Key to
Atgebra Polynomials
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Key to
Atgebra Polynomials
tn,U.r.,jh :,,;i:;i,
i ilit
^(
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By Iulie King and PeterRasmussen
Name
Class
TABLEOF CONTENTS . . . . . . .1. . .-...........2 .....'.3 '........4 .................7 ..............11 ............15 ...........15 ...........30 ......31 ............33 ".'...."'.......35 ..........36
Polynomials............. A d d i n gP o l y n o m i a l s . . . . . . . . . . . . . of Polynomials.......... Opposites ......... Polynomials Subtracting Principle TheDistributive Factoring Outa CommonFactor...... Binomials andTrinomials Monomials, Polynomials Multiplying andFactoring Rule........ TheZeroProduct Equations Quadratic Equations UsingQuadratic W r i t t eW n ork P r a c t i cTee s t . . . . . . . . The Searchfor the Fifth about2000s.c. The weresolvedby the Egyptians Linearequations of the MiddleEastdevelopedmethodsfor solving ancientBabylonians certainquadraticequationsat aboutthesametime. Generalmethodstor Mohammed solvingthemweredescribedby the Arabicmathematician mathematicians ibn-Musaal-Khowarizmi about800 r.o. Renaissance discoveredformulasfor solvingequationsof degreethreeandfour,and theywerediscussedin JeromeCardano'sbooklhe GreatAftin 1545. All of theseequationsaredescribedinthetable.Thetablealsolists of degreefive. equations Equation Linear Quadratic Cubic Quartic Quintic
Degree First Second Third Fourth Fifth
Historicalnoteby DavidZitarelli by lllustration Jay Flom
Example 7x+4=0 x2+4x+4=O 2x3-6x2+7x+1=O 3xa+ x3 -2x2 + x- I = 0 Zxs+ 3xa+ 5x3+ x2+ 4x- 5 = 0
After 1545the quest was to find a formulato solvefifth degree whotook equations.We willtellyouaboutfourfamousmathematicians up thequest. All foundrevealingpropertiesof equations,butnotoneof themwasableto finda formula. day-CART Firstcame Ren6Descartes(pronounced ; 1596'1650), on Method who is bestknownas a philosopher.His bookA Discourse to this I aml An appendix includes thefamousphrase:"l think,therefore greatworkcontainsa sectionon algebra,and one of its resultsis now ruleof signs. knownas Descartes' NextcamethefamousscientistSir lsaacNewton(1642'1727),who describedthe first universallawsof gravitation,createdcalculus,and discoveredthe secretof color. AlthoughNewtonwasn'tableto solve equationsof degree five, he did discoversome propertiesof their solutions. For instance,the sum of the solutionsof the fifth degree equationin the tableis equalto -312andthe productis equalto -C5y2' Leonard Euler (pronouncedoileri 1707:1783)wrote more worksthan anyoneelse,one of whichwas a booktiied mathematical Atgebra.Towardthe endol hislifehe wentblind,yet eventhisdisability writings-helustdictatedhis did notdiminishhisoutputof mathematical
amazingcalculating findingsto otherpeople.AlthoughEulerpossessed abilities,he too wasunableto solveequationsof the fifthdegree. (1736'1813)' wasa of thisteam,J. L' Lagrange Thefourthmember realsurvivor.He managedto surviveinfancy,theonlychildof 1'l in his in spiteof thefact tamilyto do so. Healsosurvivedthe Frenchrevolution thathispatronwasKingLouisXVIanda veryspecialadmirerof hiswas MarieAntoinette,who was executedby guillotine. ln'1770 Lagrange of solutionsof fifthdegree on properties wrotetwoverylongmanuscripts and he vowedto retumto the subiectlater,but althoughhe equations, livedanother43 yearshe nev'eragainwroteaboutalgebra. Allofthesemenattackedthesameproblemat differenttimeperiods withoutsuccoss.Couldtheproblembesolved?Theanswerawaitedthe workof two brilliantteenagers. On the coverof this bookars the portraitsof Newton'Lagrange' Eulerand Descartes.
IMPoRTANTNOTICE:This book is sold as a studentworkbookand asnotto be used as a duplicating master. No part of this book may be reproducedin any form without the prior written permissaonof the publisher. Copyrightinfringementis a violationof FederalLaw. lnc.All rightsreserved' Proiect, @1990by KeyCurriculum Copyright A Xeylo Fractions,Key to Decimats,Key to Percents,Key to Atgebra,Key to Geometry,Key to Measurement'and of KeyCuniculumPress. registeredtrademarks Key to MetricMeasurementare CA 94608 Press,115065thStreet,Emeryville' by KeyCuniculum Published ISBN1'55953-004-9 08 07 06 05 23 22 21 Printedin the UnitedStatesof America
Polynomials In Book2we saidthata termis a verysimplekindof expression is the wheremultiplication onlyoperation. Thisbookis aboutexpressions thataremadeby addingandsubtracting polynomials. terms.Theseexpressions arecalled -UT y. -' - 1 i s a p o l y n o mi n i ayl. | -1 I- JX + L Xz
* 5c
3a * 2b
3X
3ar*
at *2a'*6a
I
is a polynomial inr.
isa potynomiat ina, bandc. isa polynomial inr.
i s a p o t y n oi nmoi .a t
Mostof the polynomials in thisbookarepolynomials in thevariable r. We canlistpolynomials in r is in r according to theirdegree.Thedegreeof a polynomial equalto the highestpowerof r thatis in the polynomial. Fourth Degree
Third Degree
3x+* 5x3- 2x, * 5x - rf
6xt* 5rt + {x-2
6x*- 3r'*7 tt**Xr-X+
|
SecondDegree
First Degree
Zero Degree
x z+ 5 x + 6
3 x+ T
1
xl +x
$x'-9
6x'
3 x 3 - 5 x -| rfx'
3xa
*t-3x
5x x-6 ?(
Tellthe degreeof eachpolynomial.
