Key to
&ebra
Multiplying snd Dividing Rationsl Expressions
By Iulie King and PeterRasmussen
TABLEOF CONTENTS
Review
Rational
Expressions
...........
Multiplying
Fractions
Equivalent
Fractions
Rewriting
Fractions
in HigherTerms
S i m p l i f y i nFgr a c t i o n s . . . . . . . . . . . . .
D i v i d i nPgo l y n o m i a l s . . . . . . . . . . . . .
Rewriting
Fractions
in Simplest
Form
Simplifying
Multiptication
Problems........
Reciprocals
..........,.
Dividing
Fractions.
Written
Work
Practice
Test........
.............1
............2
........6
g
........
..............11
.........14
. . . . . . .2. 2
............24
.......26
.......gO
.........
91
.................
35
.........
96
Youngand GiltedI
Numerous
mathematicians
triedtofindmethodsforsolvingalltypesoffifth
degreeequations
from 1545until1820,including
Ren6Descartes,
Sir
lsaacNewton,LeonardEulerandJ, L. Lagrange.
Allfailed.Twobrilliant
teenagerssucceededin the 1820's.Unfortunately
neitheronelivedpast
25 yearsof age. We willtell you aboutNielsHenrikAbel.
Abel (pronounced
AH-buili1902-1829)
was bornand raisednear
Oslo,the capitalof Norway.Hisfamilywasverypoor. Nielsearnedthe
reputation
in highschoolas thebestmathematics
studenteverproduced
by Norway.However,he hadto dropoutol schoolat age1gto helpraise
hisfamilyafterthe deathof hisfather.
Eventhesemisfortunes
didn'tpreventNielsfromworkingon mathematicsineverysparemoment.Hewasdetermined
to finda solutionto the
generalfifthdegreeequation.At onepointhethoughthehaddiscovered
a formula,andhe submittedit for publication,
butbeforeit waspublished
he foundan errorin his solution.
Nielsthentookan entirelydiflerentapproach.Hethought,,,perhaps
no solulionexists."Althoughstill in his teens,that is exacflywhat he
proceededto prove.
This was a revolutionary
result. lmagine,it is impossible
to ever
constructa formulato solveeveryequationof degreefiveor higher.
Thegovemment
of Norway,at theurgingof Abel,shighschoolmaffr
teacher,BerntHolmboe,paidfor Nielsto travelto thelearnedacademies
in Europeto presenthisworkto theleadingmalhematicians
of theworld.
Theaimwasto establishhisreputation
to helphimobtaina professorship
at a leadinguniversity.
Unfortunately
Abel'sproofwasso novelthatnotoneof theleaders
in thefieldwasableto understand
it at first. Duringhismanytravelsfrom
countryto countryNielscaughtpneumonia.He returnedto Norway
withoutthedesiredteachingpositionanddiedattheageof 26. Tragically,
a letterwithan oflerfor sucha distinguished
professorship
in Germany
arriveda few daysafterhisdeath.
On the coverof this bookyoungNielstries in vain to explainhis
mathematical
ideasto a professorat a Frenchuniversity.
Historicalnoteby DavidZitarelli
lllushationby Jay Flom
fMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a dupticating
master. No part of this book may be reproducedin any form without the prior written permissionof the
publisher. Gopyrightinfringementis a violationof FederalLaw.
Copyright@1990by KeyCuniculumProject,Inc.All rightsreserved.
@Key to Fractions,Key to Decimals,Key to Percents,Key to Atgebra,Key to Geometry,Key to Measurement,and
press.
Key to Metic Measurementare
registered
trademarks
of KeyCurriculum
Published
by KeyCurriculum
Press,115065thStreet,Emeryville,
CA 9460g
Printedin the UnitedStatesof America
23 22 21
08 07 06 05
lsBN 1-55953-006-5
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Review
Remember
thata rationalnumbercan alwaysbe writtenas a fractionwithintegersas the
numerator
anddenominator.
In thisbookwe willstudyalgebraic
expressions
thatcan
standfor rationalnumbers.Youwill haveto usewhatyou knowaboutpolynomials
as wetl
as whatyou knowaboutrationalnumbers.
Writeeachrationalnumberas a fraction.Thengraphit on the numberline.
