ALGEBRAIN

WORDS

AGuideofHints,Strategiesand

SimpleExplanations

GregoryP.Bullock,Ph.D.

Copyright©2014GregoryP.Bullock,Ph.D.AllRightsReserved.Thisbookmaynotbeusedorreproducedinpart,inwhole,orbyanyothermeanswhatsoeverwithoutwrittenpermission.Bullock,GregoryP.Algebrainwords:aguideofhints,strategiesandsimpleexplanationsMATHEMATICS/Algebra/GeneralSTUDYAIDS/StudyGuidesFirstEditionTheUnitedStatesofAmerica

TableofContentsINTRODUCTIONWhatIsThisBook?WhyDoYouNeedAlgebra?

REVIEWOFTHEBASICSTheRealOrderofOperations:GEMATheTruthaboutPEMDASTheUnwritten1PropertyCrisesofZeros,Ones&NegativesIntegers&WholeNumbersPrimeNumbersIs51aPrimeNumber?WhatisaTerm?Whatisa“Like-Term”?WhatisaFactor?FactoringTheProcedureforPrimeFactoringThePrimeNumberMultiplesTableTheGreatestCommonFactor(GCF)TheLeastCommonDenominator(LCD)GCFvs.LCD

FRACTIONSProcedureforAdding&SubtractingFractionsMultiplyingFractionsDividingFractions

OPERATIONSOFBASESWITHEXPONENTSMultiplyingBasesWithExponentsDividingBasesWithExponentsExponentsofExponents(a.k.a.PowersofPowers)

SOLVINGSIMPLEALGEBRAICEQUATIONS

SolvingaSimpleAlgebraicEquationwithOneVariable(FirstDegree)Arrangement:DescendingOrderExpressionsvs.Equations

LINEAREQUATIONSADiagonalLine:AHorizontalLine:AVerticalLine:WhatDoes“Undefined”Mean?HowtoGraphaLinearEquationTheSlopeEquationThe4ImportantEquationsforLinesWhenx1=x2:Wheny1=y2:Parallel&PerpendicularLinesonaGraph

SOLVINGASYSTEMOF(TWO)LINEAREQUATIONSWhatDoes“SolvingInTermsOf”Mean?Graph&CheckTheSubstitutionMethodTheAddition/EliminationMethodExamplesforChoosingtheMethod

Interpretingthe“Solutions”OneSolution-ConsistentNoSolution-Inconsistent,ParallelInfiniteSolutions-Dependent

TRINOMIALS&QUADRATICSWhatAre“Solutions”toQuadraticEquations?SolvingQuadraticEquationsFactor&SolveTrial&Error/ReverseFOILMethodTheac/GroupingMethodCompletetheSquareTheQuadraticFormula

ThePartEveryoneForgets(TheLastStepoftheQuadraticEquation)Graph&Check

QuadraticswithZeroWhencis0:ax2+bx=0WhenBothb&care0:ax2=0Whenbis0:ax2+c=0“TheDifferenceofTwoSquares”ConjugatePairBinomialsTakingtheSquareRootofBothSidesTheSumofTwoSquaresSpecialWordsforSpecialCasesPerfectSquareTrinomialTheDifferenceofTwoSquares

Primevs.NoSolutionClarification:WhentheSolutionis0

RATIONALEXPRESSIONSProcedureforSimplifyingRationalExpressionsProcedureforAdding&SubtractingRationalExpressionsSimplifyingaComplexRationalExpressionAll-LCDMethod(detailedversion):SimplifyOverallNumerator&OverallDenominatorSeparatelyMethod(detailedversion)All-LCDMethod(shortversion)SimplifyOverallNumerator&OverallDenominatorSeparatelyMethod(shortversion)AnnotatedExample1UsingtheAll-LCDMethodAnnotatedExample2UsingtheOverallNumerator&DenominatorMethod

TheWrongWaytoSimplifyaRationalExpressionExtraneousSolutionsProcedureforSolvingEquationswithRationalExpressions&ExtraneousSolutionsCrossMultiplication

Cross-Multiplicationvs.CrossCancellingRADICALS,ROOTS&POWERSPerfectSquares&AssociatedSquareRootsListofPerfectSquares&AssociatedSquareRootsCommonPerfectCubes&AssociatedCubeRootsOtherPowers&Relationshipsof2,3,4&5Manipulating&SimplifyingRadicalsListofCommonRadicalFingerprintsExtraneousRootsinRadicalEquations

FMMs(FREQUENTLYMADEMISTAKES)TheTwoMeaningsof“CancellingOut”CheckingYourAnswersMiscellaneousMistakesScientificNotationonYourCalculatorWhatDoes“Error”onaCalculatorMean?

CLOSING

INTRODUCTION

WhatIsThisBook?

Thisbookisaguideofcommonmathandalgebratopicsthatareexplainedinanon-traditionalway.Itisnotatextbook,norisitaconventionalstudyguide.Thisisabookwherebasicmathematicalandalgebraictopicsareexplainedinlaymen’sterms,sometimesevenpurposefullyredundantterms,tomakeyourunderstandingeasierandyourlearningcurvefaster.It’smoreofaguideofsupplementalinformationandperspectivesonthemathyoumustlearn.

ItutoredCalculusfortheMathDepartmentinundergraduateschoolasapart-timejob,thenbeganteachingmathatthecollegiatelevel(BasicMathandArithmeticthroughCollegeAlgebra/Pre-CalcI)in2009toawiderangeofstudentsofvariousagesandmatheducationbackgrounds.Duringthattime,Ibegannoticingtrendsamongmystudentsandclasses.OnemajortrendInoticedwasthedivideamongpeoplewhoseemedto“getit,”andthosewhodidn’t“getit”aseasily,asquickly,orinthesamewayasthosewhodid.Althoughit’spointlesstoclassifystudentsintogroups,myjobasaninstructoristohelpbridgethegapandfindmechanismstohelpeveryone“getit.”

Asmyteachingstyleevolved,Inoticedthatalotofmath(eitherinthebooksortraditionallectures)wastaughtinasortof“mathlanguage,”meaningmostlyinnumbers,variablesandlinesofequationsandsimplificationsteps…whichisallwellandgood,becausethat’swhatmathis.ButIfoundthatmuchofitwaslefttointerpretation,whichsomewouldgetandsomewouldn’t.SoIstartedtranslatingthemathintowordedstepsandnotesandfoundthatstudentsrespondedpositivelytoit.ThiswasthebridgeoverthegapIwaslookingfor.Sincethen,Ibegangivingexplicitlywordednotesincluding,butnotlimitedto,stepbystepinstructions.Throughobservingcommonlearningpatternsamongstudents,Ialsowasabletopredictcommonquestionsorareasofconfusion,soIwouldgivenotestopreemptivelyanswerquestionssuchas“WhatdoIlookfor?”or,

“WhenshouldIusethis?”or,“Whatwillitlooklike?”

andpreparestudentsforfrequentmistakeareasbyalsoshowingwhatnottodo,alongwithwhattodo.Theseexperiencesinspiredmetorecordmynotesandmakethemavailabletoanystudentwhowishestheyhadanotherresourcetomakelearningmathandalgebraeasier.AsIstated,thisbookisnotatextbook,andbythatImeanIdon’tgiveextensiveexamplesandpracticequestionsthewaytextbooksdo.Mathtextbooksaregenerallyverygoodatgivingthemandcontainawealthofinformation.Buttraditionaltextbooksalsoteachinaveryrigidandoften

bottom-upway.I’vefoundthatmanytextbooksteachcertaintopicstosuchasub-categoricallevelofdetailthatstudentslosesightofhowitconnectstothebiggerpicture.SowhatIofferareotherperspectivestothemathfromthetextbooks,andIsometimesunveiltheminamoretop-downway.

Ibelievethat:givingmultipleperspectives,givingdetailed,step-by-stepinstructions,showingexampleswithcommentary,connectingkeytopics,translatingmathterminology,answeringfrequentlyaskedquestions,highlightingcommonmistakes,anddrawingattentiontosomeofthemoreminute(yetimportant)detailswhich

sometimesslipunderstudents’radar,willexpediteyourabilitytorealizeandabsorbmaterial.Thiswillhelpyougetbettergradesandsaveyouvaluabletime,frustration,andevenmoney(ifitmeansyoudon’thavetorepeatacourse).Sometimesstudentslearnfasterbyseeingthingsinwords.In2010,IpublishedmyfirstbookGRADES,MONEY,HEALTH:TheBookEveryCollegeStudentShouldRead,whichisacollectionofadviceandstoriesonavarietyofcollege-relatedtopicsgearedtowardshelpingstudentsexcelandgetthemostoutoftheircollegeexperience.Init,Idedicated10shortchapterstohowtogetthebestgrades.Inoneofthosechapters,Iexplainhowimportantandhelpfulitistolistentothewordsofyourprofessors,andhowtakingnotesonwhattheysayissometimesbetterthanjusttryingtoquicklyrecordwhattheyputontheboard.That’sthebasisofthisbook:Puttingthemathintowords.WhenIwaswritingGRADES,MONEY,HEALTH,Ioriginallyintendedtoincludeachapteronsomebasicmathandalgebrahints,butmyamountofmaterialkeptgrowing,andIwantedtokeepthebookasshortaspossible,soIkeptthemathsegmentsout.Thenmymath-help-noteskeptgrowingandultimatelydevelopedintoitsownbook…thisbook,dedicatedentirelytomathandhelpinganyandeverymathstudent.

ThisisAlgebrainWords.

WhyDoYouNeedAlgebra?Manystudents,especiallythosewhosemajorsarenotmathorscience,askthequestions,“WhydoIneedalgebra?”or,“WhenamIevergoingtousethis?”Thesearecompletelyvalidquestions.Ononehand,thatquestioncanbeaskedaboutanygeneraleducationclasswhichdoesnotseemtorelatetoone’smajor.Thegeneralstockansweris:Becauseitmakesyoumorewell-rounded.Andthat’strue,butformany,thatanswerisstillvagueandunacceptable.Withregardstomathspecifically,manyinstructorsputforthgreateffortandcreativitytoexhibitreasonsandscenariosastohowmathisusedineverydaylife.Iapplaudanyandallinstructorswhocanconvincinglyconveytheseanswers.Unfortunately,mostdonothavetheanswers,andevenmoreunfortunately,theytendtobe:

Willhelpyouwithmoneyandpersonalfinances,Willhelpyouwithmeasurementsaroundthehouse,Willhelpyoubeamoreefficientshopper,Youcanhelpachildwiththeirhomework,Willhelpyouunderstandtimesignaturesinmusic,Willtellyouhowlongitwilltakeyouandafriendtosplitajob,Etc.

Theseanswersareclichéandunsatisfactory.Thesetasksrequirelittletonoalgebra(althoughtheydorequireanunderstandingofdivision,fractions,anddecimals).Butthetruthis:thereareresources(likesmart-phoneapps,orjustaclassiccalculator)thatdomost,ifnotallofthosefunctionsforus.Anothertruthis:Thosewhoreallyneedandusealgebrafortheircareersareaselectfew.However,thatdoesn’tmeanyouhavenouseforit.Here’sthebetteranswertoyourquestion:Learningalgebramakesyoubetteratproblemsolving.Italsomakesyouawarethatthereisordertotheworldwelivein.Algebraisbasicallyaseriesofrules;youmightevenconsiderititsownlanguage.Everythinginourlivesrelatetounderstandingrulestohelpussolveproblemsandnavigatethroughlife.Thatincludes:

Laws,thelegalsystem,andcontracts;rules,regulations,andstrategiesofsportsandgames;negotiations;learninganewlanguage;criticalreading;computers,smartdevicesandelectronics;managingandworkingwithpeople;giving,takingandfollowingdirections;

readinggraphs,chartsanddatatables;accomplishingtasksandgoalsofallsizes;andyes,evenfixingthingsaroundthehouse.