T x ' - 6 x "+ 4 x - 7
rsa
x.-r+ i s a xt - 5x*2 6xa x +5
I 5xt * 2x'- 4x 01990by KeyCunic1rlum Project,lnc. Do notdupli:atowilhoutp€rmission.
third
in r. degreepolynomial in r. degreepolynomial
isa
in r. degreepolynomial
isa
in r. degreepolynomial
isa
in r. degreepolynomial
isa
in r. degreepolynomial
isa
in r. degreepolynomial
-r5 5 2
Adding Polynomials we canalways in manyways.lf we havetwopolynomials Polynomials aretikenumbers onlyto combine Wejusthaveto remember to getanotherpolynomial. addthemtogether fketerms.Hereis anexample:
( 6 x - 7 )+ ( T x + 5 ) =l O x - 2 for youto add: Hereare somepolynomials
(3x+5)+({x*l)=
( 2 x .+3 x * 7 \ + ( 8 x ' * 3 x - 4 ) = (x" * 5) * (r'+ 4) =
( 5 x - 3 ) + ( 4 r* 7 ) +( 2 x - 6 1 = (x. * 6x- 5) + (n.-8r -t{) = ( 3 a+ { b + c ) + ( 5 a- 4 b + 2 c )= ( 3 x * L l ) + ( 5 x" 2 ) + 2 t = (5x' + 4x -7) + (-5x' - 4x + 7) =
2
+
3x" + 5x +2 txz +3x+2
+
xz+6x+9 X z+ t X * t
5x'-{x-8 +
6xt-9t,+7
3x-9 + |xz + 2x+2
O19S by l(eyCurdorlumProieci,Inc. Oo notduplbatewithoutp€rmission.
Oppositesof Polynomials Thereis anotherwaythatpolynomials hasan arelikenumbers.Everypolynomial opposite.To findtheoppositeof a polynomial wejustfindtheoppositeof eachterm inthepolynomial andthensimplify. 5 x" - 3x + I
Forexample, theopposite of
is
-5x" --3x +-l
whichsimplifies to
-5x" * 3X - 7
Youfinishupthetablebelow. Polynomial
OppositePolynomial
-3x'+6x-2
3x"-8r+2 6a*7b
+{
5x'-2x-9 ?(a- 16 z(5+X{+Xt-12+X-
|
-7x-8 ?(z+5x-llt
- x 1- 5 x + 1 4 Let'ssee whathappenswhenyou add oppositepolynomials together.
(x. * 5x - lel)+ (-x. - 5x + 14)= f7x-8)+(7x+8)= ( 5 r . - 3 x + 7 ) + ( - 5 x .+ 3 x - 7 ) = Ol9$ by KeyCuniculumProl€ct.Inc Do nol dudicelewtthoulpennbslon.
3
SubtractingPolynomials problems when problems to addition Remember howwe usedto changesubtraction of the secondnumber. weworkedwithintegers?Wealwayshadto addthe opposife ol Well,polynomials workthesameway- onlythistimewe haveto addlhe opposite Hereis an example: thesecondpolynomial.
(5x3-3x -7) - (Ex3+6x-n -problem.Therearethreetermsin the Firstwe haveto changethisto an addition secondpolynomial, so we haveto besureto changeeachofthem.
( 5 x 3 - 3 x- 7 ) + ( - 8 x e + % x + 2=) Nowwejusthaveto writetheanswer:
(5xt -3x -7) * (-8xt+-6x+2) = -3et3-9t - 5 Yousubtract thesepolynomials.
( 3 x ' + 5 x- Z ) - ( 7 x ' - 5 x * 4 ) = ( x"+4r- l) = (2y.-, +3)-(3y.-t-T) =
(6x"+2x-n
(y'+y-6)-(y"*5y-6)= (ttx'+ 3x *5) - ( txz-3x * 5) = 3x'+5x- 2 2*-3x+7
az - 6a+ 3
q*-3x+ 2
8x'- 8x- 3 4x+Z
tlx2+ 3x + 5
4
5a' - 2a -8
@19S by Key CurriqJlumProioci,Inc. 0o not duplicalo wilhout p€rmission.
Intheseproblems youhaveto addandsubtract polynomials. Change thesignson polynomials youaresubtracting, butnoton onesyouareadding.
( 6 x "- 4 x + 7 ) + ( 2 x ' - 3 x - 9 ) = (3x"* 5x - l) - ( {x. -2x+ t) = ( 3 x + 5 ) + ( 2 x- 3 ) * ( 4 x - 6 ) = ( q * b - c ) * ( q + b * 2 6 ) - ( q+ b * c ) =
(2x"* x -3) + (x2-2x+ 3) -( lz +x+3) - ( x. -3x - l2) =
( x- y - z ) * ( x - y - z )- ( x - y - z ) + ( ? ( * y * z = ) ( 3 a '+ 2 b * 4 ) - ( a . + f , - l ) - ( G r+1 2 b - l ) - ( a ' + b - 2 ) = Solveeachequation.
8 x - ( 5 x - t +=) 2 5 6x-(tfx-5)=13 6x +(-5r+t) =25 3r +9(r=25-+
lOx-(3x+6)=8
3r =21 x=7 ( 6 x* 9 ) - ( 2 x - 5 ) = 3 8
01990ry KeyCuniorlumProl6cl,lnc. 0o not duplloarewilhoutpermis8bn.
( c l x *l O ) - ( 3 x* 2 ) = 7 4
5
Writea polynomial for the perimeterof eachfigure.
2k-1
3k +lt 3k+tl
2k- |
Angwer.e-ffi 4 m+ 7
5 m+ 7
6
019$ by lcy CurdorlumPrcisct,Inc. Do ncr dupllcstewithoulpomlrsbn.