- +=
-6
16=
-5
-4
-1.9=
-2i=
3.H=
O=
-3
Factor36 fivedifferentways.
36
36
/\
/\
36
36
A,
3b
A
/\
Writean equivalent
expression
usingexponents.
5 ' x x x r (=
7y'7y'7y=
3 n ' 5 n=
p ' $ P '2 P=
rr'2t.2t-2t=
(rxXyyy)=
Multiply.
x(x*4)=
5 h ( h* 3 ) =
( x * 2 ) ( r* 6 \ =
c8'c=
02' o'5=
. 6y,=
2,1*
6 ( e - 5=)
(2x+3)(3x-l)=
Factor.
3x-12=
8Y-12=
Xt+6X*9 =
a ' - 8 1=
3nr - n - tt =
t'-llt+10=
Solveeachequation.
+=30
# = 15
@1990by Key Curric1rlumProject, Inc.
Do not duplicato without permission
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+ =12
RationalExpressions
Remember
thata termis an algebraic
expression
in whichmuttiplication
is the
onlyoperation.
A polynomial
canbea singletermor canbe madebyaddingand
subtracting
terms.
P o l y n o m i axl sz :+ l
l?
a'-4a +5
3xt
-5x * 2y
A rationalnumberis a fractionwithintegersfor the numerator
anddenominator.
A rationalexpressionis a fractionwilhpolynomiats
for the numeratorand denominator.
jnleger
Rational
Number:
' Integer
Rationat
Expression:
#ffii:i
Of course,the integeror polynomial
on the bottomcannotbe 0 (sincewe cannotdivideby
0). Hereare someexamplesof rationalexpressions:
7
x- 5
x
x + 5
y"-1
O
x'-4x +r{
n+6
x+6
Makeas manyrational
expressions
as youcanby usingoneofthesepotynomiats
forthe
numerator
andoneforthedenominator.
xa
to
x-5
2
x'+3x*2
@19S by Ksy Curiculum Proiaql,Inc.
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Findthe valueof eachrationalexpression
when x = 3. lf the answeris not an integer,leave
it as a fraction.
x + + = 3 + += Z
x+7
3+7
x-5
lO
-
x
x+z
=
*-l=
x +|
x+3
x-8 =
xz*l
=
x + l
x t - 5 x +=l O
?(-5
-
x'* 2x+ l
xt+4x+4
-2^t
x'-4x+4
4x-l
- 5
=
Findthevalueof eachrationalexpression
when x = -2 and y = 5.
3^=
3y
- =
T
+x
a*2 =
Y*2
l =
ir=
xY
xzy'
+ =
xy
x*#
z
5
3
O19$ by KeyCurdcllumProloct,Inc.
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Youmighthavenoticed
thateveryansweron page3 wasa rational
number.
Oncein a whilewedo nofgeta rational
number
forananswer
whenwe replace
variables
withnumbers
in a rational
expression.
Lookatthesubstitution
tablebelow.
o+5
=
6
6
=
-5
r
=
l+5
6
O+5
6
1
6
=
r
|
6
5
-5+5
Doyouseewhathappened
when a = -5? Whena = -5, +
hasno valuebecauseit
equals6 + 0, whichhas no answer.we saythat ;fo. is*uhfefinedwhen a = -s.
Youfinishthesesubstitution
tables.Be on the lookoutforsubstitutions
whichgiveno
answer.Write"undefined"
youfindone.
whenever
n + l
n+$
019$ by Key CurriculumProiocl,Inc.
Do not duolicals without o€rmissior
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alwaysstandsfor a number,we haveto set the
To be surethata rationalexpression
cannotequalzerc.
conditionthatthe denominator
#o
ooo
i,'r
a
a+2
N ' + 5 + * 6 .X r - g + O
' a+Z+O
^t-?
'
Youwritethecondition
foreachfraction.
a
To r
5
T'
2x
x+v
x-y
5X- 2
Fromnowon we shall assumethat
of eachfractionis not zero.
the denominator
That'sso we won'thaveto writedownthe conditioneachtime.
of 1.
You alreadyknowthateveryintegercan be writtenas a fractionwitha denominator
of 1. Thismeansthat
Everypolynomial
can alsobe writtenas a fractionwitha denominator
everypolynomial
is a rationalexpression.