Isitallsolvingfor“x”?Obviouslynot,oratleastnotliterally,butwecanallbenefitfromlearninghowtonavigatethroughrules(and)tosolveproblems.Algebrawillhelpyoubemorelogicalandseelifemorelogically.There’sonemorething.Uponlearningandpassingalgebra,youwillfeelinspiredandempowered.Algebracanbeacomplicatedsubject.Ifyouconquerit,youwillthenhavetheconfidencetotakeonotherseeminglycomplicatedobstacles.Regardless,algebraisasubjectyouhavetotake.Noonesaysyouhavetolikeit,butyoumightaswellacceptitanddoyourbesttosucceedinit.Thisbookwillhelpyouthroughit.

Throughoutthebook,Ioftenreferyoutootherrelatedtopics,soI’veincludedhyperlinkstoallowyoutoquicklyjumptosuchreferredsections.Alsothroughoutthisbook,Iusethevariable“x”astheuniversalvariable,eventhoughmanyproblemsyouwillencounter(inclassorinlife)mayfeatureanunknownvariabledifferentthan“x”.

REVIEWOFTHEBASICSFirst,wemustlookatafewbasicsthatwillbeusedandreferredtothroughoutthisbook.Fromyeartoyearandclasstoclass,youmayhavegraspedthemajorityofthematerialyou’velearned,andbuiltagoodfoundation.But,say,overasummerorholidaybreak,orjustfromnotusingitenough,youmayhaveforgottenafewofthemoreobscureprinciples.Theseareheretoquicklybringyoubackuptospeed.

TheRealOrderofOperations:GEMAYoumayrememberthatwhenaddingterms,youwillgetthesamesumregardlessoftheorderinwhichyouadd.Youmayalsorememberthatthesameappliestomultiplication:Youwillgetthesameproductregardlessoftheorderofthefactorsyoumultiply.Theseare“commutativeproperties.”However,thisdoesnotapplytosubtractionordivision.Forsubtractionanddivision,ordermatters.Ordermatters,includingwhensubtractionand/ordivisionismixedinwithtermsbeingaddedand/ormultiplied.Sinceordermatters,thereareasetofrulesinplacetohelpuscalculatenumbersandtermsintheproperorder,andtoputconsistencyintothewaywedomath.Thesearetheorderofoperations.Often,booksorinstructorsdonotteachthiscompletelycorrectly.Here,itwillbeexplainedcompletely,withnothinglefttobemisinterpreted.1.SimplifyinsideGroupsfirst,ifpossible,frominnertoouter.Agroupisasetof(parentheses),[brackets],{braces},overallnumerators,overalldenominators,andradicands.2.Exponentsorroots,whichevercomesfirst,fromlefttoright.3.Multiplicationordivision,whichevercomesfirst,fromlefttoright.4.Additionorsubtraction,whichevercomesfirst,fromlefttoright.

TheTruthaboutPEMDASManystudentsaretaughttheacronymandmnemonicdevicePEMDAS,whichstandsfor“Parentheses,Exponents,Multiplication,Division,Addition,Subtraction.”ImustwarntoyoubecarefulofPEMDAS;itismisleadingandincomplete.Ifyoulearntorelyonit,itcanfailyou.Letterbyletter,here’swhy:P:Thefirstorderisgroups,ofallsorts,asdescribedabove.Ifyouthinkofparenthesesonly,andyougettoothergroups,youmightthinkitappliestoparenthesesonly.AmoreappropriatefirstletterandwordshouldbeGforGroups.Groupsinclude,butarenotlimitedtoparentheses.E:Thismakesyouthinkofexponentsonlyinsteadofrootsaswell.Thisisn’tabigdeal,sinceradicalscanbeconvertedintoexponentform(whentheyare,theyarecalledrationalexponents),buttheyareofteninrootorradicalform,soyoumustbepreparedforthat.Whenbothexponentsandrootsappearinanequation,dowhichevercomesfirst,fromlefttoright.Also,rememberthatradicandsshouldbesimplifiedfirst,ifnecessary,astheyaretechnicallyagroup,asmentionedinthefirststep.MD:Thereasonthisismisleadingisbecausesomepeopleinterpretthistobechronologicallyliteral.Inotherwords,someseeMbeforeDandthinkmultiplicationmusthappenbeforedivision,butinfact,itmeansthatanyMultiplicationorDivisioncomebeforeanyAdditionorSubtraction.Butmultiplicationordivisionshouldbetreatedwiththesamepriority,andyou’resupposedtoperformwhicheverofthetwooperationscomesfirstfromlefttoright,inthedirectionyouread.Forinstance,if,inanequation,adivisionsigncomesbeforeamultiplicationsign,fromlefttoright,youdividefirstandmultiplynext.AS:Manyofteninterpretthisas“AisbeforeS,”(astheythinkMcomesbeforeD)butinfacttheyareofequalpriorityinthewayM&Dare.Additionorsubtractionshouldonlybeperformedafterallotheroperationsarecompleted.Thenyouperformeitheradditionorsubtraction,whichevercomesfirstfromlefttoright.Ifsubtractioncomesbeforeaddition,youwoulddothesubtractionfirst,thentheadditionnext.Ifanything,considerPEMDASaloosereminderofthecompleteOrderofOperations,althoughifitwereuptome,PEMDASbethrownawaycompletelyandreplacedwithGEMA:

1. Groups(simplify,innertoouter)2. Exponentsorroots3. Multiplicationordivision4. Additionorsubtraction

Actually,since:

Rootsaretechnicallyaformofexponents,whenconvertedtorationalexponents*;Divisionistechnicallymultiplicationofafraction;andSubtractionistechnicallyadditionofnegativenumbers,

itcouldevenbecondensedtojust:

1. Groups,2. Exponents,3. Multiplication,4. Addition

*Note:Rootsareconvertedto“rationalexponents”whentheradicalsignisremovedandtheroot-numberismovedtothedenominatoroftheexponent.

TheUnwritten1Youmustrememberthat“1”isoftennot(requiredtobe)writtenorshown,butisstillthere.Igiveanexampleshowingthe“1”writtenasboththecoefficientinfrontofx(couldbeanyvariable),thedenominatorofx,thepowerofx,andthedenominatorofthepower1(ofx).Movingright,Ishowitwiththedenominatorsremoved,andthenallthe1sremoved,showingjustx.

Howeverfundamentalthismayseem,itisaconceptstudentsoftenquestionorforget,andforthatmatter,sometimesfailtoimplementwhennecessary.Herearereasonsitishelpfultorememberthat“1”isstillthere:

Asacoefficientsoitsassociatedvariablecanbeaddedtootherlike-terms,suchas:x+3x=1x+3x=4x

orasinaddingradicals:

Asadenominator,especiallyfor(fractionconversionstolike-fractionsduring)addition/subtractionoffractionsasin:

Asadenominatorfor(inverting,thenmultiplyingafractionduring)divisionoffractionsasin:

Asapowerorroot,especiallyformultiplyingfactors(ofacommonbase)withexponents(inwhichyouaddtheexponents),asin:

Asapowerorroot,especiallyfordividingfactors(ofacommonbase)withexponents(inwhichyousubtracttheexponents),aswhensimplifying:

AsavalidplaceholderafteraGreatestCommonFactorhasbeenfactoredout,asin:3x+3=3(x+1)

Itisalsoworthremindingyouabouttheunwrittenoneassociatedwithanegativesign.Takealookatthefollowingexamples:-4,-x,- .Thesecanbethoughtofas:(-1)(4),(-1)(x),and(-1)( ),respectively.

PropertyCrisesofZeros,Ones&NegativesTherearemanyfundamentalpropertiesinvolvingvariousoperationswith0,1andnegativenumbers(Iwillfocusmostlyon“-1”).Someareeasytoremember,however,someareeasytoconfuseorforget,buttheyarevitaltogetright.Intextbooks,theseareoftenthrownatyoufromdifferentdirections,atdifferenttimes,oftenwithvocabularyordefinition-likelabels.Thesearepropertiesinvolvingmultiplication,division,exponentsandroots.Foreaseandconvenience,I’vesummarizedtheimportantoneshereinthissection,leavingoutthelabels,butshowingtheproperty,thenexplainingitinwords,thewayyoumightsayit,hearit,orhearitinyourhead.Hearing(orreadinghowyoumighthear)theseshoulddrivehomeanextradimensionintoyourbrain,soyoucanmoreeasilyrecallthemlater.

(#)(1)=#.Inwords:Anynumbertimesoneequalsitself,always,withnoexceptions.#÷1=#,

alsoseenasafraction: =#.Inwords:Anynumberdividedbyoneequalsitself,always,withnoexceptions.Forfractions:Anynumberoveroneequalsitself(thetopnumber).Likewise,anynumbercanbeassumedtobeoverone,andcaneasilybeconvertedtoafractionbyputtingitoverone.Any#orterm÷itself=1,

alsoseenasafraction: .Inwords:Anynumberortermdividedbyitselfequalsone,exceptwhenthatnumberiszero.Forfractions:Anynumber(orterm)overitselfequalsone,exceptwhenthosenumbersarezero.Anotherwaytosayitis,“anynon-zeronumberoveritselfequalsone.”Seethenextexample.

#÷ 0=undefined,alsoseenasafraction: =undefined.Inwords:Anynumberdividedbyzeroisundefined.Orasafraction,anynumberoverzeroisundefined.Anynumberdividedbyzerodoesnotequalzero.Youmightsay“youcan’tdivideanynumberbyzero.”

1÷0=undefined,alsoseenasafraction: =undefined.Inwords:Onedividedbyzeroisundefined,becauseanynumberdividedbyzeroisundefined,asshowninthepreviousexample.0÷ any#=0,withtheexceptionofwhenthedenominatoris0;Alsoseenasthe

fraction: =0,exceptwhenthedenominatoriszero.Inwords:Zerodividedbyanynumberequalszero,withoneexception.Theexceptionis:Zerodividedbyzeroisundefined,becauseanynumberdividedbyzeroisundefined.Anotherwaytothinkofitis:0÷ anynon-zero#=0,and

=0.0÷1=0,

alsoseenasafraction: =0.Inwords:Zerodividedbyoneequalszero.Infractionform:zerooveroneequalszero.Thisexemplifiestwootherproperties,previouslyshown.Thisexamplefollowsthat:

Anynumberoveroneequalsthatnumber(thetopnumber),andZerooveranynon-zeronumberequalszero.

0÷ 0=undefined,

alsoseenasafraction: =undefined.Inwords:Zerodividedbyzeroisundefined,becauseanynumberdividedbyzeroisundefined.

Thisistheexceptiontotherulethat“anynumberoveritselfequalsone.”Itisalsotheexceptiontotherulethat“zerodividedbyanynumberequalszero.”Zerodividedbyzerodoesnotequalzero,assomemistakenlythink.