The DistributivePrinciple Remember theDistributive Principle fromBook1? Distributive Principle : It a, b andc are integers, then a(b+ c) = ab + ac and (b+ch,=ba+m,. - we Thisis the principlewe usewhenwe wantto multiplya singletermtimesa polynomial justmultiply the singletermtimeseachtermin the polynomial. Herearesomeexamples: ,-,
3(x*Ll)=3x+12 /^,
x(x+5)=xz+5x
5 x ( 2 x * 6 ) = l O x z+ 3 O x 3 x ( x e - 5 x * 2 ) = 3 x 3- l 5 x ' +6 x L t x ( 2 * + 3 x . - x + 6 )= 8 x + +1 2 * - 4 x "* 2 4 x To multiply a polynomial timesa singletermweusethesecondpartoftheDistributive Principle.
G
( x* q ) b = 6 x * 2 +
.> (x-5)x=
x z- 5 x
G
( l O x* 3 ) 5 x = 5 O x ' + 1 5 x ( 3 x " + x - 7 ) 2 x = 6xt + 2x" - l4x o1990by KeyCurridrlum Proiect,Inc. Do notduplicate withoutp€rmis8bn.
7
UsetheDistributive Principle problem to do eachmultiplication below.
ft
3Qx- 5)= 6x-15 5(6x- 4) = -
/<=>{
t+(3x-y + 5) = l2x - rly + 20 5 ( 3 x- y + 5 ) =
(3a* 1il2= 6a + 8b
6 ( 3 x- y + 5 ) =
(x*5)lO=
(3x-y+5)7=
frr -Set_ Ll)= -lor +ZO - { ( 3 y+ 5 )=
-t+( 3x. - 6x * Z) =-|,2r,+24r.- g
-3(2o5b)=
-l(3?(z-6x+2)=
(q* x)-8 =
(3x"-6x +2)(-lO)=
e3^
=2*-7x
B y* 5 ) y=
-5(3rt?'6x + 2) =
-
- tt) = + YQVz 3x
r(5-8y)=
t..(Zy"+ 3x - t+)= (7y"+3r - 4) t(y=
ty (3x+{y) =
x ' ( 2 y '+ 3 x - 4 ) =
,rcr
Sx(afx-7) -3a (rta +2) = ttx (xz- 5) = (2x*l)2x'=
5 y ( 3 x +{ y - 8 ) =
Zxy(3x*4y-8)=
B,l -6 X-5y)=
(7xt-5x-6)5x'= - t + b ( a - 3 b * c= )
7 x ( 3 x + 4 vI ) =
7^'Y(3x'y + 2xyt * x3)= 019S by Key CurioJlum Proisct, Inc. Do nol duplbato wilhoul p€rmission.
Solveeachequation.
5 ( x +3 ) = 3 5
3 ( x * 6 )= 3 ?
23x*15)=18
5x*15-€ 35ns
5r = 2 0 x =t+ 8 ( x- n = 3 2
5 ( a + 3 )= 8 c
7x=4(x*6)
t{(x*5)= 3(x-6)
3Qx-5)+{ =3l
l O= 1 8* t t ( 3 x* 7 )
3 ( 3 x + 5 )= 2 G x - 3 )
Ol99Oby KoyCuriculumProjed,Inc. Oo not duplixto withoutpemission.
9
Writea polynomial fortheareaof eachrectangle.
fi = 3x(2x+5) = 6x" + l5x
2 r+ 7
3r- 2
7n+4
10
O'199 by Koy Cufiiculum Projoci, Inc. Do not duplicate wlthoul p€rmi66bn.
FactoringOut a GommonFactor Sometimes we needto breakdowna polynomial intoa product of otherpolynomials. Thisis calledfactoring.lf youcoulddotheprobtems onthelastthreepages, thenyouwon'thaveanytroubte factoring singletermsoutof polynomials. Hereis a polynomial thatwe willfactor:
Bx+20= 4 goesintoboth8 and20. Sofactorout4:
8x+20=4(
)
Nowit'seasyto fill in theparentheses:
8x+20=4Qx+5) Here'showto checkyouranswer:
4 Q x + 5 )= 8 x + 2 0 Herearesomepolynomiats for youto factor.
6 a- 3 0 = 3 ( 6a-30= 6( 6 a- 3 0 = Z ( l8x-27= 3( l8x-27= q( 4x - 32=
)
5x-5y * lOz= 7x-7v-72--
)
lAa - 28b + lOc =
)
I
)
2x" + l8x + ltt =
)
fOx'*15x-ZO= 6a- l8b* lZc=
8y-lO=
9x+lly+lJ=
3Oa-10=
8q + l}b * { =
Didyougetthelastproblem?lf not,seethe nextpage. @1990by Key Cunidjlum ploi€d, Inc. Do nol dupli:ate without perricrbn.
11
on page11: Here'showSandyandTerrydidthelastproblem
Terry
Sandy
8a* l}b+ t{= tl(2a+3b+l)
8q+l|b+4 = 1(2q+3b)
whentheycheckedtheiranswers: Here'swhathappened
Terry
Sandy
o =8a+l2b+t{
4Qa+3b+l)
"X = 8 a + l 2 b lea+3b)
Sandy'sanswerchecked.Terry'sdidn'tbecauseTerryjustfactoredoutthe 4 without leavinganytermin itsplaceto multiplyby. Sandy'sway. Factoreachpolynomial
l5y+$=
l t { n+ 2 l e + 1 =
7a*7= l L n- 3 =
?x-l8y+3= ll -22a +44b=
5*2Ox= 6 x- 2 =
5x"+lox+5= 28x'+28x*7=
Factorthebiggestnumberyoucanoutof eachpolynomial.