X+3=
x+3
l
a ' - 6= #
y ' *3 y - 5
l
Writeeachpolynomial
as a fraction.
5
x =
xz + 4^ =
a2
2=
l+-Y'=
3a+ =
x3+xt=
n z+ l n + l ? =
5 xI v=
5c'+lO=
X l+ 7 x * l O =
5
01gfn by KeyCurrlorlum
Prol€ct,lnc.
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MultiplyingFractions
Multiplying
fractions
is easy,whethertheyare numbersor expressions.
Youjustmultiply
the numerators
and multiplythe denominators.
Multiply.
+
5 1 = 3- 5
3 2
6
5=
- . -
i l 3
5 ' z
-6
1=
- . _
1
8
- 9. - =3
r o z
-t
:1 .-9 =
z . + . 2=
3 3 5
5 Z +
x + =4 - *
3y 3y ef,
-a. - =a
-3. ; d
=
4
a
f O x_2= x
1
3
3x'
6'a=
5
b
5
7
b
x?
q'4"=
y
'ry
3r =
5x3 7x2
3 . -3=
b
q
2 x 2 x -2=x
3y 3y 3y
Multiplying
a fractionby an integeror a polynomial
is easy,too. Writethe integeror the
polynomial
as a fractionby puttinga 1 on the bottom.Thengo aheadand multiply
numerators
and denominators.
3 . 2 =I
-x-3
x'l! =
6
. X
e r - =
T s
\ ) - =
5
q
v
2
3x"'t =
y"
5 a . 5at =
3b
6.-7=
5
6 =
*
x ' Jq =
g.x=
x3' ?( t =
2 a .3 c =
Y
5
7b
Somemultiplication
problems
lookhardbutareeasyif youchangeeachnumberto a
fraction.
=8.
2E. 5 +
7
2
5+'H=
(3.5aX+b)
llr = f T
I
33 ' 2 +=
l+=
Qtr)(.oe*
)
-r5 '
11
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Multiply.
3
x-2 __
?(
x+5
-x =-
! , -
3 x- 6
x
ra+5a
x-8
-x =+
5
6
4
2x
2 a +3
z
Y x-Y
3o+tl
3
5
(a"5)
a
a-2
x
?(+3
?
(+v
4 =
( x - + )=
=
r + 3 . ? (+ +
x
?(
I +5
^.'.'''..'-=
-b
?(+t
x-6
a-3
a-7
1 1 +6 )
,+
.
(y-5)=
7
@1990by Koy Curiculum Poitrl, Inc.
Do not duplicate wilhout p€rmisSbrr.
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Herearesomerationalexpressions
whichhaveexponents.Writeeachoneoutthe long
way. Thenmultiply.
zs .sz =2 :5l - (+)'=
(?)'=
(+)'=
Gr=
(t)'=
(t)r=
(i)-=
(!")'=
/x-5\t -
\ml
x- - -- .5 x- =- 5
x+5
t, +6
x'- lox +25
-
x'+ l2l * 36
l x * 2l \ 2 =
1
\x*T/
It,- 3\2=
\r-l
/5
)t=
\x*5/
8
@19$ by Kay CurriculumProjecl,Inc.
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EquivalentFractions
(except$1is equalto1.
anddenominator
Everyfractionthathasthe samenumerator
goesintothe numerator
onetime. Eachfractionbelow
That'sbecausethe denominator
is equalto 1.
2
2
3
3
4
4
8
E
t
l
O
O
O
O
-
-
l
l
-50
-50
Z -5
2 -5
Whenwe multiply
a numberby 1, we alwaysendup witha numberequalto the numberwe
startedwith. Forexample:
n
5 JL=5
%&=-6
3qz
3qz
[=
Thesamethinghappenswhenwe multiply
a fractionby anotherfractionthat'sequalto 1.
We endup witha fractionthat'sequivalentto the fractionwe startedwith.
Herearesomeexamples:
n
2 J3\
4
t -/ f
,
t
n
6
r
t
\ "'?i:Iil::'l
L
2 J+h
E
r
f
\ "':;:1il72'/
to t.
equivalent
to | . Findsomemorefractions
f , $ anof areallequivalent
t .