Thispropertyisespeciallyusefulwhenlookingat:

SlopesofVerticalLines(see:AVerticalLine,and:Whenx1=x2),andExtraneousSolutions(see:SolvingEquationswithRational

ExpressionsandExtraneousSolutions).

Sofar,wehavelookedatmanypropertiesandexampleswhichresultas:“Undefined”.Whendoingthesefunctionsonacalculator,youmightget“error.”Foramoreonthat,see:WhatDoes“Error”Mean?

Allbutoneofthepropertiesshownsofarinvolvedivisionwith0and1.Thefollowingsectionwillfocusonexponentsandrootsinvolving0,1andnegativenumbers.Base1=BaseInwords:Anybasetothepowerof1=thebase…whichisthesameastosay:Any#1=itself.Inwords:Anynumber(base)tothepowerofoneequalsitself(thatbase),asinthenextexample:1Any#=1.Inwords:Onetothepowerofanynumberequalsone,withnoexceptions.01=0.Inwords:Zerotothepowerofoneequalszero.10=1.Inwords:Onetothepowerorzeroequalsone.Any#0=1.Inwords:Anynumbertothepowerofzeroequalsone(withoneexception;seenextexample).Anotherwaytorememberitis:Anynon-zero#0=1.Inwords:Anynon-zeronumbertothepowerofzeroequalsone.00=undefined(anddoesnot=1or0).Inwords:Zerotothepowerofzeroisundefined.Likewise,zerotothepowerofzerodoesnotequaloneorzero.

Itisimportanttorememberthenamesofthecomponentsofaradical.Theterminsidetheradicalistheradicand.Theterminthe“v”istheroot.Also,althoughthesymbol,shapeandsetupofaradicalmaycloselyresemblelongdivision,theyarenotthesameinanyway.

Inwords:Thesquarerootofoneisone.Note:Thesquarerootmeans“totherootoftwo,”butthe“2”iscommonlyunwritten.Whentheradicalsignhasnonumberwritteninthe“v”area,itisimpliedtobetwo,meaningthesquareroot.

Inwords:Anynon-zeronumberrootofpositiveoneequalsone.

.Inwords:Anyradicandtotherootofzeroisundefined.The“rootofzero”canalsobecalledthe“zerothroot”orthe“zeroethroot.”Also,sometimesthisanswerisgivenas“infinity( ),”insteadofundefined.

Inwords:Thesquarerootofzeroequalszero.

Inwords:Anynon-zeronumberrootofzeroequalszero.

norealsolution…andmaybeexpressedas“i".Inwords:Thesquarerootofnegativeonehasnorealsolution.Itissaidtohaveno“real”solution,becausethesymbol(letter)“i”(for“imaginary”…asopposedto“real”)canbeusedinterchangeablywith .Inthatway,youcanconsider

ashavingasolution,butsinceitisnotarealnumber,itissaidtobe“norealsolution.”Thesymboliisusefultomanagemultipleoccurrencesofthesquarerootofnegativeonewithinanexpressionorequation.

Thefollowingexamplesdemonstratetheimportanceofthesecondorderofoperations,aswellasthecompleteandproperwordingofthatrule,mainlythatrootsaretobecomputedfirstbeforeanyothermultiplicationanddivisionoradditionandsubtraction.Noticespecificallythattherootis(tobe)computedfirst,andanyfactorsornegativesignsoutsidetheradicalareappliednext.

Inwords:Negativeofthesquarerootofpositiveoneequalsnegativeone.Itisimportanttonoticethatthenegativesignisoutsidetheradicalandisthusattributedtotheresultoftheradical,whichiscalculatedfirst,accordingtoorderofoperations.

norealsolutionInwords:Anyevennumberrootofnegativeonehasnorealsolution…andmaybeexpressedintermsof“i".

=norealsolution…andmaybeexpressedas“-i".Inwords:Thenegativeofthesquarerootofnegativeonehasnorealsolution,andthusmightbeansweredas“norealsolution,”oras“negativei.”Itisimportantnottomistakenlyseethisas“anegativetimesanegativeequalsapositive,”becausetheevenrootofanynegativenumberhasnorealsolution,firstandforemost,regardlessofifitismultipliedbyanegativeoutsidetheradical.Rememberingthatthesquareroot(oranyevenroot)ofanynegativenumbercan’tbefound(yieldsnorealsolution)isimportanttorememberwhen:

solvingquadraticequations,usingthequadraticformula,andunderstandingthegraphsofparabolas.

Inwords:Thecuberootofnegativeoneequalsnegativeone.

Inwords:Anoddnumberrootofnegativeoneequalsnegativeone,asinthelastexample.

Inwords:Thenegativeofthesquarerootofpositiveoneequalsnegativeone.Thisfollowstheorderofoperationsbytakingtherootfirst(ofpositiveone),thenattributingthenegativesignfrominfrontoftheradicalnext,whichisreallymultiplying(theresultoftheradical,whichhere,is1)bynegativeone(the“1”outsidetheradicalassociatedwiththenegativesignisunwritten).

Thefollowingexamplesinvolvingexponentsalsodemonstratetheimportanceoforderofoperations,andtheprioritizationofgroups(here,parentheses)andexponents,whichprecedemultiplication(ordivisionandadditionorsubtraction).Theplacementofanegativesignwithregardstotheparenthesesandthebasenumberisveryimportant.(-1)2=1Inwords:Thebasenegativeonesquaredequalspositiveone.Forthistobetrue,thenegativemustbeassociatedtothebase:oneinsidetheparentheses.(-1)even#=1Inwords:Negativeonetothepowerofanyevennumberequalspositiveone.(-1)3=-1Inwords:Thebasenegativeonecubedequalsnegativeone.(-1)odd#=-1Inwords:Thebasenegativeonetothepowerofanyoddnumberequalsnegativeone.-(1)2=-1Inwords:Anegativeoutsidetheparenthesestimespositiveonesquaredequalsnegativeone.Orderofoperationsdictatesthatexponentsmustbeperformedfirst,andpositiveonesquaredequalspositiveone.Sincethenegativeisoutsidetheparentheses,it’slikemultiplyingnegativeonetimespositiveone(positiveonebeingtheresultofonesquared).Finally…-12=-1Inwords:Negativeonesquaredequalsnegativeone.Orderofoperationsplaysaprominentroleinthisanswer.Specifically,theexponentmustbeappliedtothebaseofonefirst,andthenegativeisappliednext.Thisexampleisoften

mistakenlyanswered“positiveone,”becausesomemistakenlyseethisasthesquareofnegativeone,ornegativeonetimesnegativeone.Thiswouldbedifferentifthenegativewereinsidetheparentheseswiththeone,andthepoweroftwowasoutsidetheparentheses,asshownfourandfiveexamplesback.Thisexamplenotonlyshowstheimportanceoftheplacementofsignsandparentheses(orlackof),butitalsoshowstheimportanceofhowthemathisheardorspoken.Ifyoulookbackathowthiswastranslatedinwords,youcan’tassumethatthenegativeisinparentheseswiththebase“1”.Forthatreason,whenspeakinganequationcontainingparentheses,youshouldbespecificastowheretheystart,end,andwhatisinsidethem.Istartedyouoffwithsomefundamentalproperties.Thisreviewwillhelpyoubrushuponthelittlethingsmanypeoplecommonlyforgetfromtheirpreviousmathexperience.Havingthesereminderswillgiveyouanextraedgeinsolvingproblems(especiallyproblemsthatappeartobecomplicated,butaren’t,onceyouapplytheseequalities).Formoreonradicalsandcommonlyusedroots,seethe:Radicals,Roots&Powerssectionlaterinthebook.

Integers&WholeNumbersThewords“integers”and“wholenumbers”arecommonlyconfusedandmisusedamongstudents.Thesewordsareusedtoproperlycommunicatemath,soitisimportantyouusethemproperly,aswellasunderstandwhentheyareused.Wholenumbersarepositive,non-decimal,non-fractionnumbers.Themainaspectstudentsoftenforgetaboutwholenumbersisthattheycanonlybepositive.Theycanalsobedefinedas“positiveintegers.”Oneplaceyouwillseewholenumbersusedareinchemicalformulasandchemicalreactionequations.Thesubscriptsinachemicalformulacanonlybewholenumbers.Andthecoefficientsinabalancedchemicalreactionequationalsocanonlybewholenumbers.Integersarealsonon-decimal,non-fractionnumbers,butcanbepositiveandnegative.Anyandallwholenumbersarealsointegers(wholenumbersarethepositiveintegers).Theword“integers”isoftenseenandusedduringfactoringoftrinomialsandquadratics.WhenfactoringusingtheTrial&Error/ReverseFOILMethod,youaretoldtolookatthe“integerfactorsofthefirst(ax2)andlast(c)terms”(although,sometimesfractionsanddecimalsarepermittedtoo,whereapplicable,whichisgenerallywhenthefirst(ax2)andlast(c)termsarefractionsordecimalstobeginwith).Also,theabsolutevalueofanyintegerresultsinawholenumber.

PrimeNumbersAprimenumberisanumberwhichisnotdivisiblebyanynumberotherthanitselfand1(withoutresultinginanon-integernumber).Meaning,ifyoudivideaprimenumberbyanythingotherthanitselfor1,theresultwillbeadecimal(orequivalentfraction).Thefirstthirtyprimenumbersarelistedhereasareference,soifyoueverneedaplacetoquicklycheckanumber,youcanreferhere.Youcaneasilyfindmoreontheinternet.1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109…Whyaretheseimportanttoknow?Becauseknowingwhetheranumberisprimetellsyouwhetheryoucanproceedtofactorit.Ifitisnotprime,youcanfactorit.Ifitisprime,youcannotfactorit.Thisisusefultoknowwhenyouarefactoringnumbers(sometimesusingafactortree;seeyourtextbookformoreonfactortrees);whendoingfactortrees,yourgoalistofactorallnumbersintoprimenumbers.ThismaybeusedwhenfindinganLCDorintheprocessoffactoringtrinomialsintobinomials.Youalsousefactoringwithradicals,butinthosecases,youdon’talwaysneedtofactortoprimenumbers.Formoreonthat,see:Manipulating&SimplifyingRadicals.Polynomialscanalsobeprime,whichintheircasemeanstheycan’tbefactoredintopolynomialsanysmallerthanthemselves.Foranexampleofaprimepolynomial,see:TheSumofTwoSquares.Itisalsoworthnotingthattheoppositeofaprimenumberisacompositenumber,whichisanumberwithfactorsotherthanitselfand1.

Is51aPrimeNumber?Thenumber51isnotaprimenumber,butisoftenmistakenforbeingprime.Iguessitjustsomehowlooksprime…whateverthatmeans.It’sclearlynoteven(soit’snotdivisibleby2),andit’sjustanumberwedon’tuseasregularlyasthenumbers0through50.Plus,manyotherprimenumbersendin1,asseeninthelistofPrimeNumbers.Butdon’tletthisnumberslipunderyourradarwhenyouneedtoknowwhetheritisprime.Howcanyoutellitisn’tprime?Becauseitfollowsthetechniquethat:ifyouaddthedigitsofanumber,andthatsumisamultipleof3,thenthat(original)numberisalsodivisibleby3(andanotherinteger).Thedigits5+1=6,and6isclearlyamultipleof3,therefore51isalsodivisibleby3.Thenumber51canbefactoredto(3)(17),whichisactuallytheprime-factorizationof51.ThesefactoringstrategiesarediscussedmoreinTheProcedureforPrimeFactoring.Also,youwillnoticethenumber51inThePrimeFactorMultiplesTable.