71x + 28= lOa+ 25b= 40n-21 = 3 3 p+ 5 5 = l8x - 3Oy= 100+ lOz = 56x' + 42 = 12
Zox * 60y- lOOz= llta - l\b * 6c = 24 * 48x + 4?x' = 5Oa- 20b * 3Oc = 6c'+27c-15= l2r + 36s- 5Ot = l8x - lly -3 = O19$ by Key Cwric,tlum Ptoiecl, hc, Oo nd duglbate withoul petn{8Ebn-
Anynumberthatgoesintoallthetermsof a polynomial is a commonfactorof those terms.Oftenwe needto factoroutthebiggest possible number.Thisis calledthe greatestcommonfactor. Factoring outthe greatest commonfactorcanbe done in stepsif youdon'tfindit onthefirsttry.
1 2 a + 5 6 = 2 ( 2 l a + Z B )o o o = / . 7 ( 3 q + + ) oo o = ltt(3q+q) Factoroutthegreatestcommonfactor.Doit in stepsif youneedto.
l6x - t+8=
Z O O x+ 8 O y l 2 O z =
3Ox +
72a - q6b - t+8c=
7 5 x 1 5=
32x, + t+O1+ 160=
commonfactorscanbevariables as weilas numbers. x z + 8 t = * @ * g @ o o o o -o
=x ( x * 8 )
2 a 2 + a t= 2 @ @ * a @ @ o = ot'(?*a) Herearesomeforyoutotry. x2+3x=
43*5a.+3Cr=
5at + 2a=
2x?*xz-8n=
vt z - 7 ut =
ab * 2b * b" =
lZx - ?(z=
E - r
f , x ' y * x I *- ?l Y
3x3 + 2x" =
X3 - {rt. *X =
?(4- 5x" =
ab+ b"*Zbc=
01900 by KeyCuricutumplol€d. IrE Do not duptlcatewfihoutpemlcebn.
13
haveseveralcommonfactors. thetermsof a polynomial Sometimes
oo 5 x q+ l 5 x 3= @ . @ . " * O . 3 ' @ o
=5x'(x*3) foryouto factor.Alwaysfactoroutthe biggestsingle Herearesomemorepolynomiats termyoucan.
5 x ' + l O x= 5 x (
*6+x5-xr=
8a"-2a=
a?-o5-ot =
4yt * 3y'
5ct * 3a* + [qr =
x^Y'* x'Y=
6^',1-xyt * 2x'Yt=Xy(
x"y' * *I =
xtl **1 *r'rl' =
3x + lZx' =
Crbt+ A.b.*Cb =
6x' - 9x' =
l?a, -9a' - 6a = 3q (
ttx' + 6xy
fOxt + 4xt +6x =
lOxt - 15xa =
8"/ * 8)" -8f, =
l Z a "- l B q b=
ttxt - tlx+ * E?rt=
7xt -7x' =
53r+ + Slxt- 72x' =
Z 5 x + 3 O x y=
60a. * 30ab- 90ac=
14
?('|(
Projscl,Inc. ot9m by KeyCutticulum Oonol dlpllcatewilhoutpetmissbn.
Monomials,Binomialsand Trinomials Fromnowon wearegoingto callsingtetermsmonomiats.polynomials thathavetwo termswillbecalledbinomiats; andpolynomials withthreetermswillbecalledtrinomials. Seeif youcanaddsomemoreexamples to theonesthataregivenbelow. Monomials
Binomials
Trinomials
3x
x+5
x2 +7x + lO
v
7a-1
3x'- ?x + 6
6x" -B
jat
x * Y- z
+ /*z
a + b
2x'++ 8xe-x'
#i" t soms' t We alreadyknowhowto multiplymonomials (singleterms)timesotherpolynomials by usingthe Distributive Principle.Areyoureadyto multiply binomials timesbinomials? Well,herewe go . . . All we haveto remember is to multiplyeachtermin the firstbinomial timeseachtermin thesecondbinomial.Hereis an example:
(x*5)(x+3)= Firstmultiply r timeseachtermin thesecondbinomial. ,'-\*
( x* s x x * 3 ) = x . + 3 x Thenmultiply 5 timeseachtermin thesecond binomial,
(x* 5Xx
= X " + 3 x *5 x * 1 5
Nowsimplifythe answerby addingliketerms.
(x+5)(x+3) = xa +3x + 5 x * 1 5 = X2 *8X prolect,lnc. Ol99Oby KeI Cuniculum Oo notdupli:atewfihoutperiisgion.
+
r5 15
outyourself. foryouto multiply Belowaresomeproblems
(rfrzt v
= xz+2x+3x+6 (x* 5)(rt*6) =x'*5x+6
( x* 5 ) ( x* 2 1 =
( x +l ) ( x * 6 ) =
(y*3Xy* 5) =
( a + 8 X a* 9 ) =
* l0)= (y*'+)(y
( z + 3 ) ( z + 3=)
terms,so be extracareful. belowhavenegative Theproblems
=x'-5x-{x+zo 1"ffi5) \<-/ = x. - ?r*2O
(r_rft)= X l + 3 x - ] 1 - Z l = 13 - 4r( -21
(x-A'r-Lt)=
( x- 5 X x * 3 ) =
( x - 3 ) ( x- 5 ) =
(x*D|.r.-6)=
( x - 6 ) ( x - 6=)
(x+8Xx-5)=
(x-l)(r-8)=
( x - 8 ) ( x * 5 )=
16
0199 by KoYCutriqrlum Proiocl, Inc' Oo not duglbate withoul permission.
Multiply.