J
2
2
_t-3_=
2
g
6 _
6
l- .-32 - 1
| . 7=
z 7
I
.ry=
2 50
t
Z
8=
8
l
- . - =
2
(Youpickthenumber.)
Findsomefractions
to |.
equivalent
3 . 2_
q
z
9._=
+
3
3
4 3
=
3 . -+ =
3 - 5
-
- . - =
4 L f
4 - 5
3.-=
+
3
1.-=.
4
. -
=
4
(You pick numbersfor these.)
9
@1990by Koy CufriculumProiscl,Inc
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Labelthe pointsshownon eachnumberlinebelow.
I
,
' l
, ,
iano;
l,land]
the fifths
thesixths
the eighths
thetenths
thetwelfths
Listallthe fractionswhosegraphsaredirectlybelowthe graphof
,,
Whatis trueaboutall of thesefractions?
Listallthe fractionsshownabovethatareequivalent
to eachfractionbetow.
I
3
3
q
2
3
I
5
I
2
-:
+
2
+
t+
-:
5
I
6
2.
6
5
g,
3
5
6
10
@199 by Ksy CurriculumProiect,Inc.
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'udser!/d
moqln elPrldnp pu oo
'cul 'lr€lold urnlmrrnc loy Iq
ot6i6!o
It
'1e;urou'{;od
puE
ro requnuouPSoq}^q Jo}euluJouop
'stutolteq6tqut uotlcpllp altJMaJ
oI
ot{l{1dr11nu
Jolelotunu
q
=
q- |
=
q
E
l/l
'u€
=
D
h+D
zv
= *k ez
:uzAq
= g
iqr- Aq
e
z
=
I
=
f-
= I2
=
E
h-
A
t-
+t
__s'€
g.x
_ s L,
s,Z-
=
E
b
I
e
= T
L
=
of
b
9.9
97
g'9
o€
:e Aq
:r^q
:e-^q
:zAq
.^r
JOlPUil,llOUOp
puP
rolPJaunu
I1d11nyn1
'uollceJltlcpo Jol
'sullal req6;qu! uollcpJl
lualp^lnbaue pull
'sural ratlblrlursrpue ol luapnrnbe
sgqclq^'# qil^ dn pepuaa14
I
&.=e ' 8= I
tz
L
E.L
:e)p!d
'ql$ pauels
sotuocJaAsue aql os 'l ^q
aqt of
o/n
uolpell
no
luelenlnba
lo Joqunu oql ^q uo!}ceJl
uollcplleqt 6ul{ldl1nulo} slunoulpleql 'palcgdo^ uorssardxa
pue O lou s! tlclqn uotssaldxaJo Jaqulnu
pue lolpJournuaql {gdr11nu
orll lo Jolputulouop
'srulolraq61qu! uo!]cpllp otpmoro] {ern {see uE s! alaql'aos uec no{ sy
e >1c;d
1sn[ep1
sural raq6;g u! suollcprJFultlrmau
Findfivefractions
2
equivalent
to i.
Findfivefractions
to $.
equivalent
Findfivefractions
1
equivalent
to 7.
l'z =
4'z
6 . =
2.3=
t'
4.
3'
2.5 =
5.s
l '
2.2
5.2
= .:!
lo
5.s
6 . =
z,to
l '
4.
Findfivefractions
2*
equivalent
to f .
Tr
7.. = 6
2x'z
2x's
3 . s
3'
=
4.
=
=
c.to
3'
6.
3'
=
E
Findfivefractions
equivalent
to i.
Findfivefractions
equivalent
to f .
-5 .2
'X'2
x.2
''o
3
Y.z
X'-b
?( 'to
2x'z<
1.x
y .4,
-!'tr+41
2x'gx
3.3r
?(
(x*3)'z
(fr1.,
(r +3)'r
iEi''
=
=
(r( + 3) .3i
(Tz).t*
(jjO
r"'1r
k-Zl
(r+r+)
_
x.aa+21 =
.6+2l
Y
5'(a-l)
X .(r-t)
Find5 fractions
equivalent
to #
.(a+l)
y.(1+r)
x'(a+91
2x ''t
c'y
=
V
'-b
I
5.a
x.1
5 . X
=
ind5 fractions
to
equivalent
(- r - 6 ) ' z
=
( x- 6 ) ' r
[.a- 1.1
=
(,1+ {) .z
r-6
* 4 .
