b)2
8
m(m
2 -
8)
8-6
Exam
ple
1Exam
ple
2
Perf
ect
Sq
uare
Tri
no
mia
la t
rinom
ial of
the f
orm
a2 +
2ab
+ b
2 o
r a
2 -
2ab
+ b
2
Sq
uari
ng
a B
ino
mia
lF
acto
rin
g a
Perf
ect
Sq
uare
Tri
no
mia
l
(a +
4)2
= a
2 +
2(a
)(4)
+ 4
2
=
a2 +
8a +
16
a2 +
8a +
16
= a
2 +
2(a
)(4)
+ 4
2
=
(a +
4)2
(2x -
3)2
= (
2x)2
-2(2
x)(
3)
+ 3
2
=
4x
2 -
12x +
9
4x
2 -
12x +
9 =
(2x)2
-2(2
x)(
3)
+ 3
2
=
(2x -
3)2
Answers (Lesson 8-5 and Lesson 8-6)
Co
pyrig
ht ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
pan
ies, In
c.
Chapter 8 A18 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
38
Gle
ncoe A
lgeb
ra 1
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Qu
ad
rati
c E
qu
ati
on
s:
Perf
ect
Sq
uare
s
So
lve E
qu
ati
on
s w
ith
Perf
ect
Sq
uare
s F
act
ori
ng a
nd
th
e Z
ero
Pro
du
ct P
rop
ert
y
can
be u
sed
to s
olv
e e
qu
ati
on
s th
at
involv
e r
ep
eate
d f
act
ors
. T
he r
ep
eate
d f
act
or
giv
es
just
on
e s
olu
tion
to t
he e
qu
ati
on
. Y
ou
may a
lso b
e a
ble
to u
se t
he s
qu
are r
oo
t p
ro
perty
belo
w
to s
olv
e c
ert
ain
equ
ati
on
s.
S
olv
e e
ach
eq
ua
tio
n.
Ch
eck
yo
ur s
olu
tio
ns.
a. x
2 -
6x +
9 =
0
x
2 -
6x +
9 =
0
Origin
al equation
x
2 -
2(3
x) +
32 =
0
Recogniz
e a
perf
ect
square
trinom
ial.
(x
- 3
)(x -
3) =
0
Facto
r th
e p
erf
ect
square
trinom
ial.
x -
3 =
0
Set
repeate
d f
acto
r equal to
0.
x =
3
Solv
e.
Th
e s
olu
tion
set
is {
3}.
Sin
ce 3
2 -
6(3
) +
9 =
0,
the s
olu
tion
ch
eck
s.
b.
(a -
5)2
= 6
4
(a
- 5
)2 =
64
O
rigin
al equation
a
- 5
= ± √ ##
64
Square
Root
Pro
pert
y
a
- 5
= ±
8
64 =
8 $
8
a
= 5
± 8
A
dd 5
to e
ach s
ide.
a
= 5
+ 8
or
a =
5 -
8
Separa
te into
2 e
quations.
a
= 1
3
a =
-3
Solv
e e
ach e
quation.
Th
e s
olu
tion
set
is {-
3,
13}.
Sin
ce (-
3 -
5)2
= 6
4 a
nd
(13 -
5)2
= 6
4,
the s
olu
tion
s ch
eck
.
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
1. x
2 +
4x +
4 =
0 {-
2}
2. 16n
2 +
16
n +
4 =
0 {- 1
−
2 }
3. 25d
2 -
10
d +
1 =
0 { 1
−
5 }
4. x
2 +
10x +
25
= 0
{-
5}
5. 9
x2 -
6x +
1 =
0 { 1
−
3 }
6. x
2 +
x +
1 −
4 =
0 {- 1
−
2 }
7. 25k
2 +
20
k +
4 =
0 {- 2
−
5 }
8. p
2 +
2p
+ 1
= 4
9
9. x
2 +
4x +
4 =
64
{-
8,
6}
{-
10,
6}
10. x
2 -
6x +
9 =
25
{-
2,
8}
11. a
2 +
8a
+ 1
6 =
1
12. 16
y2 +
8y +
1 =
0 {- 1
−
4 }
{-
3, -
5}
13. (x
+ 3
)2 =
49 {-
10,
4}
14. (y
+ 6
)2 =
1 {-
7, -
5}
15. (m
- 7
)2 =
49
{0
, 14}
16. (2
x +
1)2
= 1
{-
1,
0}
17. (4
x +
3)2
= 2
5 {-
2,
1 −
2 }
18. (3
h -
2)2
= 4
{ 4
−
3 ,
0}
19. (x
+ 1
)2 =
7
20. (y
- 3
)2 =
6
21. (m
- 2
)2 =
5
{-
1 ±
√ $
7 }
{3 ±
√ %
6 }
{2 ±
√ %
5 }
8-6
Exerc
ises
Sq
uare
Ro
ot
Pro
pert
yF
or
any n
um
ber
n >
0,
if x
2 =
n,
then x
= ± √ #
n .
Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-6
Ch
ap
ter
8
39
Gle
ncoe A
lgeb
ra 1
Sk
ills
Pra
ctic
e
Qu
ad
rati
c E
qu
ati
on
s:
Perf
ect
Sq
uare
s
Dete
rm
ine w
heth
er e
ach
trin
om
ial
is a
perfe
ct
sq
ua
re t
rin
om
ial.
Writ
e yes o
r no.
If s
o,
facto
r i
t.
1. m
2 -
6m
+ 9
2. r2
+ 4
r +
4
y
es;
(m -
3)2
y
es;
(r +
2)2
3. g
2 -
14g +
49
4. 2w
2 -
4w
+ 9
y
es;
(g -
7)2
n
o
5. 4
d2 -
4d
+ 1
6. 9
n2 +
30n
+ 2
5
y
es;
(2d
- 1
)2
yes;
(3n
+ 5
)2
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
, w
rit
e prim
e.
7. 2
x2 -
72
8. 6
b2 +
11b
+ 3
2
(x +
6)(
x -
6)
(2
b +
3)(
3b
+ 1
)
9. 36t2
- 2
4t +
4
10. 4
h2 -
56
4
(3t -
1)2
4
(h2 -
14)
11. 17a
2 -
24
ab
12. q
2 -
14
q +
36
a
(17a -
24
b)
pri
me
13. y
2 +
24y +
14
4
14. 6
d2 -
96
(
y +
12)2
6
(d -
4)(
d +
4)
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
15. x
2 -
18x +
81 =
0 {9}
16. 4
p2 +
4p
+ 1
= 0
{- 1
−
2 }
17. 9g
2 -
12g +
4 =
0 { 2
−
3 }
18. y
2 -
16y +
64 =
81 {-
1,
17}
19. 4n
2 -
17 =
19 {±
3}
20. x
2 +
30x +
150 =
-75 {-
15}
21. (k
+ 2
)2 =
16 {-
6,
2}
22. (m
- 4
)2 =
7 { 2
± √ %
7 }
8-6
Answers (Lesson 8-6)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he M
cG
raw
-Hill C
om
pan
ies,
Inc.
Chapter 8 A19 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
8
40
Gle
ncoe A
lgeb
ra 1
Practi
ce
Qu
ad
rati
c E
qu
ati
on
s:
Perf
ect
Sq
uare
s
Dete
rm
ine w
heth
er e
ach
trin
om
ial
is a
perfe
ct
sq
ua
re t
rin
om
ial.
Writ
e yes o
r no.
If s
o,
facto
r i
t.
1. m
2 +
16m
+ 6
4
2. 9r2
- 6
r +
1
3. 4
y2 -
20y +
25
y
es;
(m +
8)2
y
es;
(3r -
1)2
y
es;
(2y -
5)2
4. 16p
2 +
24
p +
9
5. 25b
2 -
4b
+ 1
6
6. 49
k2 -
56k
+ 1
6
y
es;
(4p
+ 3
)2
no
y
es;
(7k
- 4
)2
Fa
cto
r e
ach
po
lyn
om
ial,
if
po
ssib
le.
If t
he p
oly
no
mia
l ca
nn
ot
be f
acto
red
, w
rit
e
prim
e.
7. 3
p2 -
147
8. 6x
2 +
11x -
35
9. 50
q2 -
60q
+ 1
8
3
(p +
7)(
p -
7)
(2
x +
7)(
3x -
5)
2(5
q -
3)2
10. 6
t3 -
14
t2 -
12
t 11. 6d
2 -
18
12. 30
k2 +
38k
+ 1
2
2
t(3t +
2)(
t -
3)
6(d
2 -
3)
2(5
k +
3)(
3k
+ 2
)
13. 15b
2 -
24
bf
14. 12h
2 -
60
h +
75
15. 9
n2 -
30
n -
25
3
b(5
b -
8f)
3
(2h
- 5
)2
pri
me
16. 7
u2 -
28m
2
17. w
4 -
8w
2 -
9
18. 16a
2 +
72a
d +
81d
2
7
(u -
2m
)(u
+ 2
m)
(w
2 +
1)(
w +
3)(
w -
3)
(4
a +
9d
)2
So
lve e
ach
eq
ua
tio
n.
Ch
eck
th
e s
olu
tio
ns.
19. 4
k2 -
28k
= -
49
20. 50b
2 +
20
b +
2 =
0
21. ( 1
−
2 t
- 1
) 2
= 0
{ 7
−
2 }
{- 1
−
5 }
{2}
22. g
2 +
2
−
3 g
+ 1
−
9 =
0
23. p
2 -
6
−
5 p
+
9
−
25 =
0
24. x
2 +
12
x +
36 =
25
{-
1 −
3 }
{ 3 −
5 }
{-
11, -
1}
25. y
2 -
8y +
16 =
64
26. (h
+ 9
)2 =
3
27. w
2 -
6w
+ 9
= 1
3
{-
4,
12}
{-
9 ±
√ $
3 }
{3 ±
√ $$
13 }
28. G
EO
METR
Y T
he a
rea o
f a c
ircl
e i
s giv
en
by t
he f
orm
ula
A =
πr2
, w
here
r i
s th
e r
ad
ius.