( x - 3 ) ( -x2 )
(x*5Xx-8)=
( x - l ) ( x - l )=
(x + r{)(r -4) =
( x- 7 ) k - 2 ) =
(n*lXr-5)=
(x-3)(x- 1) =
(r-3)(r*tl)=
( x * 8 ) ( r- 3 ) =
(x- 5Xx -2) =
( x * 5 ) ( x- 2 ) =
( x - 6 ) ( x + 6=)
( a - 4 ) ( q - 6 )=
( y* 3 ) ( y - 3 ) =
( x - 3 ) ( x" 5 ) =
( x * 9 ) ( x- 2 ) =
Multiply. ( x * 5 ) t = (x +5Xr+5) 2
(x*t)t
( x- 5 ) ' =
X3+5t+5r.25 ?(3+lOx+25
=
(r - 7)t=
(x * l0)' =
(x-l)'=
01990 by Kot Cunlculumproiert, lnc. Oo nd dgplk)atewlttroutpernirsnrr.
17
for the areaof eachrectangle. Writea polynomial x,+4
A = (a,-2)(x+4) s xa+Tx-2x-6
x-2
x-2
= 13+2x-8
x+4
r+7 x-4
x-4 x+7
x+2
x+4
x+4
x+2
r+3
r+3
r+3
r+3
r+6
x-1
x-1
r+6
18
@199 by KeyCuriculumProjscl,Inc Do notduolicalewilhoulpsrmission
To do the nextset of problemsyouwillneeda pencilandan eraser.Youwillneedto makesomeguesses- andif yourguessturnsoutto be wrong,you will needto erase it andguessagain. Eachproblemis a seconddegreetrinomial inr. Yourjob is to factorthetrinomial intoa productof two binomials.Hereis an example:
x'+$a+|2=(
)(
)
Firstfactorthe12as manyoignt waysas youcan.
fl ra' 'r+z\\
t\,
x'+Bx*lZ=(
) (
)
Nowtestout the differentcombinations untilyoufindonethatworks. ( r * 3 ) ( x* { ) X , ' +4 X +3 x+1 2 x'+1x+12
(n*l)(r+12)
( x " 2 1 ( x+ 6 )
X . " l / x + l r t+ l Z :("+l3x+12
X.+6x+2r+12 'x/+8z-+l
As you can see,all threecombinations gaveus the correctfirstandlastterms of thetrinomial, butonlythethirdcombination gaveus 8r forthe middleterm. So hereis the answer:
Yus!
x"+Bx+lZ=(x*n(x+6) Lookat the problems below.Canyoufillin eachpairof blankswiththe rightnumbers? Checkeachanswerto be sureyou are right. o o O
fiGr"oo(-*^ma ( of 20 o,M r/
lz'ro)
Xl+7x+|'2=
(x+3)(a+t)
r(+tfi+3x+12 rl+7x+12 qv X2 + l}x +20 = ( x + _ ) ( x + _ )
tj,
vw
xt+Zlx+20=(x+
)(r* )
?(2+lt{x+24=(X*
) ( x* _ )
,'t'5 \ I z ' t o1
m
x' + 9x + 2 0 = ( x + _ ) ( x * _ )
@1990by Key Curic1llum project, tnc Do not duplicate without p€mission.
f;n leu
r(z+ llx * 21 = (x +_ )(x *
19
in r. Factoreachoneintoa Product Eachof theseproblemsis a seconddegreepolynomial of two binomials.Checkanyansweryou arenotsureof.
? ( 2 + 9 x * 1 8=
(x +3)(r+6)
x'+l3x*36=
c,heck: t 3 + b t t + 3 x + 1 8 ? ( 1+ 9 x + l E
r ( ' + l l x + 1 8=
r,e + l9x + 1 8 - -(
(
X X
)
aa+ }Ox+36=
x2+l?r.+36=
I,L+7x+10=
x 2+ 3 7 x + 3 6 =
x" + llx + l0 =
* t l5n + 36=
, r ( *1{ r + t
rta+l7x+30=
?(" + 5l
=
* H =
X.*2X+ | =
1 ( z+ 3 x + 2
20
=
xz+ llr + 30 =
az+l3x+30=
x't3ll
+30=
o19m by KeyCuticulumPrcioct,Inc. Do nol drplicalewllhoutpermBsion'
Thesepolynomials havenegative youcanmake terms.lf youfactorthemverycarefully, thenumbers andthesignscomeoutrighteverytime. Gn
@i\."nf*_,---tjre) LIls
h)
[';'1il,.@ w
h/hot foctors of
-28 cdd fo'3?
[d
x 2- l O x+ 1 6= ( r - 2 l ( x - 8 )
x"- 3x -28 = (r + r+)(x-7)
chclr : rt - gr - 2x +lb xt - lOut+ 16 a 2 ' - 8 x + 1 =6 ( X )
x.+3x-26=(
xr-l7x+16=(
x2-2x-2*=
)(
)
check: ,(t + {x -7x- 28 xz - 3x -28
1 z - 8 x * 1 5=
'^2 +2* -21 =
A z - l 6 x + 1 5=
re +8x-20=
xr -l7x+72=
-z -A
a'-lb = (x+4)(x-r+)
x2-{=(
- Cl- - 2A 1\t \.r r\
check: 1e- tlx + 4r - l5 xa-16 x" -25 =
xt-l=
xz -81=
?(2'- l0O =
O1990by KoyCunlcalum Proiect,Inc. Do notduplbatgwllhoutp€rmission.
X
I
-
X
)
21
Don'tforgetto checkyouranswers. Factortheseseconddegreepolynomials. rr+l{x*tl5=(
x'-{r-{5
=
l(
a r + f 7 a+ 7 2 =
cr- a -7?=
x' + lEn + t'15 =
or + a -72
r'-l2r-15=
X r - 5 X + 4 =
a 1- l t r + 4 5 =
P"
aa+l|x-tl5=
ri+
xa+4r
-el5 =
a1-l8r +{5
22
=
=
+3P-4 =
4x-lZ=
a3-elX+t
=
Sa - lOs +25=
=
?(1 + rl51 + +5 =
Y''6y-*1
xa -t&fzt - $5=
o.t+2s+l
a8 + t+rla - 45 =
11rs+{m+3 =
?(t - 46r + tl5 =
c a - c - Z
=
=
Projed,Inc. o19{p by KoyCur.ic1Jlum Oonotdupliralowithoulp€rmission.