(x-6) (r-l)
(r+T) (a-t)
0,- cl
(r.3)
(e+ {) (t+31
12
-
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we mustfigureoutwhat
lf wewantanequivalent
fraction
witha certain
denominator,
to multiply
Herearetwoexamples:
thetopandbottombyto getthatdenominator
2
6
=
3
21"
2 ' 8_ 1 5
3.e 21
x
3xto
6 'tr =
l8r
x .tr
3x2
Youtry these. Firstfigureoutwhatthe denominator
hasbeenmultiplied
by.
Thenmultiply
the numerator
by the samenumberor expression.
+ 'ro= +o
5 .to
-3'
50
5.
-t'
=
1.
1'5
y'b
-5
8.
27
=
5r
2t'
6y
3.
2a' =
b.
b
5o.
+'
b_.5)' =
1
' =
2
=
=
35
7'
2.
=
32
a.'
3.
=
lz
n ' =
2m.
lOm
r '
-2r.
4p'
=
(x+l)'
( x +2',,
-
8
50
63
8r2
=
6(x+21
iplyby.
( multiplvthc lop ondbollon [r 3.)
l ' 3
3(r -21
5
(x"4).
I
l O x* t l O
x
(n-3).
*' - 3l<
lZr * 20
2n
Cn-Tt'
nt* n
3
2
(r + 5).
(3x*5)
az +J1 + lQ
(y-+).
Y,-16
13
019gOby KeyCurric1rlum
Prcjoct.Inc.
Oonol duplicate
withoutp.rmi6sbn.
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SimplifyingFractions
and
We knowhowto rewritea fractionin higherterms.Wejustmultiply
the numerator
denominator
by the samenumber.
20
35
+5
7's
,"*iiiT-idrrerms
To simplifya fraction,
or rewriteit in lowerterms,we dotheopposite.Wefactorthe
numerator
anddenominator
of thefraction
sothatoneof thefactorsontopis thesameas
oneof thefactorsonthebottom.Thenwecancelthesamefactorfromthetopandbottom.
westartedwith.
Whatis leftis oursimplified
fraction,
whichwillbeequivalent
to thefraction
Hereis an example:
o
oo
o
o
20
35
=
5 +
5'7
\
lrewritten
l
n
=
151.4
J lb t - l
=
l-l
a
4
1
tnr*#
Simplifyeachfraction.Do eachproblemin two steps.
-
[ 3 is o foclorof -3 ond12.l
H.2
t + .3
-4
=
l4
-12
=
2l
20=
22
tb
3
25
6
t8
-4
- 3 ^w
8.-t
-r5
I
t2
-
=
2
t5
I
t'
1 8 =
2+
-8o
=
30
14
019$ by Koy CurriculumProiecl, Inc.
Oo nol duplicato without parmission.
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the
thetop andbottomandcanceling
by factoring
Simplifyeachrationalexpression
commonfactors.
2 is ttre onlyfoctor
left onthe top. On
thc bottom I hove3,
=3t'. o o o
yyd r. ?.Y.v
2'z'{'{'{ = - z
3'z'{'{'{'y'y 3 t '
6vt
1y'
W
=
r
y
CD'z'x'yf.f
lOx3v'
3 5x y r
7y
ryJ- =
5x'
6x'
3a' =
xz
xY
a5
lOa'
=
5ab
=
arb
4Oa
x'y'
+
x"yz
x*v =
? x " Y=
xy*
loy"
3xt Y'
2 lx y
27 xtv
-1F
=
2Ox"v
=
I
30 xyz
=
36 xy3
-60 a'bz
LtOab
15
Ol99O by Key Cunicllum Proioct, Inc.
Do nol duplicale wilhoul pormissbn.
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youcando in yourhead,thelessyouwillhaveto write
andcanceling
Themorefactoring
a fraction.
outeachtimeyousimplify
o
The ye on top corrcls
two of the y's on the
bottom. 5o / is left
on the bottom.
withoutwritingout
foryouto simplify.Tryto do the canceling
Herearesomemorefractions
allthe factors.
