美国加州数学教材代数08

'ACTORING418Chapter8FactoringFactoring8Real-WorldLinkDolphinsFactoringisusedtosolveproblemsinvolvingverticalmotion.Forexample,factoringcanbeusedtodeterminehowlongadolphinthatjumpsoutofthewaterisintheair.1Foldinthirdsandtheninhalfalongthewidth.3Open.Cutshortsidealongfoldstomaketabs.2Open.Foldlengthwise,leavinga1_2”tabontheright.4Labeleachtabasshown.FactoringMakethisFoldabletohelpyouorganizeyournotesonfactoring.Beginwithasheetofplain81_2”by11”paper.�Standard11.0Studentsapplybasicfactoringtechniquestosecond-andsimplethird-degreepolynomials.Thesetechniquesincludefindingacommonfactorforalltermsinapolynomial,recognizingthedifferenceoftwosquares,andrecognizingperfectsquaresofbinomials.�Standard14.0Studentssolveaquadraticequationbyfactoringorcompletingthesquare.(Key)KeyVocabularyfactoredform(p.421)perfectsquaretrinomials(p.454)primepolynomial(p.443)0418-0419COCH08-877852.indd4180418-0419COCH08-877852.indd4189/21/065:58:40PM9/21/065:58:40PMChapter8GetReadyForChapter8419RewriteeachexpressionusingtheDistributiveProperty.Thensimplify.(Lesson1-5)1.3(4-x)2.a(a+5)3.-7(n2-3n+1)4.6y(-3y-5y-5y2+y3)5.JOBSInatypicalweek,Mr.Jacksonaverages4hoursusinge-mail,10hoursofmeetinginperson,and20hoursonthetelephone.Writeanexpressionthatcouldbeusedtodeterminehowmanyhourshewillspendontheseactivitiesoverthenextmonth.Findeachproduct.(Lesson7-6)6.(x+4)(x+7)7.(3n-4)(n+5)8.(6a-2b)(9a+b)9.(-x-8y)(2x-12y)10.TABLETENNISThedimensionsofahomemadetabletennistablearerepresentedbyawidthof2x+3andalengthofx+1.Findanexpressionfortheareaofthetabletennistable.Findeachproduct.(Lesson7-7)11.(y+9)212.(3a-2)213.(3m+5n)214.(6r-7s)2EXAMPLE1Rewriten(n-3n2+2+4_n)usingtheDistributiveProperty.Thensimplify.n(n-3n2+2+4_n)Originalexpression=(n)(n)+(n)(-3n2)+(n)(2)+(n)(4_n)Distributentoeachterminsidetheparentheses.=n2-3n3+2n+4Multiply.=-3n3+n2+2n+4Rewriteindescendingorderwithrespecttotheexponents.EXAMPLE2Find(x+2)(3x-1).(x+2)(3x-1)Originalexpression=(x)(3x)+(x)(-1)+(2)(3x)+(2)(-1)FOILMethod=3x2-x+6x-2Multiply.=3x2+5x-2Combineliketerms.EXAMPLE3Find(3-g)2.(3-g)2=(3-g)(3-g)LawsofExponents=32-3g-3g+g2Multiply.=32-6g+g2Combineliketerms.=9-6g+g2Simplify.GETREADYforChapter8DiagnoseReadinessYouhavetwooptionsforcheckingPrerequisiteSkills.Option1TaketheQuickCheckbelow.RefertotheQuickReviewforhelp.Option2TaketheOnlineReadinessQuizatca.algebra1.com.0418-0419COCH08-877852.indd4190418-0419COCH08-877852.indd41910/10/063:53:43PM10/10/063:53:43PM1182936420Chapter8Factoring8-1MonomialsandFactoringInthesearchforextraterrestriallife,scientistslistentoradiosignalscomingfromfarawaygalaxies.Howcantheybesurethataparticularradiosignalwasdeliberatelysentbyintelligentbeingsinsteadofcomingfromsomenaturalphenomenon?Whatifthatsignalbeganwithaseriesofbeepsinapatterncomposedofthefirst30primenumbers(“beep-beep,”“beep-beep-beep,”andsoon)?PrimeFactorizationNumbersthataremultipliedarefactorsoftheresultingproduct.Numbersthathavewholenumberfactorscanberepresentedgeometrically.Considerallofthepossiblerectangleswithwholenumberdimensionsthathaveareasof18squareunits.Thenumber18hassixfactors:1,2,3,6,9,and18.PrimeNumbersBeforedecidingthatanumberisprime,trydividingitbyalloftheprimenumbersthatarelessthanthesquarerootofthatnumber.Awholenumberexpressedastheproductofprimefactorsiscalledtheprimefactorizationofthenumber.Twomethodsoffactoring90areshown.Method1Findtheleastprimefactors.90=2·45Theleastprimefactorof90is2.=2·3·15Theleastprimefactorof45is3.=2·3·3·5Theleastprimefactorof15is3.PrimeandCompositeNumbersWordsAwholenumber,greaterthan1,forwhichtheonlyfactorsare1anditself,iscalledaprimenumber.Awholenumber,greaterthan1,thathasmorethantwofactorsiscalledacompositenumber.Examples2,3,5,7,11,13,17,194,6,8,9,10,12,14,150and1areneitherprimenorcomposite.MainIdeas�Findprimefactorizationsofmonomials.�Findthegreatestcommonfactorsofmonomials.PreparationforStandard11.0Studentsapplybasicfactoringtechniquestosecond-andsimplethird-degreepolynomials.Thesetechniquesincludefindingacommonfactorforalltermsinapolynomial,recognizingthedifferenceoftwosquares,andrecognizingperfectsquaresofbinomials.NewVocabularyprimenumbercompositenumberprimefactorizationfactoredformgreatestcommonfactor(GCF)0420-0424CH08L1-877852.indd4200420-0424CH08L1-877852.indd4209/21/066:01:05PM9/21/066:01:05PM™ä™ÊÊÊ


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xLesson8-1MonomialsandFactoring421EXAMPLEPrimeFactorizationofaMonomialFactor-12a2b3completely.-12a2b3=-1·12a2b3Express-12as-1·12=-1·2·6·a·a·b·b·b12=2·6,a2=a·a,andb3=b·b·b=-1·2·2·3·a·a·b·b·b6=2·3Thus,-12a2b3infactoredformis-1·2·2·3·a·a·b·b·b.Factoreachmonomialcompletely.1A.38rs2t1B.-66pq2GreatestCommonFactorTwoormorenumbersmayhavesomecommonprimefactors.Considertheprimefactorizationof48and60.48=2·2·2·2·3Factoreachnumber.60=2·2·3·5Circlethecommonprimefactors.Thecommonprimefactorsof48and60are2,2,and3.Theproductofthecommonprimefactors,2·2·3or12,iscalledthegreatestcommonfactorof48and60.Thegreatestcommonfactor(GCF)isthegreatestnumberthatisafactorofbothoriginalnumbers.TheGCFoftwoormoremonomialscanbefoundinasimilarway.�TheGCFoftwoor