Hereare somemoreseconddegreepolynomials to factor. b.+t+b-60=
72 -2r - 53 =
A1 - 3a -28
xl - 8X + lZ =
=
m 2- 2 m - 6 3 =
xz-1x
l8 =
m L + m - 3 0 =
cr
c
-2O=
cr'+a-2O
z t + z
=
- l Z =
xl -jl
+ E =
sa+ l3s +42=
xl -ltl1+18=
r(a-9=
a.-36=
f-b+=
x'+ x -17=
xz- | =
x z - 2 t - 1 5 =
? ( r- + 9 =
x 2 - 2 x - 3 =
:ta- f6=
x 2 + 2 * - 3 =
ml' - ll*t =
@1990by Ksy CuriculumProjod, Inc. Do nol duplicats withoul D€rmission.
23
a a binomialtimes youhaveto multiply a monomraltimes fn eachof theseproblems first. together thebinomials binomial.lt'seasiestif youmultiply
3(x- 7)k +2)= 3(x'+2x-7x-lt) 2(x*3)k- 3) =
=3(*'-5x- f+)
=3xt-f5x-42 L { ( x _ 3 X+x5 ) =
a ( x * l ) ( x * 4= )
6 ( x * 5 ) ( x * 4=)
x(x- 5)(x- 4) =
youcan. foryouto factor.Firstfactoroutthebiggestmonomial Belowaresometrinomials timesa binomial. that'sleftintoa binomial Thenfactorthetrinomial
6 x '* l B x - 6 0 = 6 ( N " + 3 x - f o l x ' + E x "- 2 O x =x ( x l + 8 x - 2 O l = x (r +IO)(x-21 = 6 h - Z ) ( x +5 ) 2x'+fOx-28=
x3+]xr*lOx=
5x. r 35x+60 =
a3 - 7a, + 6a =
24
Project,Inc otgS by KeyCutticulum Do notdudlcatswithoutpormigsbn.
Multiply eachpairof binomials. Remember to multiply eachterminthefirstbinomial times eachtermin thesecond binomial.
(2x-3X5x* {) =lOx'+8x-lSx-lZ =fO x r- 7 x - 1 2 ( 3 x - fX t { x + 3 )=
( 3 x- { ) ' = ( 3 x - q X 3 x- t l )
= 9* -lax-l}x + 16 ( 6 X * l ) Q x + 3 )=
=?t/-2tl*+lb
(5x* 2l'= (5x*3X3x+4) =
( 2 x *5 X x- 3 ) = ( { x - l X 3 x- l ) =
Qx-3)" =
( t l x *f ) ' =
(3x- 5)(2x-5) = (3x-7)"= ( a { x - 3 X x + 3=)
(3x + 4)(3x - tf) =
01990 by KeyCuniculumP?oleci,Inc. Do nd dupllc€l€wlthoutpermi$ion.
25
Let'sfactorthistrinomialintothe productof two binomials.You mayneedto do a lot of guessingon this problem,so usea pencilanderaser.
l}x'+ llx + 6 = (
X
Firstfactorthe 12 andthe 6 as manydifferentwaysas you can.
lZx' + 11x
@ + 6
=
(
)(
Nowstarttestingout differentcombinations on somescratchpaperuntilyou findonethat makesthe middletermcomeoutto be 17x.
$x+ZlQx+3) l2xz+l8x+1x*6 l?x" +22x "6
$x*31(2x*Z) l?x"+l2x*6x+6 l?x"+l8x+6
(Ix*A (3x*3) l/x' +l|x* 6x+6 llx" * 18r*6
(atr* 3)( 3x *2) l | x " + 8 X+ 9 x +6 2a,'+l7r +6
(6x"1\(2x*6) 177z+ 36x+2x+ 6 l ? t " + 3 8x + 6
$x*6)(21 + 1;
l}x"*6x*l/ x+6 l|x"+l8x+6
This is it !
So hereis the answer:
lZx' + l]x + 6 = ( { x + 3 ) ( 3 x + 2 ) Nowhereis oneforyouto factor:
6 x ' + l ? x *1 5 = ( 26
X
)
ol9S by K€y CurriculumProject,Inc. Do not duplicalswilhoutp€rmissior
Herearesometrinomials foryoutofactor.Checkyouranswereachtime. 3r'+{r+l= (3r+l)(x+l) c h e c k :3 : 13+3t +a+ | =\t+tr+l
2* * 7r +3 =
7x.+3x+l= (
2*'+ 5x +3 =
X
'
chcck: 5xz+6r*l=
2 * ' - 7 7 .* 3 =
*r"*5X*l =
2r'-5x-3=
lir.+tfi+l=
2t'+5x-3=
Zx'-3x+l=
2t'-x-3=
3x'-2x-l=
*l'-t{y+l=
3r'*21-l=
ta'+34- l=
5x.+5x*l=
9x'-6r+l=
01990by KeyCurriqrlum Projoci,Inc. Do nol duplbatewithoutpermi8sbn.
27
Factoreachtrinomial.U sescratchpaperto checkyouranswers.
9x'+6x+l
6x" + lllx + 4 =
9 x ' * [ 0 x *I
6x'-lOx+4=
9x' lOx+l-
6x"*25x+4 =
4x' - Ilx + 6 =
6x.-Z3x-t{=
It
lbl'
.
- /Xt^t
? = ?