WhatisaTerm?Atermisanynumberorvariableorcombinationofthemthateitherstandsaloneorissetapartbyothertermswitha“+”or“-“sign.Termsmaybemadeupoffactors,buttermsarenotfactorsthemselves.Theycan’tbefactorsbecausefactorsaremultiplied,notsplitapartbyplusorminussigns.Becarefulnottouse“factor”and“term”interchangeably.

Whatisa“Like-Term”?Alike-termisatermwithacommonbaseandcommonexponent.Termsmusthaveboththesamevariable(letter)andthesameexponenttobeeligibletobecombined.Onlylike-termscanbeaddedorsubtracted(*seenotebelow).Iftherearetermsthatarenotvariables,theymustonlybeconstants(numbers)whichcountaslike-termsandcansimplybeaddedorsubtractedtogether.“Like-terms”alsoapplytoaddition&subtractionofradicals.Sinceanyradicalorrootcanbeconvertedtoandexpressedasanexponent(whenitis,itiscalledarationalexponent),thisfollowsthedefinitionof“like-terms.”“Like-radicals”canstillbeaddedandsubtractedevenifthey’renotinrationalexponentform.Youmaynotdealwithradicalswhenyoubeginthebasicsofalgebra;youusuallylearnabouttheselater.Formoreonhowlike-termsapplytoradicals,see:Manipulating&SimplifyingRadicals.*Note:Thisdoesnotmeanlike-termscannotbemultipliedordivided,becausetheycanbe.Thestatementreferredtoaboveismeanttoaccentuatethatnon-like-termscannotbeaddedorsubtracted.Tobeclear,anytermscanbemultipliedanddivided;theydon’thavetobe“like.”Non-like-termscanbemultiplied(however,whentheyaremultiplied,theyaretechnicallyfactors).Considertheterms:3x2and4x2.Theyarelikeandcanbemultipliedtoget12x4.

WhatisaFactor?Afactorisanumberorvariablethatisorcanbemultipliedbyanothernumberorvariable.Factorscombineviamultiplicationtomakeaterm(andyes,factorsaremultipliedtogiveaproduct,butthissectionismeanttohelpdistinguishbetweenfactorsandterms,astheyareoftenusedincorrectlyinterchageably).Butoftentimes,toservetheirfunction,factorsarenotmultipliedtogether,rathertheyarefactoredfromlargernumbersortermsandshownasunmultiplied,individualfactors.Unmultipliedfactorsjustlooklikenumbersorvariablesstandingnexttoeachother(ofteneachinparentheses).Convertingalargernumberortermintofactorsisdonebyfactoring.

FactoringFactoringisawayofbreakingdownalargernumberortermintoitsfactors.Factoringisperformedusingdivisionandtrial&error(explainedinTheProcedureforPrimeFactoring).Thisisoftendonetocomparefactorstootherfactors,sowhencommonfactorsarefound,theycanbecancelledoutorgroupedtogetherbyadjustingtheexponent,dependingonthesituation.Also,termsareoftenfactoredintoprimefactorswhenyoudoafactortree.Formoreonfactoringnumbers,see:PrimeNumbersandThePrimeNumbersMultipleTable.Factoringtermswithvariablesisabitdifferent.ItisexplainedinTheGreatestCommonFactor,afewsectionslater.

TheProcedureforPrimeFactoringForsmallernumbers,findingtwostartingfactorsmaybeeasytofigureout.Butforalargenumber,itmaynotbeaseasy,andthisisaplacestudentrunintotrouble.Inthiscase,youcanuseaprocesswhichstartsfromaneasyplace,butyoumustknowtheprocessandfollowitproperlyandsequentially.Thisprocessinvolveslookingforasmallprimenumberthatthenumberyouarefactoringisdivisibleby(actually,youarelookingforthesmallestprimenumberthatthenumberyouarefactoringisdivisibleby).Thisistheprocess:Askyourselfifthenumberyouarefactoringisdivisiblefirstbythesmallestprimenumber(2),thenthenextlarger(3),andkeepworkingyourwayupuntilyoufindit.YoumayneedtocheckbacktothelistofPrimeNumbersonceyougobeyondtheprimenumbersyouknowbyheart.Withinthisprocessaresub-processesthatyoushouldapplyasyouworkyourwayuptheprimenumbers.Thefollowingarethosehelpfulsub-processes:

For2:Checktoseeifthenumberyouarefactoringiseven.Specifically,checktoseeifthelastdigitofthenumberiseven.Ifitiseven,thentheoriginalnumberyouarefactoringisdivisibleby2.Thenyoushoulddivideitby2tofindtheotherfactor.Ifitisnoteven,thenthenumberisnotdivisibleby2,norisitdivisiblebyanyotherevennumber:4,8,10,12,etc.Next,testifitisdivisibleby3…For3:Addupthedigitsinthenumberyouarefactoring.Ifthesumofthedigitsisdivisibleby3,thentheoriginalnumberyouarefactoringisalsodivisibleby3.Thenyoushoulddivideitby3tofindtheotherfactor.Ifthesumofthedigitsisnotdivisibleby3,thentheoriginalnumberisalsonotdivisiblebyanyothermultipleof3suchas6,9,12,15,18,etc.Next,testifitisdivisibleby5…

For5:Ifthenumberyouarefactoringendsineither5or0,thenitisdivisibleby5.Ifdoesn’tendin5or0,thenitisalsonotdivisiblebyanyothermultipleof5.Next,testifitisdivisibleby7…For7andhigher:Takethenumberyouareattemptingtoprimefactor,divideitby7,andanalyzethequotient.Ifthequotientcomesoutasawholenumber,thentheprimenumber7youdividedbyisafactor,andthequotientistheotherfactor.

Ifthequotientcomesoutasadecimal,thenthenumberyoudividedisnotafactor.Repeatthisprocessbyattemptingtodividebythenexthigherprimenumberuntilyougetaquotientwhichisawholenumber.Youmayneedtorepeatthisprocessonthewholenumberquotientuntilyouendupwithallprimenumbers(factors).YoumightalsorefertomyTableofPrimeNumberMultiplestoexpeditetheprocess.Onceyouhavefoundallthefactorsneeded,orhavecompletedthefactortree,presentyourfactorsaccordingtoyourbookorteacher’sinstructions.

ThePrimeNumberMultiplesTable

Whenstudentslearnmultiplication,theyaresometimesgivenatableshowingthenumbers1-10(orhigher)alongthetop(row)anddownalongtheleft(column),toeasilyfindtheproductoftwonumbers.Studentsarealsotaughtprimenumbers,especiallywhenlearningto(prime)factornumbers.Asintheexampleof“51”discussedpreviously,theretendtobesomehighernumberswhicharenotprimenumbers,butmayeitherappeartobe,orstudentsgiveuptryingtofigureoutiftheyarethroughdividinguptheprimenumberslist,accordingtothesuggestedprocedureforfactoringlargenumbers(andfindingwhattheyaredividedby).SoIcreatedahelpfultoolIcallaPrimeNumberMultiplesTable.Itisimportantyouunderstandwhatitisandhowtoreadit.Thistableismeanttobeanextensionfortheprocedureusedtoprimefactoranumber,accordingtoTheProcedureforPrimeFactoring.

3 7 11 13 17 19 23

3 9 21 33 39 51 57 69

7 21 49 77 91 119 133 161

11 33 77 121 143 187 209 253

13 39 91 143 169 221 247 299

17 51 119 187 221 289 323 391

19 57 133 209 247 323 361 437

23 69 161 253 299 391 437 529

29 87 203 319 377 493 551 667

31 93 217 341 403 527 589 713

37 111 259 407 481 629 703 851

41 123 287 451 533 697 779 943

43 129 301 473 559 731 817 989

47 141 329 517 611 799 893 1081

53 159 357 561 663 867 969 1173

First,youwillnoticethatthenumbersacrossthetoprowanddownthe

leftcolumnareprimenumbersonly.Also,noticethattheprimenumbers2and5arenotshowninthetable.Thatisbecauseitwouldbeawasteofspacebecauseyoucaneasilytellifanynumber,nomatterhowbig,isdivisibleby2(becauseitwouldbeeven)or5(becauseitwouldendin0or5).Eventhoughthemethodfordeterminingifanumberisdivisibleby3issimple,Iincluded3anyway,becauseyoucan’talwaystellifanumberisamultipleof3justbylookingatit,thewayyoucanforanevennumber.

Ifandwhenyouareattemptingtofactoralargenumber,youshouldlookforthatnumberinthistable.Ifyoufindit,thenyouautomaticallyknowitisaproductofthetwoprimenumbersfromthetoprowandleftcolumnitcomesfrom.Ifyournumberis:

lessthan1173,notfoundinthistable,(andnotamultipleof2,3or5,asyoushouldhavecheckedinthebeginning),

thenyournumbermustbeprime.Ifyournumberisgreaterthan1173,thenyou

mustcontinuecheckingit(bydividingit)withprimenumbersbeyondtheprimenumber53.

Inshort,thistablewillsaveyoualotofguess-workandtrial&error.Althoughthisismeanttobeatime-saverandatooltohelpyoufamiliarizeyourselfwithproductsofprimenumbers,youstillmustbeabletofigureitoutthelongway(withoutthetable),asI’msureyouwon’tbepermittedtousethistableorbookduringatest.

TheGreatestCommonFactor(GCF)TheGCFisthelargestfactorthatcanbefactoredoutofeveryterminvolved.Anotherwaytothinkofitis:thebiggestfactorthateachtermcanbedividedby,withoutresultinginafractionordecimal.TheGCFisfoundandusedintheoverallsimplificationprocessfor:

Reducingfractions,and/orFactoringaseriesofterms,

Morespecifically,theGCFismainlyfoundandusedfortworeasons:

1. Tocancelcommonfactorsinafractiontoreducethatfraction.Inthiscase,youwouldfindthegreatestfactorthatiscommoninthenumeratoranddenominatorandproceedtocancelthemout(to1).

2. ToextracttheGCFoutofaseriesofterms,which,onceyoufindit,

youfactoritoutofeachterm,andtheGCFgoesoutside(usuallytotheleftof)asetofparentheses.

TheLeastCommonDenominator(LCD)TheLeastCommonDenominator(LCD)isalsoknownorusedinsomecontextsastheLeastCommonMultiple(LCM).Actually,allLCDsareLCMs,butLCDisjustspecifictodenominators.BooksusuallyintroducetheconceptandprocedureforfindingtheLCMinpreparationforlearningLCDs.Inthisbook,IwillrefersolelytotheLCD.TheLCDiscommonlyfoundandusedforthreemainreasons:

1. Toconvertfractionsinto“likefractions”foraddingandsubtractingfractions(including:rationalexpressions),bothofwhicharediscussedlaterinthebook.

2. Toeliminatealldenominators(andtherebyallfractions)inanequationbymultiplyingeachfractionbytheLCD.Inequationswithfractions,thissometimesmustbedoneinordertosolve.