If
in
creasi
ng t
he r
ad
ius
of
a c
ircl
e b
y 1
in
ch g
ives
the r
esu
ltin
g c
ircl
e a
n a
rea o
f 100
π
squ
are
in
ches,
wh
at
is t
he r
ad
ius
of
the o
rigin
al
circ
le?
9 i
n.
29. PIC
TU
RE F
RA
MIN
G M
ikaela
pla
ced
a f
ram
e a
rou
nd
a p
rin
t th
at
m
easu
res
10 i
nch
es
by 1
0 i
nch
es.
Th
e a
rea o
f ju
st t
he f
ram
e i
tself
is
69 s
qu
are
in
ches.
Wh
at
is t
he w
idth
of
the f
ram
e?
1.5
in
.1
0
10
x
x
8-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 8-6
Ch
ap
ter
8
41
Gle
ncoe A
lgeb
ra 1
Wo
rd
Pro
ble
m P
racti
ce
Qu
ad
rati
c E
qu
ati
on
s:
Perf
ect
Sq
uare
s
1. C
ON
STR
UC
TIO
N T
he a
rea o
f L
ibert
y
Tow
nsh
ip’s
squ
are
pla
ygro
un
d i
s re
pre
sen
ted
by t
he t
rin
om
ial
x2
- 1
0x
+ 2
5.
Wri
te a
n e
xp
ress
ion
u
sin
g t
he v
ari
able
x t
hat
rep
rese
nts
th
e
peri
mete
r. 4
x-
20 o
r 4(x-
5)
2. A
MU
SEM
EN
T P
AR
KS
F
un
tow
n
Dow
nto
wn
wan
ts t
o b
uil
d a
vert
ical
moti
on
rid
e w
here
th
e p
ass
en
gers
are
la
un
ched
str
aig
ht
up
ward
fro
m g
rou
nd
le
vel
wit
h a
n i
nit
ial
velo
city
of
96 f
eet
per
seco
nd
. T
he r
ide c
ar’
s h
eig
ht
h i
n f
eet
aft
er
t se
con
ds
is h
= 9
6t
- 1
6t2
. H
ow
m
an
y s
eco
nd
s aft
er
lau
nch
wou
ld t
he c
ar
reach
144 f
eet?
3 s
eco
nd
s
3. B
USIN
ESS
S
ain
i S
pri
nk
ler
Com
pan
y
inst
all
s ir
rigati
on
syst
em
s. T
o t
rack
m
on
thly
cost
s C
an
d r
even
ues
R,
they
use
th
e f
oll
ow
ing f
un
ctio
ns,
wh
ere
x i
s th
e n
um
ber
of
syst
em
s th
ey i
nst
all
.
R
(x)
= 8
x2 +
12x
+ 4
C(x
) =
7x
2 +
20x
- 1
2
Th
e m
on
thly
pro
fit
can
be f
ou
nd
by
subtr
act
ing c
ost
fro
m r
even
ue.
P(x
) =
R(x
) -
C(x
)
Fin
d a
fu
nct
ion
to p
roje
ct m
on
thly
pro
fit
an
d u
se i
t to
fin
d t
he b
reak
-even
poin
t w
here
th
e p
rofi
t is
zero
.
P(x
) =
x2-
8x+
16;
x=
4
4. G
EO
METR
YH
oll
y c
an
mak
e a
n o
pen
-to
pp
ed
box o
ut
of
a s
qu
are
pie
ce o
f ca
rdboard
by c
utt
ing 3
-in
ch s
qu
are
s fr
om
th
e c
orn
ers
an
d f
old
ing u
p t
he s
ides
to
meet.
Th
e v
olu
me o
f th
e r
esu
ltin
g b
ox i
s V
=3
x2
-36x
+108,
wh
ere
x i
s th
e
ori
gin
al
len
gth
an
d w
idth
of
the
card
board
.
a.
Fact
or
the p
oly
nom
ial
exp
ress
ion
fro
m
the v
olu
me e
qu
ati
on
. 3(x-
6)(
x-
6)
b
. W
hat
is t
he v
olu
me o
f th
e b
ox i
f th
e
ori
gin
al
len
gth
of
each
sid
e o
f th
e
card
board
was
14 i
nch
es?
192 i
n3
c.
Wh
at
is t
he o
rigin
al
sid
e l
en
gth
of
the
card
board
wh
en
th
e v
olu
me o
f th
e b
ox
is 2
7 i
n3?
9 i
n.
x
x
3 in
3 in
8-6
Answers (Lesson 8-6)
ERROR: undefined
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