6 x ' - Z x - L +-
6x'*l7r+5
6x' +lOx-t1=
6 x ' * l 3 x* 5
6x"* 5x
6x'-l3x-5
6 x "- l l x + { =
6 x '- l 3 x * 5 =
l b x '- l l x - 5 =
6x'-1x- 5 =
l 6 x ' - 3 8 x -5 =
6xt+7x-5 =
l 6 x " - l 6 x - 5=
8 x . + l L t X* 3 =
f 6 x '+ 7 9 x - 5 =
Bx'-fOrt+3=
l 6 x "- 2 x - 5 =
Bxt * 25x *3 =
l6x'+llx-5=
28
+=
019g) by lGy Cu?hulumP'oiecl,Inc. withoulp€rmi$irn. Do nol duplicatg
Multiply.
Factor.
( 3 x* 2 ) B x - Z \ = 9 x . - 6 x + 6 t - 1
l6x"-9=(
X
= ? X .- 1 ( 5 x + 7 X 5 x - 7 )=
4r' -25 =
Q s - 3 X 2 s+ 3 ) =
3br'- l0O=
(8x- t+XBx+4) =
E l y z- l =
Multiply.
Factor.
(2x* 3yXx-2y)=2x, - T"I * 3ry =2N.-xy-61"
2x,* 5ry*3y'= (
(5x*yX3x*2\\ =
ttr. + l6xy* l5y"=
(ta-3b)(a* b) =
9a'- 3ab- 2b"=
(2x+5yXZx- 5y)=
?x,- l6y"=
@1990 by KsyCurriculum Pfoiocl,lnc. Do nol duplicato withoulpormissbn.
)(
29
The Zero ProductRule numbers.Then Writeallthepairsof integerfactorsyoucanfindforeachofthefollowing writethenumberof pairsyoufound. -50 0 14 19 16 2ondB -2 ond'6 Tond +
-{
ond -tf lb ond l -15 cnd -l
6 pairs
pairs
parrs
pairs
parrs
pairsof factorsfor0, youwouldn'tbefinishedyet. The lf youtriedto writea//possible willdoforonefactoras longastheotherfactoris number Anynumber of pairsis unlimited! Rule: 0. Thisleadsusto theZeroProduct lf a productis 0, thenat leastoneof the factorsmustbe 0. terms: if a. b =0 , Or, in algebraic t h e n a = 0 o rb = 0 . like(r - 3Xr + 2) = 0' All we needto do is find we canusethisruleto solveequations numberswhichmakeeitherx - 3 ot x,+ 2 equalto 0. Therearetwo solutions.Canyou seewhattheyare? Finishthe tableto showthatthe solutionsare3 and-2.
x-3)(x+Zl
Rule. Solveeachequation usingtheZeroProduct
( x- 7 ) ( x- 2 ) = O x - 7 = Oo r x - 2 = O
( x - 6 ) ( r - t )= O
(x+3)(x-3)=O
x(x-t{)=0
3 x ( x - 5 )= O
5x(x+8)=O
(x-A(x-9)--0
( x - 3 ) ( x * 6=) O
(x+f)(x+5)=O
*=7
30
orl=2
Projoci,Inc. Ot9q) by KeyCurriqrlum Oonol duplbalewhhoulpe?mlsslon,
Quadratic Equations Equations like (r - 3Xr + 2) = 0 arecalledquadraticequations.A quadratic equation canalwaysbewrittenwitha seconddegreepolynomial on onesideanda 0 ontheother side.(r- 3)(r+ 2) =0 is equivalent lo 12- x -6 = 0,so it is a quadratic equation. lf we canfactorthepolynomial in a quadratic equation, we cansolveit by usingthe ZeroproductRule.Hereis anexample: Xz _ gx _ ZO = O F i r s t w e f a c t o(rx: - l O ) ( x * 2 ) = O product ThenweusetheZero Rule: X - lO = O Thenwefindthesolutions: x = l0
or x + 2 = O -2_ or x =
To checkthesolutions, wetrythemintheoriginal equation.
( l O ) ' -S ( t O- 2 0 = t O O- e O - 2 0 = e f n ' - 8 f 2 \ - 2 O = q + 1 6 - Z O= O Solveeachequationby factoringand usingthe ZeroproductRule.
0=x'+7x-16
O= x'-25
xt+4x =O
019S by KeyCunlollumPrclect,Inc. 0o not dudicatendthoutpermis8bn.
*z+4x
5=O
31
by to makeonesideequalto 0. Thensolvetheequation FirstusetheAdditionPrinciple factoring. lToqcta0h:lc\ ^/'-^^t
x8 - 5 x
( f d?edlo odd,'z.l :+--r-z1.--rz.z -2 no --2
* d 2 =/ ' o o
x'-6 t?
x2+7X+l=l
x'+l6X+42=3
xl +36=l?x
32
xa-5x-lO={
x L+ 7 N = 8
x a+ 4 8 = 4 9
x'-7x
-tz
xL-lOx+5=21
1 2- 5 x + l l = l l
=-2X ?("- lO^r.
x z+ 3 x - 1 2 = 4 x
3 x ' - 3 x - 8 = 2 x 2+ 2
1'+5X+5=2q+l
o19e by ftt CwdorlumProlcct,Inc. D,ond dwllcde wtthoutp.|tt$rbn.
Equarions llqing Quadraric Makeanequation foreachproblem.Thensolvetheequation andusethesolutions to find theanswer. lf I multiply a number by4 andadd5, lf I multiply a number by 10andsubtract 9, I getthesquareof thenumber. I getthesquareofthenumber. Whatcouldthenumberbe? Whatcouldthenumberbe? alr +5 = x' Equation: Equation:
t = xr -thc
Q= 1(3-{t-5
O=(r-5Xr+l) A - 5 = Oo r r + l = O
a =5 orr=-l
Answer:5 or -f
Answer:
lf I addthe squareof a number to the numberitself,I get 30. Whatcouldthe numberbe? Equation:
lf I addthe squareof a numberto 2 timesthe number,I get63. Whatcouldthe numberbe? Equation:
Answer:
Answer:
I'mthinking of a number.Thesquare of thisnumber is thesameasthenumber times4. Whatcouldmynumber be? Equation:
I'mthinking of a number.This numberis 12lessthanitssquare. Whatcouldmynumberbe? Equation:
Answer:
Answer:
O19$ by KeyCuniculum Prciecr,Inc. Do nol dupli;etewithoutpermirsirn.