3. ToreduceComplexRationalExpressions.WhendoingproblemsinvolvinganLCD,Irecommendyouwrite“LCD=(thenshowthefactorshere)“onyourpaperandfillinthefactorsasyougatherthem.Thisgivesyouaplaceofreferencetokeeptrackofyourfactorswhileyoulookbackatthefractionsintheproblem.SometimesyoushouldusefactortreestohelpyoufindtheLCD,especiallyiftheLCDisn’tobvioustoyou.Actually,usingfactortreesisagoodhabit.IoftenfindthatstudentswanttoavoiddoingfactortreesbecausetheythinktheLCDismoreobviousthanitoftenis,butthisisamajormistake.YouhaveahigherchanceoffindingthecorrectLCDbydoingfactortrees.FormorehowtofindtheLCDandfactortrees,pleaserefertoyourtextbook,asIdonotgivetheproceduresinthisbook.Butalso,see:TheProcedureforPrimeFactoring.

GCFvs.LCDStudentscommonlyconfusethemeaningsandapplicationsoftheGCFandLCD,probablybecausetheybothinvolvefactorsandbothareusedforsimilarreasons.Bothareusedforsimplificationpurposesandbothcanbeappliedinsomewaytofractions.Inthissection,Iwillgothroughafewbriefhintstohelpyoudifferentiatebetweenthetwo.First,youmustunderstandtheirgeneraluses,asyoucanreadintheprevioussection.Next,let’sbreakdownthewordsandpayattentiontosomecommon,associatedkeywords.ThebestkeywordstoassociatewithGCFsare:“out,”“smaller,”“found”and“division.”

GCFsarealwaysfactoredoutofatermoranumberofterms,tomakethosetermssmaller.Whenusedwithfractions,GCFsarefactoredoutofthenumeratoranddenominatortomakefractionssmaller.GCFsarefactoredoutbydividingtermsbytheGCF.AGCFcanonlybefound;itisnotmade.Itisafactorthatisalreadythere,oralreadywithinthetermsofinterest.IfthereisnoGCFpresent,thenthefractionisalreadycompletelyreduced,orthetermsarealreadycompletelysimplified.

WhereasanLCDmayalreadybethere(ifthelargestdenominatorisalreadytheLCD),butitcanalsobemade…

ThebestkeywordstoassociatewithLCDsare:“bigger,”andsometimes“multiply,”and“made.”Thestartingpointislookingatthelargestdenominator.TheLCDusedmaybethelargestoftheoriginaldenominators(youwouldhavetoevaluatetomakethatrealization).Butifitisnot,then:

TheLCDwillultimatelybebiggerthanalltheoriginaldenominators.TheLCDwillmadebymultiplyingtheappropriatefactors.Thefractionsinvolved,whicharetobeconverted(ifnecessary),willhavetheirnumeratorsanddenominatorsmultipliedtomakefractionswithbiggernumbersthanbefore.

Thesekeywordscanseemabitmixedwheninreferencetosimplifyingcomplexrationalexpressions.Inthiscase,theLCDofallmini-fractionsiseitheralready

there,ormade(bymultiplication),thenmultipliedbyallmini-fractionsinvolved.Thisultimatelyreducesthecomplexrationalexpression,butalongtheway,someorallofthenumeratorsofthemini-fractionsbecomebigger.Lastly,itisimportanthowyouusetheword“find.”IntheGCFcontext,“findit”,means“lookforit.”IntheLCDsense,youlooktoseeifthelargestdenominatorisalreadytheLCD,otherwise,youhavetofind(meaningmake)it.

FRACTIONSItisimportantyouknowhowlongdivisiontranslatestofractionform,sincefractionsareaformofdivision,andyouprobablylearnedlongdivisionfirst.Also,youwillrevisitlongdivisionlaterwhendoing(long)divisionofpolynomials(notcoveredinthisbook).Itisalsoimportanttogettheproperterminologydown,whichisoftenoverlooked.Anexampleofalongdivisionsetupisbelow,withthewordsintheproperplaces.

Thewayyouwouldsaythisis:“Thedividenddividedbythedivisorequalsthequotient.”Sometimespeopleusetheword“in”whendescribingdivision.Inthatcase,itwouldbe“Howmanytimesdoesthedivisorgointothedividend?”Answer:Thedivisorgoesintothedividend(the)quotientnumberoftimes.

Considertheexample:

whichisthesameas:Inthiscase,youmightask:“Howmanytimesdoestwogointosix?”Inthiscase,youaresaying:“Howmanytimesdoesthedivisortwogointothedividendsix?”Answer:Twogoesintosix3times.Whichisthesameas:“Sixdividedbytwoequals3.”Noticehowthedivisoranddividendtranslateintoafractionandlongdivision,andviceversa:

Thedividendisthenumerator.Thedivisoristhedenominator.Andthedenominatorgoesintothenumerator.

ProcedureforAdding&SubtractingFractions1)FindtheLCD.2)Convertallfractionstolike-fractions(unlessafractionisalreadyincorrectform)bymultiplyingthenumeratoranddenominatorofeachfractionbythemissingfactorwhichwillmakethecurrentdenominatortheLCD.3)Totherightoftheequalsign,writetheLCDinthedenominatorandperformtheoperations(additionandsubtraction)ofthenewlyconvertednumeratorsfromtheleftinthenewnumerator,overtheLCD,ontherightoftheequalsign.4)Simplifythefractioncompletely.

Simplifythenumerator,ifpossible,bycombininglike-terms,thenfactorifpossible,andthenreducethefraction,ifpossible.

MultiplyingFractionsStartbyattemptingtocross-cancelcommonfactors.Then,separately,multiplyallthenumeratorstogether,andmultiplythedenominatorstogether.Thenevaluatewhatyouhave.Itisoftentaughtthatfractions,oncecombined,shouldbeexpressedinlowestterms(alsocalledreducedorsimplifiedform).Ifyouproperlycross-cancelledbeforemultiplying,youranswerwillcomeoutinlowestterms.Butsometimesstudentseitherforgettodothisstep,orjustmissasetofcommonfactorstocancelout.Ifyouforget,that’sok,youranswerwon’tcomeoutwrong,butthenumberswillbebiggerandyouwillhavetocontinuetofactor.Evaluateyouranswerandlookfora(greatest)commonfactortocanceloutattheend.Torecap,youcaneither:

Lookforcommonfactorsinthenumeratoranddenominatorfirst,priortomultiplying,andcancelthemout(thisiscalledcrosscancelling)–whichresultsinareduced,moremanageablefraction,oryoucan

Dothemultiplicationfirst,thenfactorthenumeratoranddenominator,andcanceloutcommonfactorslast.

Thetruthis,youcandoiteitherway,andsometimesyoujustendupdoingamixofbothtoachievethecorrect,reducedformofthefraction,whichisfineandnormal.Butideally,itisbettertocrosscancelfirst.Formoreonthis,see:Factoringand:WhatisaFactor?Also,formoreoncrosscancelling,see:CrossMultiplicationvs.CrossCancelling.

DividingFractionsDividingfractionsismuchdifferentthanmultiplyingfractions.Actually,onlythefirstmajorstepisdifferent.Thenitbecomesmultiplyingfractions.Inshort,youmultiplythefirstfractiontimesthereciprocalsofthefractionsbeingdivided.Buthereisamorespecificstep-by-stepprocedure,followedbysomecomments.

1. Simplifynumeratorsanddenominatorsseparately,andeachfractionseparately.

2. Keepthefirst(left-most)fractionthesame(meaning:donotinvertit…actually,youshouldre-writeitasisonthenextlinedown).

3. Invert(flipupside-down)allfractionsthataretobedivided(theywillhaveadivisionsigntotheleftofthematfirst).Oncefractionsareinverted(theyarenowcalledreciprocals)…

4. Changewhatweredivisionsignstomultiplicationsigns(adot,orputeachfractioninparentheses).Manystudentsoverlookthissimplestep.

5. Nowmultiplyallfractions,usingtheprocedureformultiplyingfractions.

Note1:Justtobeclear,whendividingastringoffractions,keepthefirstoneontheleftthesameandflipeachremainingfractionupsidedown,beforemultiplying.Whenafractionisflippedupsidedown,itiscalledthereciprocal.Itisalsoreferredtoasthe“inversefraction”orsimplythe“inverse.”)Note2:Donotattempttocrosscancelfactorsbeforeflippingthefractionsandmultiplying.Savecrosscancellinguntilafterthefractionsareflippedandthedivisionsignsarechangedtomultiplicationsigns.Note3:Ifyouattempttodividenumeratorsacrossthetopanddividedenominatorsacrossthebottom(inthewayyouwoulddowhenmultiplyingfractions),youwillnotice…itworks!However…youarenotencouragedtodoitthatwayforonesimplereason:itcangetverycomplicatedalongtheway,givingyoustrangefractionstomanage,andmanyplacestomakeamistake.Forthisreason,youarehighlyencouragedtocloselyfollowtheprocedureofflipping,thenmultiplying.It’seasier,andifnothingelse,itismuchfaster.

OPERATIONSOFBASESWITHEXPONENTS

MultiplyingBasesWithExponentsWhenmultiplyingnumbersorvariableswithacommonbase,keepthebasethesameandaddtheexponentstogether.Remember…whenmultiplyingfactors(withacommonbase)withexponents,youdonotmultiplytheirexponents;thisisafrequentlymademistake.

DividingBasesWithExponentsWhendividingnumbersorvariableswithacommonbase,keepthebasethesameandsubtracttheexponentofthedenominatorfromtheexponentinthenumerator.Remembertokeepthesignsoftheexponents.Youmaysubtractanegativeexponent,yieldingapositiveexponent.Youmayalsogetanegativenumberastheexponent,whichisfine,butinthefinal,simplifiedformofyouranswer,youshouldn’tleaveanexponentnegative.Ifanexponentisnegative,movethefactor(thebase)withthatexponenttotheoppositepartofthefractionandchangethesignoftheexponenttopositive.Remember,anyandeveryfactorhasanunwrittenexponentof“1”.Alsorememberthatwhenexponentsaddorsubtracttoequalzero,anybasetothepowerzeroequals1.(Reviewthisin:TheUnwritten1,and:PropertyCrisesofZeros,OnesandNegatives).

ExponentsofExponents(a.k.a.PowersofPowers)Whenyoutakeapowertoapower,multiplytheexponents.Rememberthatiftherearemultiplefactors,youmustdistributetheouterexponenttotheexponentofeachfactorintheparentheses,includingthecoefficient.Todistributetheouterexponenttoeachexponentoffactorsintheparenthesesmeansyoumultiplythoseexponents.Remember,avariablewithnoexponentshownreallyistothepowerof1,andmustnotbeforgottentobemultipliedbytheouterexponent.Thisisafrequentlymademistake.Also,whendistributinganexponent,donotforgettoapplythatpowertothecoefficientifthereisone.Thisisanothercommonmistake,oftenforgottenbystudents.Thismaybebecausestudentslookfortheconspicuousexponentswrittenwiththeobviousvariables,butwhencoefficientsdon’thaveexponentsassociatedwiththem,theyarejustshownwithaninconspicuousunwrittenpowerof1.