20xv
hi
5x"
?y'
1 7a +
l Oa '
ia'
l5:r'
25x"
=
as
1f,
x+v
loy'
,(y*
3x'y"
2hy
20x"y
30 xy'
27x3y
2 7* v ,
-60 a'b"
l0a'
404
xz =,
xY
4-
36x?y3
-
ab
tb
=
40ab
l.5x'
3x'
36 xy3
xtY
x"y
Didyougetthe lastthreeproblems?lf you'renotsure,lookat the nextpage.
@199 by Kay CurriculumProjscl,Inc.
Do nol duolicalswithoulDsrmission.
16
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youcancancelout allthe factorson the TOP . . .
Sometimes
aV
fb
=
I
x =
5x
q1
3x
xy
1x
x t vI z
2x'
lOx'v
I
youcan cancelout allthe factorson the BOTTOM. . .
Sometimes
5
ait
Ef
/*
= 5!'=5xs
I
ry
x
6x'
=
3x
W=
4x'
5ab
3x'y'
x'y
=
youcancancelout allthe factorson the TOPANDBOTTOM. . .
Andsometimes
=*=l
ry
I
*,{
a'b'
17
0199Oby Key CurliculumProieci,Inc
Do not duDlicatswilhoul o€rmission.
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Simplifytheserationalexpressions.Eachoneis alreadyfactoredfor you.
Youjust haveto cancelandwritethe answer.
2a'
2a*l0
4x*12
4 ( x *3 )
Bx-4y qQx-y)
xt * 3x
\ Ltr3J
3J;( 2x* 3)
3r y - lZx' 3 x (y - 4 x )
6x' * 9x
x+4
= -
l(x*tl)
=
5 x * 1 5 5(-1{-gT
2x* I
3 x - 1 2= -3=(x - 4 )
x ' - 4 x x ( x- 4 )
xY*2Y = y ( x * 2 1
7 x *1 4 7 ( x * 2 )
2(x*tl)
Youwillhaveto factorthe polynomials
in eachrationalexpression
below.
Thencancelfactorsandwriteyouranswer.Showeachstep.
3v
3'
^-6
- - -
- - = - -
3y*6 3(
x2*2x
5x*f0
^
) o
Zt'- lZx Ll-(
)
(
l(
-
)
s ( )
2x-8
3x-tZ
-
2(
)
3(
)
lOx
5x * t{0
x'+ t{{
t{x + f 6
18
O19$ by Key Curriolum Proiocl, Inc.
Do nol duplicato withoul permissron.
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thatis a
Whenwe aresimplifying
rational
we canonlycancelan expression
expressions,
by all
factorof boththe top andbottom.An expression
is a factorof the top if it is multiplied
the restof the top. lt is a factorof the bottomif it is multiplied
by allthe restof the bottom.
Hereis howSandyandTerrydidthe lastproblemon page18.
Terry
Sandy
x" + r{x
L { x+ l 6
^lp-+t
+brq1
=
x2 + Lh(
x
-
-tf"(+
rl
Right!Sandyremembered
to factorfirst.
x + 4 is a factorof bothtop andbottom.
tb
=
{z
16
Wrong!4r is nota factorof eitherthetop
or the bottom.
you cancel.
Simplifyeachfraction.Remember
lo factorbefore
a'*3a
3a*1
3x*5y
6x" l)y
Lfx"B
4x*lZ
6a,* ?a
Zaz+ Zab
4ab
x2+3x
2 x ' +6 x
3a
x
2x'* 8r
3x'+ 9x
xz+3d
19
Ol99O by Key CurriculumProjoct, Inc.
Eb not duplbate without pormission.
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lacloredlor you.
Somearealready
expression.
eachrational
Simplity
I
x' -2x'B
l-r41(x+D = ^*2
(r+ s)I-/-
x'+ x -2O
k-Z)k- +)
(2x+!h-2)
x ' + l O x* 2 5
x2 -25
4#-T
fu-rtfQ+/)
x-5
x z* x - 3 0
xt+41+4
X
x++
x'*2x-2+
x"-16
x.*llx+28
4x'-25
2x"-3x-5
=
x-5
x"*7t*l?
xz*13'7+42
(r + Qk-5)
k-5)(x+2)
)(
x'*8x*16
=
(r +5)(x+ 5)
(rt+5Xr-5)
x ^+ 6 N * 8
x'- 16
?