33
for the areaof eachrectangle.Thenwritean equationandsolveit. Writea polynomiat Useyoursolutionsto findthe lengthandwidthof eachrectangle. r-4
*+5
r-4
A=84
Equation:
r-4
x-4
r+5
Polynomial: x ( x + 5 )
A=64
s Xa+5x
?(2+5r=61 xa+5x-84=O
(x-7)(x+lZ)=O r - 7 = O o ra + 1 2 = O
@or
lr='lZ
I = X + 5 =1 / . r w = X = 7
Answer:
x+2
r-2
A=96
2x+1
4=210
x-2
2x+1
x+2
Polynomial: Equation:
Answer:
34
Answer: Proioct,Inc" Ot9S by KeyCutticrrlum Do not duplicalewithoutp€rmission.
Written Work Dotheseproblemson somecleanpaper.Labeleachpageof yourworkwith yourname,yourclass,the date,andthe booknumber.Alsonumbereachproblem. Keepthiswrittenworkinsideyourbookandturnit in withyourbookwhenyou'refinished. Pleasedo a neatjob. in r. lf it can be factored, is a seconddegreepolynomial Eachof the followingpolynomials go aheadand factorit. lf it cannotbe factored,write"can'tbe factored"afterit. 1.
6x2- %,
2. x2-2x-48 3. 3x2+9r+6
you canjustfactorout a monomial. Sometimes 6x2-9x=.3lx(2r-3)
youcanfactorit intoa binomial Sometimes timesa binomial.
4 . x 2+ 9 5. x2+5x 6. 12+6r +9 7. 3xz+ 12 8.
3x2-5x
9. x2-3x 1A. xz - 25
x2-2x-48=(r-8)(r+6)
you cantactorout a monomialand Sometimes thentactorit intoa binomialtimesa binomial. 3 x 2 + 9 r + 6= 3 ( x 2 + 3 x + 2 ) = 3(r + 1)(r+ 2) Andsometimesit can'tbe factoredat all! can't be factored x2+ 4 x2+ x + 1 can'tbe factored
11.24xz-8
1 4 . x 2- 2 x + 1
17.2x2+8r+8
1 2 .x 2 + 1 M , + 2
15. 9x2-6x
18.4x2-9
13. x2-6r+5
1 6 .2 x 2 + 1 k + 1 5
1 9 .4 x 2 + 4 x - 1 5
20.x2-36=0
22.x2+12x+36=0
2 4 .x 2 - 1 2 x + 3 6 = 0
21.x2+13r+36=0
23.x2+12x=0
25.x2+36=20x
Solveeachequation.
26. Makea sketchandwritean equationfor the problembelow.Thensolvethe equation and useyoursolutionsto answerthe question. fieldis 5 metersmorethanitswidth. The lengthof a rectangular Itsareais 300squaremeters.Howlongis thefield? 27. Writean equationand use it to solvethe followingproblem. The squareof a numberis threetimesthe numberitself.Whatis the number? 01990by KeyCuriqrlumProiecl,Inc. Do not dupli:at€wilhoulpermissirn.
35
PracticeTest Writeasa polynomial.
( 5 x . * 3 x - t + ) + ( 3 x 2 -. 8 x - 2 ) =
(ttx'+ 5x -8) - (2x" - 6x +21= ( 5 x+ 1 ) + ( 2 x - t + )- ( 3 x * 2 ) = 3 ( 5 x+ 2 ) =
Z ^ y ( y - x=)
aQ-a)=
-4(q - b) =
(a*b*5)3b=
3 x ( x ' + x )=
Factor.
3a' _ 6a.b=
2 x ' + 4 x ' + 6 x=
4x - tl =
vt 3 - vJ z - vl =
l O x- 5 y +3 5 =
3 x " y- 5 x y ' =
Writeas a polynomial.
( x * 3 ) ( x + 7=)
k - 2 ) ( x + 3 )=
(x - r{)" =
(2x* 3)' =
( 7 a - 3 ) ( 3 q - t + )=
36
0199 by KeyCutriqrlumProiect,Inc. Do notduolbatewithoutpormission.
Factor. -
arl
t I
xn L _ + xI
n
_./l - r
=
3X'+4x*l=
q'
- l O a* 2 1
3 y '* z q y - 6 0=
at
-8x * 16
l Q x "* l 7 x + 3 =
v'- 6 4 =
xz-4x-60 =
Solve.
6x-Qx-3)=31
3(x-5)=45
ira+2x-15=O
x2'-7x = O
Writea polynomial fortheperimeter.
4x2+4x+4
Writea polynomial forthearea.Thenwritean equation anduseit to tindthelength andwidth. 3r +2 A=120 3 x+ 2 otgl bt l(.t O,'barm P|ol.d, Inc. Do nol ducb- .tur r'rbb.r.
37
Book l: Operationson Integers Book 2: Variables,Termsand Expressions Book 3: Equations Book 4: Polynomials Book 5: Rational Numbers Book 6: Multiplying and Dividing Rational Expressions Book 7: Addingsnd Subtrscting RationqJExpressions Book 8: Graphs Book 9: Systemsof Equations Book lO: SquareRootsand QuadraticEquations Answersand Notesfor Books 1-4 Answersand Notesfor Books5-7 Answersand Notesfor Books8-10
Key to Key to Key to Key to Key to Kev to
Fractions@ Decimals@ Percents@ Geometry@ Measurement@ Metric Measurement@
dtr
CURRICULUMPRESS KEY ^ \ Education* krtovators in Mathematics
rsBN 1-55953-004-9
ilff llililili