SOLVINGSIMPLEALGEBRAICEQUATIONS

SolvingaSimpleAlgebraicEquationwithOneVariable(FirstDegree)Thegoalistocompletelyisolatethevariableandtohaveitequalanumber,whichistheanswer.Althoughthismayseemeasy(anditwillbecomeeasierasyoupractice),itcanalsobecomplicated(ifnotjusttedious),andstudentswhoarelearningthisforthefirsttimeoftenunderestimatetheimportanceofdoingthisinanorderly,systematicway.Ifyoudon’tlearntodothisproperly,youwillquicklygetleftbehindinclass.Thisisdoneintwogeneralsteps:

1. Isolatingthe“termwiththevariable,”Usingadditionandsubtraction,andthen

2. IsolatingthevariableUsingdivisionormultiplication.

Thefollowingisachronologicallistofdetailedinstructionstohelpyou.1.Ifthereareanydenominators,findtheleastcommondenominatorandmultiplyalltermsonbothsidesbytheLCDtoeliminatealldenominators.2.Simplify:Identifyandcombinelike-terms,ifany.Note:#s1&2areinterchangeable.Youcaneliminatedenominatorsfirstandcombinelike-termsnext.Itisusuallyeasierto“getrid”offractionsfirst,soyoudon’thavetogothroughAdding&SubtractingFractions.3.Isolatethe“termwiththevariable.”Useadditionorsubtractionto“move*”theconstants(non-variablenumbers)totherightoftheequalsignand…

Useadditionorsubtractionto“move*”theterm(s)containingvariablestotheleftoftheequalsign.

*Iuse“move”inacontextwhichindicatestheuseoftheadditionprincipleofequalityinwhichyouaddtheoppositeofthetermyouwanttomove(becauseaddingoppositesequalzero,cancelingoutaterm),andwhatyouadd/subtracttoonesideoftheequalsign,youmustdototheothersidetomaintaintheequality.

4.Simplifybycombininglike-terms.Thereshouldbeoneterm(thetermwiththevariable)ononesideandanumberontheotherside.5.Isolatethevariable:Multiplybothsidesbythereciprocalofthecoefficientinfrontofthevariable.Ifthecoefficientinfrontofthevariableisnegative,youshouldmultiplybothsidesbythenegativereciprocalinordertoeliminatethenegativesignandmakeyourisolatedvariablepositive.Youshouldnowbeleftwithavariableequalinganumber.

Arrangement:DescendingOrder

Itisalwaysbesttoputalltermsindescendingorder–fromhighestpowertolowestpower,fromlefttoright.Thisorganizationfacilitateseasiersimplification.Putalltermsindescendingorder,eventermswithinparenthesesandgroups.

Onereason(descending)ordermattersisforfactoring.Itiseasiesttofactorpolynomials(liketrinomialsintobinomials)whenyouseethetermsindescendingorder.Itwillalsohelpyouidentifyandcanceloutcommonfactors(whentheyarepolynomials)whenthefactorsinsidetheparenthesesareindescendingorder.

Anotherreasontermsneedtobeindescendingorderisforlongdivisionofpolynomials.Becauseofthesystematicprocessoflongdivision,thedivisoranddividendmustbothbeindescendingorder.

Expressionsvs.EquationsItisveryimportanttoknowthesubtledifferencebetweenanexpressionandanequation.Simplyput,expressionsarenotequations.Expressionsarecombinationsoftermsandoperationsymbolswithnoequalsigns.Sinceexpressionsdonothaveequalsigns,theycannotbesolved,theycanonlybesimplified.Equationsaresolved.Booksoftenfocusonexpressionstostressandpracticesimplification.Thisisnecessary(althoughsometimesmisleading…I’llexplainwhyshortly),becauseequationscontainexpressions.Equationsaremathematicalsentencesthatcontainanexpressionorexpressions,andequalssigns,andcanbesolved.Thestepstosolvinganequationinvolvesimplificationoftheexpressionswithintheequation.AsIwassayingabove,booksfocusagreatdealonexpressions,whichisfine,butthisiswhyitcanbemisleading.Thebooksgobyabottom-upapproachandnarrowfocusonsimplifying(factoring)expressionsonly,beforeincorporatingthoseapplicationstowardssolvingequations.Whathappensis:studentsgetinthemindsetofsimplifyingorfactoringanexpression(only),andstopping,that(laterduringsolving)aftersimplifying,theyforgettosolvetherest,usuallynotmorethantwosimplestepsfromtheendpoint.Thisisespeciallyevidentwhenstudentsaresupposedtosolvequadraticequations.Acommonmistakeisthatstudentswillsuccessfullyfactorthetrinomialinanequationbutthenforgettosolveforthevariables.Sokeepthisgoalinmind:Factoringanexpressionisonlypartofsolvinganequation.Onceyousuccessfullyfactoranexpression,getinthehabitofcontinuingontosolvetheequation.Thosestepsarediscussedinthenextsection.Author’sNote:Ifitwereuptome(andsomeday,Ihopeitis…Ihopetowriteanentirealgebratextbook),thelessonsonlearningfactoringandsolvingwouldbeconsolidatedintoonelesson,tobetterconnectthereasonsforlearningfactoringtosolvingequationsandgraphing.Inthemeantime,Ihopethisbookhelpsyourealizethattheseconceptsarecloselyconnectedandnotjustseparateentities.Belowaresimpleexamplesofanexpressionandanequation.Noticethesmalldetailswhichsetthemapart.Anexpression:3x2+x-10

Anequation:3x2+x–10=0,orAlsoanequation:y=3x2+x-10

LINEAREQUATIONSAlinearequationisanequationofthefirstdegree;itproducesastraightline.Linesaregenerallyknowntohave:

aslope(m),oney-intercept(b),onex-intercept(thereisnosymbol,butthex-interceptisxwheny=0),andthe(slope-intercept)form:y=mx+b.Itcouldalsobeinstandardform.

However,therearecircumstancesinwhichtheywillnothaveallofthesecriteria.Iwillsummarizetheseandthethreetypesofstraightlinesnext.

ADiagonalLine:Adiagonallinewillhaveay-intercept,anx-intercept,andaslopeofanythingotherthanzero(orundefined).Itwillbeintheformofy=mx+b(whenitisinslope-interceptform).

AHorizontalLine:Ahorizontallinewillbeintheform:“y=anumber.”Itwillhaveay-intercept,andmorespecifically,theequationwillbe:y=they-intercept.Forinstance,ifthey-interceptis3,theequationofthelinewillbe“y=3”.Studentsalsomistakenlythinktheequationforahorizontallinewillbeintheform“x=”(Ithink)becausetheyassociatethe“x-axis,”with“horizontal.”Buttheoppositeisthecase,asexplainedabove.Itisworthnotingherethattheequationforthex-axisis“y=0”becauseitintersectsthey-axisat“y=0”.Also,ahorizontallinewillhaveaslopeofzeroandwillnothaveanx-intercept.Studentsoftenmistakenlysayhorizontallineshave“noslope”(becausetheslopeiszero),butthisisincorrect.“Noslope”doesnotmean“zero”.Formoreonthis,see:TheSlopeEquation,andWheny1=y2.

AVerticalLine:Averticallineisintheformof“x=anumber.”Averticallinewillhaveneitheray-interceptnoraslope,butitwillhaveanx-intercept.Morespecifically,theequationwillappearintheform:x=thex-intercept.Inotherwords,ifthelineintersectsthex-axisat-5,thentheequationforthelinewillbe“x=-5”.Studentssometimesmistakenlythinktheequationofaverticallinewillbeintheform“y=”(Ithink)becausetheyassociatethey-axiswithbeingvertical.Itisworthnotingherethattheequationofthey-axisis“x=0”becauseitintersectsthex-axisat“x=0”.Anothercommonmistakeistouse“zero”and“no-slope”interchangeably,buttheyaresignificantlydifferent.Averticallineliterallyhasnoslope(notevenzero).Itcanalsobesaidthattheslopeofaverticallineis“undefined.”Formoreonthis,see:Whenx1=x2,and:WhatDoes"Undefined"Mean?Theconceptofhorizontalandverticallines(andtheirequations)issomethingthatstudentsoftenhavetroublewithatfirst,perhapsbecausethebooksseemtogivethemasmallsectiondisplacedfromthemoreemphasizeddiagonallines(whichisjustinevitable).Nevertheless,agoodwaytogainastrongerunderstandingofhorizontalandverticallinesistographthem(thatwayyoucanseethem,andweallknowwhathorizontalorverticallooklike),andtohaveagoodunderstandingofcalculatingslope,asshownin:TheSlopeEquation.

WhatDoes“Undefined”Mean?

Thereareavarietyofcircumstanceswhere“undefined”isusedtodescribetheoutcomeofanequation.“Undefined”cansometimesbeusedinasimilarcontextas“NoSolution,”suchaswhencomputinganoperationthatcan’tbedone,likedividingbyzero.Sometimes“Undefined”isusedfortimesthatacomputationcan’tbedone,butisn’treferringtotheanswerofaproblem.Thiscouldbethecasewhenlookingattheslopeofaverticalline.Theslopeisn’tthe“solution,”so“NoSolution”isn’tappropriate…youwouldsaytheslopeisundefined,orhas“noslope.”

Themostimportantthingaboutunderstanding“Undefined”isnotusingitsynonymouslywith“Zero.”Also,whenyoudoacomputationonacalculatorwhichwouldresultas“Undefined,”yourcalculatorwillshow“Error.”

HowtoGraphaLinearEquationYoucangraphanyequation…sodon’tbeafraidtodoitatanytime!Makingagraph,whetheryouareaskedtoornot,isagreatwaytogiveclaritytoyourproblemoranswer,andisespeciallyagreatwaytohelpyouunderstandaproblemorequationfromamorevisualandconceptualperspective.Graphingalinearequationistheeasiestofallthetypesofpossibleequations.Tomakealine,youneedtwoorthreeormorepoints.Twopointsaretheminimumnumberofpointsneededtomakealine,buthavingathirdpointisbetter.Havingathirdpointisagoodcheckmechanismbecauseifthethreepointsdonotfallintothesameline(andinstead,makeatriangle),youknowatleastoneofthepointsiswrong,andyoumustgobackandcorrectit.Ifthisisthecase,Irecommendstartingyourtableofpointsover,sinceyouwon’ttrulybeabletotellwhichpoint(ifnotmultiplepoints)iswrong.Also,themorepointsyouhave,themoreaccurateyourlinewillbe.

Whendealingwithlinearequations,rememberthis:Whenindoubt,makeagraph.Tomakeagraph,makeatableof3ormorepoints.Usethefollowingprocedure.First,drawthetable,thenfillin“0”forthefirstx,“0”forthesecondy,and“1”forthethirdx,asshownbelow.x y0 01 Thentakeeachnumber,substituteitintotheoriginalequationandsolvefortheothervariable.Thiswillgiveyouthreeimportantpoints:(x,y)(0,)� they-intercept,alsoknownasb,orasapoint(0,b)(,0)� thex-intercept(1,)� anothereasypointtofind,neartheoriginSometimes,thesepointsoverlap,suchaswhenthex-interceptandy-interceptarebothat(0,0);orwhenthey-interceptis(1,0).That’sfine.Justmakeanotherpointonthetable.Mynextchoicewouldbetoputin“1”fory,thensolveforx.Youcanreallychooseanystartingnumberforeitherxory,thensubstituteitinandsolvetofinditscounterpartvariable.Hereisanotherrelatedpieceofadvice:Ifyourslopeisawholenumber,writeit

over1.Forinstance,ifyourslopeisfoundtobem=3,writeitas x,becausethiswillremindyouthatthereisariseandarunwhenyoudrawthegraph.Ifyouhavetwoequations(andtheirlines)tocompare,besuretomaketwoseparatetablessoyoucandifferentiatewhichpointsbelongwithwhichline.Thiscouldbeusefulforsolvingasystemoftwolinearequations.