^-)
X
? (+ +
)(
x ,- 9
X
X
x-3
^-3
^ 2- l 4 x * 4 0
xz- 6x -4O
M
xr-4x-60
3x*l
3x"-l4x- 5
20
@19S by Koy CurriculumProjed, Inc.
Do nol duolicalswilhoutggrmission.
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Simplifyeachfraction.
aLt+A
at +2a
a z+ 3 a + 2
=
-
a
q+f
|pfll(a+l)
x/-llx-12
x'- lZt,
5at * lOo
2x'- ll,x
x'-l?x*21
a' -7a- l8
( x + 2 ) " l*rZlkt,+21
? ( z - + tHTh-2\
j
-
x+2
x-2
( x- 5 ) '
(x*6)'=
x"-36
2 x z- 3 x - 2 0
(2x*5)'
x'-llx+30
3a' - l2q
4x-6
- =
(a- 4)"
Qx-3)"
To completely
simplifyeachfractionbelowyou haveto factormorethanonce.
3t(a + 5)
31.*'5
/{.'g.''25)
{lp,5l(o-5)
3- a3 " -* =l 5 q
a3- 25a
3
c-5
2x' - 2x
2x3* 6x" -8x
x 3- 3 x z - l O r t
7x' - 35*
21
@1990by Ksy Curriqrlum Projecl, Inc.
Do not dupli:ate without permissbn.
www.pdfgrip.com
DividingPolynomials
the problemas a
by anothercanoftenbe doneby rewriting
Dividing
onepolynomial
it. Herearesomeexamples:
andthensimplitying
rational
expression
Divide.
( Z x . * l O= x2)x=
I
? x r + f O=
x t:{ x * 5 J =
T
^+5
+
( 5 a '- l O a )= 5 a =
( r t . * x - 2 0 ) = ( x - 4 =)
( 3 x .* 7 x * 4 ) + ( x * l ) =
(r*4)=
(x.-l6x)+
in the lastproblemhadbeen 3x2+ 7x + 6, we couldn'thavefactoredit.
lf thetrinomial
Thenthismethodof dividingwouldn'thaveworkedandwe wouldhavehadto uselong
of wholenumbers.
is a lotlikelongdivision
of polynomials
division.Longdivision
Firstyoumakesurethatthetermsarearrangedinorde[1*|Em
first.
exponent
withthelargest
bytheirexponents,
rr rrv
rs erilt t-:,
T htreI nyvuy owrvrvs
u d i v tri d
h et-=,ltlt-:'
f i r s t t e r m o f t h\4,e d
i v i s o r ( r ) i n t+o . . , #
7x + 6
,t:'""r":'-revr
{+ | ) 3x'
(3r2)
write
the
and
the firsttermof the dividend
result(3r) overthe termit matches.
3t
the resultby the wholedivisor, x * t f3i' + 7x + 6
Nextyou multiply
3 x' + 3x
andsubtract.
writethe answerunderthe dividend,
4x
3 x+ 4 * h
Bringdownthe nexttermand repeatthe lasttwo steps zt+t J3xa + 7r + 6
writeit
3x' * 3x
untilyouare finished.lf thereis a remainder,
+6
overthe divisorand addthisto the quotient.
la
4x+4
z
@19S by Ksy CurriculumProject,Inc
Do not duolicalewilhoulpetmission.
22
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Finisheachdivisionproblem.Checkeachanswerwithyourteacherbeforedoingthe next
problem.Besureto askfor helpif youneedit.
2x
,-31ffi
x+2JXr*9x+18
xt+2^
7r
2r/ - 6l
ffiool?t
( subtraclo polynonial,
J
( wcaddits.oppositeJ
:-a--+*
3lr
x-4)3x,-lOr*17
2x+ 3) 6xt - ?(z- 7x + 12
6 l!+gre
Hereare somedivisionproblemsfor youto do by yourself.
( x "* 7 x + t j ) + ( x + 4 )
( 5 x l+ a + 3 0 ) + ( 5 x + 6 )
( 2 x z+ 3 x + l l ) + ( x - 2 )
( 3 x 3 + 5 l r - 2 x - 4 ) + ( ? (+ l )
23
@199Oby Koy Curiculum Projeci, Inc.
Do not duplicats withoul p€rmissior.
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