TheSlopeEquationOnemajorcomponentoflinesandgraphinglinearequationsistheslope.Thefollowingshowsalltheinterpretationsofslope:

Thesymbol � isthecapitalGreekletter“D”whichstandsfor“thechangein,”commonlyusedinmathandscienceequations.

The4ImportantEquationsforLinesTheseequationsshouldbememorized,namesincluded.1.Slope-InterceptForm:y=mx+b

2.StandardForm:ax+by=c,wherea,b&care#s,includingpossiblyzero.

Note:the“b”hereisnotthesame“b”(they-intercept)asintheslope-interceptform.Althoughthesameletterisusedineach,theyareusedincompletelydifferentcontexts.Theletters“a”and“b”aretypicallyusedtorepresentcoefficientsinfrontofvariables.AlsoNote:AstandardformlinearequationisslightlydifferentthanaStandardFormQuadraticEquation.

3)Slope: fromthepoints:(x1,y1)&(x2,y2)4)Point-SlopeFormula:y–y1=m(x-x1)ImportantcommentaboutthePoint-SlopeFormula:Keepyasyandxasx!Donotattempttosubstitutevaluesinforthosehere!Youneedthemtoremain(asletters)totheendoftheprocess.Thepurposeofthisformulaistosubstituteonlyvaluesinforx1&y1(fromapoint)andthevalueofm,andthenrearrangeitintoy=mx+b,wheremandbwillbenumbers.Lookatthename…it’sthePOINT-SLOPEformula…don’toverlookthename!Youneedone(x,y)pointandtheslopetosubstituteintoit,whichcanberearrangedintoy=mx+b.Sometimes,youwillbegiven2ormorepointsandnoslope(m)andwillbeaskedtofindtheequationofaline(asy=mx+b).Inthiscase,youmustfirstcalculatembyusingthetwogivenpoints(or,ifmorethantwoaregiven,youmustselectanyrandomsetoftwopoints)toputintotheslopeformula,andcalculatem.Next,chooseoneofthegivenpointsandputthecorrespondingvaluesinfory1andx1andm(thatyoujustdetermined)intothepoint-slopeformula.Then,usethepropermethods(rulesofequality)torearrangethepoint-slopeformulaintotheslope-interceptformula,y=mx+b.

Whenx1=x2:theslopeisalwaysundefined(andsaidtohave“noslope”),thelineisvertical,andtheequationforthelinewilllooklike“x=#”.

Considerthisexampleoftheequationofalinegoingthroughthefollowingpoints:(4,3)and(4,7).Noticethex-valuesarethesame,both4,sointheequation:

theslope,m,isundefinedbecauseofthezerointhedenominator.Theequationofthelinehereis“x=4”

Wheny1=y2:theslopeisalwayszero,thelineishorizontal,andtheequationofthelinewilllooklike“y=#”

Considerthefollowingexampleofanequationofalinegoingthroughthetwopoints(2,5)and(3,5).Noticethey-valuesarethesame,both5,andinthe

equationforslope: sincethenumeratoriszero,theslope,m,ofthelineiszero,andthelineishorizontal.Theequationofthislinewouldbe“y=5”

Parallel&PerpendicularLinesonaGraphLinesareparallelwhentheirslopesareidentical.Inordertoseethis,youeitherneedto:

Rearrangebothequationsintoslope-interceptformandlookatm,orSimplycalculatemforeachequationandcomparethem.Youcanalsogetagoodideabygraphingandlooking.Ifthelinescross,itwillbefairlyobvious.

Itisnotrecommendedtoevaluatetheslope(m)whenequationsareinstandardform(oranyformotherthanslope-interceptform).Formoreonthis,see:NoSolution-Inconsistent.Linesareperpendicularwhentheirslopesareexactlybothopposite(and)

reciprocalsofeachother.Forexample,ifoneslopeis4,theothermustbe .Formoreonthis,see:OneSolution-Consistent.

SOLVINGASYSTEMOF(TWO)LINEAREQUATIONSBeforecontinuing,therearesomeimportantthingstobeawareof.Asystemofequationsmeans:linesonthesamegraphthatmayintersect.Whatdoesitmeantosolveasystemoftwolinearequations?Tosolvemeanstofindthepointofintersection,whichisliterallyintheform(x,y)…soyou’reessentiallyfindinganxandcorrespondingy.The(x,y)-pointisthesolution(whenthereisasolution,whichtherewon’talwaysbe).Formoreonthis,see:Interpretingthe"Solutions",including:NoSolution,inthenextfewpages.Youalwaysneedasmanyequationsasyouhaveunknownvariablestosolvefor.Here,thereare2equationsand2unknownvariables.(Ifyouhave3unknownvariables,youwouldneed3equationswiththosevariables,etc.).

WhatDoes“SolvingInTermsOf”Mean?Whenrearrangingequations,youwilloftenhearitexplainedas“Solveforonevariableintermsoftheothervariable.”Morespecifically,youmighthearitas“Solveforyintermsofx.”Itisimportantthatyouunderstandthecontextofthewords“intermsof.”Inthebeginningofalgebrawhenyoulearntosolvesimplealgebraicequations,youareusedtosolvingoneequationwithonevariable,andfindingthenumericvalueofthatvariable;thatistheendpoint.Fromthatpoint,studentsoftengetaccustomedto:solving,andgettinganumberforananswer,andbeingdone.Butsometimes,especiallyforamulti-stepproblem(asasystemoftwolinearequationsis),yousolvefor(isolate)avariable,butyoudon’tgetanumber-answer(atleastrightaway).Butthisisok.Somestudentswhoexpecttosolveandgetanumber(instantly)thinktheymadeamistakewhentheydon’tgetanumber.Thisisnotunusual.Thisiswhereonevariableissolvedintermsofanothervariable,meaningneithervariablegoesaway,andyoudon’tsolvetogetanumber…butyouarestillrearrangingtheequationtoisolateavariableofinterest,andwhatevervariable(s)stillremainsisshownontheothersidewiththeotherterms.Tosolveforsomethingintermsofavariablecanbetranslatedandbrokendownlikethis:“Solvefor”=isolate“Something”=whateveryou’reisolating.Itcouldbeavariable.Itcouldalsobeanumberoranentireterm.Whateveritis,getittooneside,andwhateverremainsgoesontheotherside.“Intermsof”=thevariable(s)orterm(s)whichgoontheoppositesideoftheentitybeingsolvedfor.Note:Thistypeofequationrearrangementisnotonlyusedforsolvingsystemsofequations.Youwilloftenalsoseeasmallchapterinyourtextbooksdedicatedjusttomulti-variableequationrearrangement.Theequationsusedareoftenassociatedwithgeometry,trigonometry,statistics,economics,physicsandchemistry.Exercisesinsolvingmulti-variableequationsintermsofothervariablesprepareyouforactualapplicationofthoseequationsintheirrelatedfields.

TheThreeWaystoSolveTherearethreegeneralwaystosolveasystemoftwolinearequations(twolinesinatwodimensionalspace,alsoknownasaplane):1.Graph&Check2.TheSubstitutionMethod3.TheAddition/EliminationMethodAllthreemethodswillyieldthesameoutcome.Iwilldiscussthebesttimetouseeachmethod,especiallywhenitisbettertousetheSubstitutionMethodvs.theAddition/EliminationMethod.Howmanysolutionsshouldyouexpect?Therewilleitherbe:

onesolution(madeupofonexandoney),nosolution,orinfinitesolutions

Thesearetheonlypossibleoutcomes.Forinstance,therecan’tbetwosolutions.Thisexplanationiscontinuedin:InterpretingtheSolutions,inwhichthesituationsandassociatedvocabularyforthesesolutionsareexplainedinmoredetail.Butthenextthreesectionsgiveacloserlookateachofthethreesolvingmethods.

Graph&CheckSometimesthismethodisbrokenupseparatelyintoCheckorGraph,andsometimestheymust(orcan)bedonetogether.OftentheCheck-onlymethodisintroducedfirst.Inthiscase,anorderedpair(anx,ypoint)willbegiventoyou.Tocheck,youmustsubstitutethex-valueinforxandthey-valueinforyintobothequations,thensimplify.Yourgoalistodetermineifthepointisorisnotasolution.Inorderforthepointtobeasolution,eachequationwillreducetoanumberthatequalsitself,suchas3=3.Butifonlyone(orneither)simplifiestoanumberthatequalsitself,thentheproposedpointisnotasolution.Youwillknowwhenthepointisnotasolutionbecauseanequationwillsimplifytoanumberthatappearstoequaladifferentnumber,whichrevealsitselftobeablatantinequality,andthereforenotasolution.Inanotherrelatedinstance,thepointitselfisn’tgiventoyou,butapre-drawngraphis.Inthiscase,youareexpectedtoreadthepointofintersection,andthencheckthatpointasexplainedinthepreviousparagraph.Oftentimes,apre-drawngraphwillbemadetohaveanobviouspoint,andbyobviousImean“integers,”notsomeobscuredecimalnumbers.Oryoumayhavetodoeverystep:createthegraph,interpretthepointofintersectionandcheckit,aspreviouslydescribed.Youcouldbegiventwosetsofpoints(twoorthreesetsofpointsforeachofthetwolines),whichyouwillhavetoplotandgraph.Youmightalsojustbegiventwoequationsandbeinstructedto“findthesolution,”inwhichcaseyoumust:

Determinethreepointsforeachgraph(soatotalof6points),Plotthepointsandsketchthetwolines,Makeajudgmentastothepointofintersection,thenCheckthepointbysubstitutingthevaluesintoeachofthetwoequations,simplifying,andinterpretingtheoutcome.

Afewreminders:

Usetheprocedurefor:HowtoGraphaLinearEquation.Uponmakingatableofpointstoplot,takenoticeofanyx-ypointsthatarethesameinbothtables.Ifyoufindamatchingpair,thatisyoursolution.Youshouldprobablystillgraph,thenchecktoconfirm.Neatnessisessentialwhengraphing.Trytousegraphpaper.Ifyoudon’thaveany,considerusingastraightedge.Butmoreimportantly,tryyourbesttodraweachunitonyourxandyaxeswithequal

lengths.Thiswillmakeyourpointofintersectionmoreaccurate,andwillgiveyouabetterabilitytofindthecorrectpoint,whichshouldthenallowittosuccessfullycheck.Also,evenwhenyouarenotrequiredtoGraph&Check,youstillcanifyouwanttodoublecheckyourresultsfromtheSubstitutionMethodortheAddition/EliminationMethod.

TheSubstitutionMethodThesubstitutionmethodisstartedinoneoftwoways.Onewayisbytakingoneequationandsolvingitforonevariable.Whendoingthis,aimforthevariablethatwillbemosteasilyisolated.Agoodwaytoidentifythebestvariabletoisolateisbyfindingatermwitheitherasmallcoefficient,oratermwithacoefficientthattheothertermsintheequationwillbeeasilydivisibleby.Forinstance,ifoneofthetermsis“2y”andtheothertwotermsareeven(perhapstheyare6xand-8),thensolvingforyisagoodchoicebecausetheothertermsare(easilyandnoticeably)divisibleby2.Sometimesatermalreadyhasnocoefficient(meaningithasanunwrittencoefficientof1).Inthiscase,it’sagoodideatoemploytheSubstitutionMethodbecausepartoftheworkisdoneforyou(thatpartbeingtomakeitscoefficient1).Allyouhavetodothenisisolatethatvariable.Therearemanysystemsofequationswherethisisthecase.Oftentimes,thewritersofthemathproblemssetyouuptonoticethisvariable.Sometimes,thevariableisevenalreadyisolated,soallyouhavetodoisrealizethat,thenproceedtosubstitutewhatitequalsinforthatvariableintheotherequation.Onceyouhavesolvedforoneofthevariablesintermsoftheothervariable,youmusttakethatequalityandsubstituteitintothevariableforwhichyoujustisolated,intheotherequation.Forinstance,ifyoujustsolvedforyinoneequation,thenyoumusttakewhatyequalsandsubstitutethatinforyintheotherequation.Whenyoudothis,youshouldtakenoticeofthreethings:

1. Allthevariablesintheequationyoujustsubstitutedintowillbethesame.Itisonlywhentheyarethesamethatallowsyoutosolveforthenumericvalueofthatvariable.

2. Onceyoudothesubstitution,youwillbesolvingforthenumericvalueoftheothervariable.Forinstance,ifyouoriginallysolveforyintermsofxandthensubstituteinforyintheotherequation,youwillthensolveforthenumericvalueofx.Also,

3. Becarefulnottomakethecommonmistakeofsubstitutingintotheequationyoujustsolvedforinthefirststep.Ifyoudo,thenonceyousimplify,youwillendupwithanumberthatequalsitself,andthismayleaveyouconfusedandwonderingwheretogofromthere.Soremembertosubstituteintotheotherequation.

Youarenowonestepawayfromcompletingthisproblem,butstudentssometimesgetconfusedatthisfinalstep.

Thelaststepistotakethenumericvalueofthevariableyoujustsolvedforandsubstitutethatbackinforthatvariableintoeitheroftheoriginalequations,thensolvefortheothervariable.Forinstance,ifyoujustfoundthevalueofxtobe-5,substitute-5backinforxinoneoftheoriginalequations,thensolveforthevalueofy.Ihavetwocommentsaboutthis:

Studentsareoftenconfusedby:WhichoftheoriginalequationsshouldIsubstitutemyvaluebackinto?Andtheansweris:either.Sometimesthechoiceseemstoconfusestudents,sohere’showyoucanchoose.Youcaneither

justrandomlypickone,or

Choosetheequationwhichappearstosimplifyeasier.Theonethatwillbeeasiertosimplifymaybetheonewithsmallercoefficientsortheonethatdoesnotcontainfractions.Ifbothlooklikesimilardifficulty,justchooseonerandomly.Theanswerwillcomeoutthesame.

Sometimesstudentsforgetthefinalstep,perhapsbecause,uptothispoint,youareusedtoaone-numberorone-variableanswer.Don’tforgetthisstep.Remember,thesolutionisapoint(anxanday).Themostcommonmistakemadebystudentsusingthismethodisgettingconfusedaboutwhattosubstitute.Soinsummary,yousolveforonevariableintermsoftheother…leavingyouwithonevariableisolatedononesideoftheequalsign,andtheothertwotermsontheotherside.Youthensubstituteinforthevariableyoujustsolvedforintotheotherequation.The“stuff”yousubstituteintotheotherequationwillreplacethevariablewiththetwotermsfromtheothersideofthefirstequation.Youmustthendistributeandsimplifyinordertosolveforyourfirstvalue.

TheAddition/EliminationMethodSomebookscallthis“TheAdditionMethod”andsomebookscallthis“TheEliminationMethod.”Icallitahybridofboth,becauseyoustartbyaddingthetwoequations(afteranynecessaryconversions),whicheliminatesonevariable,makingitanew,one-variableequationthatcanbesolved(fortheothervariable).Remember,toperformthismethod,termsofthesamevariableineachequationmustbeoppositessotheycancelouttozerowhen(theequationsare)addedtogether.Butmyfocusistotellyouwhenitisadvantageoustousethismethod.Herearesomecluestolookfor:

Younoticetwotermsofthesamevariableineachequationwhicharealreadyopposites[meaningsameterm(variableandcoefficient)butoppositesigns].Thesearealreadysetuptocanceleachotherouttozerooncetheequationsareadded.Allyouhavetodoisaddtheequations,thenproceedtothenextstep.

Younoticethatonetermisonemultipleawayfrommakingittheoppositeofatermofthesamevariableintheotherequation.Forinstance,ifoneterminoneequationis3x,andtheotherequationhasa-9x,then3xcanbecome+9xbymultiplyingitby3(anddon’tforgettomultiplythatfactorthroughbytheothertermsinthatequation).Oryouhavetheterm-5xinoneequationand-5xintheother.Youmustmultiplyoneoftheequationsthroughby-1,tomakea-5xbecome+5x.Or,wheneverneitherequationisgivenwithavariablewith(anunwritten)coefficientof1…mainly,theoppositeofwhatisexplainedforTheSubstitutionMethod.Inotherwords,allvariableshavecoefficients(otherthan1).

CommonMistakes:Averycommonmistakestudentsmakeisaddingthetwoequationswithoutcheckingandconvertingoneequation(tomanipulateonevariableintotheoppositeofatermfromtheotherequation).Ifyoudonotproperlysettheequationsuptohaveoppositeterms,thenaddingtheequationswilljustgiveyouathirdequation,stillhavingtwovariables.Studentsoftengetstuckhere,andrightfullyso,becausethisisadead-end;there’snothingyoucandowithit.Anothercommonmistakestudentsmakeisforgettingtoaddtheconstantswhentheequationsareadded.Theconstantsarethenumberswithnovariablesattached.Don’tforgettoaddthem,astheyarejustasmuchapartoftheproblemasthetermswithvariables.

ExamplesforChoosingtheMethodInthissectiontherearetwo“systemsoflinearequations”given:SystemAandSystemB.Eachequationisalreadysimplifiedandputintostandardformforthistypeofproblem.Iwantyoutoexamineeachset,andusingthecluesexplainedintheprevioustwosections,determinewhichmethod(SubstitutionorAddition/Elimination)wouldbebesttouseineachcase.Theanswersandexplanationswillbegivenonthefollowingpage.The“solutions”tothesystemwillalsobegivenincaseyouwanttodotheproblemforpractice.SystemA:Equation1:

Equation2:SystemB:Equation3:Equation4:

SystemAwouldbestbesolvedusingtheSubstitutionMethodbecausethe“y”inEquation2alreadyhasan(unwritten)coefficientof1.Thenextstepwouldbeto

isolatetheybyadding tobothsides,givingyou: .ThensubstitutetheinforyinEquation1.ThesolutiontoSystemAis(-2,-1).SystemBwouldbestbesolvedusingtheAddition/EliminationMethod.Therearetwowaystoapproachthis.First,takenoticeofthe“14x”andthe“-7x”.Ifthe“-7x”ismultipliedby“2,”itwillbecome“-14x”whichistheoppositeof“14x”.YouwouldneedtomultiplyeachterminEquation4by2toproperlyconvert“-7x”into“-14x”.Then,whenyouaddthetwoequations,the“x”termswillcancelouttozero,allowingyoutothensimplifyandsolvefory(andthenx).ThesolutiontoSystemBis(2,-14).

Or,youcouldsolveSystemBanotherway,bymultiplyingEquation3by“2”andmultiplyingEquation4by“3”.Thiswouldmake“3y”wouldbecome“6y,”“-2y”wouldbecome“-6y,”andThey-termscancelouttozerobecause6y–6y=0.AsImentionedbefore,somestudentstakeamistakenapproachtothisfirstbyaddingthetwoequationstogetherwithoutmultiplyingthroughthenecessaryterm(s)tomanipulatetermsofonevariabletocancel.Ifmistakenlyadded,youwouldthenget(whatIwillcall)Equation5:y+14x=8…Noticehowbothvariablesstillremainintheequation?Thisleavesyouatadeadend,becauseyoucan’tsuccessfullyusethisequationtosolveforeithervariable.

Interpretingthe“Solutions”OneSolution-ConsistentWhenthetwolinescross,thisiscalledaconsistentsystem.Inthiscase,thereisonesolutiontobefound,whichisthepointofintersection.Thelinescanbeacombinationofdiagonal,horizontaland/orverticallines.Keepinmind,allsetsofperpendicularlineshaveonesolutionandmakeaconsistentsystem.NoSolution-Inconsistent,ParallelThetwolinesdon’tcross…becausetheyareparallel;parallellinesbydefinitionnevertouch.Thisisaninconsistentsystem.Thereare3waysyoucantellthatlinesareparallel:1.Whenusingoneofthethreemethodsforsolvingasystemoftwolinearequations,whenyousimplifyandgettotheendoftheproblem,youwillgetone#thatdoesnotequaltheother#.Itwilllooksomethinglike:7=-5or0=4,whichclearlyisn’ttrue.

2.Theslopes(m)ofthetwolinesareidentical.Inorderforyoutoseethis,youmustconverttheequationsintoslope-interceptform(y=mx+b),andthensimplylookattheslopes.Theequationsoflinesmayormaynotoriginallybeinslope-interceptform(y=mx+b).Iftheyarenot,convertthemtoslope-interceptformbysolvingfor(isolating)y.

Also,besureeachequationisinsimplifiedform.IfthereisaGreatestCommonFactorinanequation,youmustfactoritout.Ifyoudon’t,theslopesmayappeardifferent,eventhough,byproportion,theyareactuallythesame.

3.Graphandlook.Youcanfindoutiflinesareparallelwithoutgraphing,asdescribedinthelastparagraph.Butthismethod(graphing&looking)shouldactasabackuptothetwomethodsabove,toconfirmyouranswer.Itmayalsobeagreathelpifyouareamorevisuallearner.Forareminderongraphingfromanequation,see:HowtoGraphaLinearEquation.Oncedrawn,lookatthelinestoseeiftheyappeartocross.

InfiniteSolutions-DependentTheentirelinesoverlap…becausetheyareessentiallythesameline.Thegraphactuallylookslikeoneline.Thisiscalledadependentsystem.Thereare3waystotellthis:1.WhenyouusetheSubstitutionMethodforsolvingasystemoftwolinearequations,theequationyousubstitutedintowillsimplifytosomethinglikethis:4=4,or

-8=-8,or0=0.Itwon’tevenletyougettothepointwhereavariableequalsanumber,revealingthatthesystemisdependent.Note:Whenthishappens,studentstendtothinktheymadeamistakebecausethisoutcomeseemssoawkward,butusuallytheyhaven’tmadeamistake…it’ssupposedtoturnoutthiswaytoindicatethatit’sadependentsystem.

2.Whenreducedtosimplesttermsandconvertedtosameform,theequationsareidentical.Ifyouaregoingtocompareequations,theymustbothbeinthesameformaseachother(eitherslope-interceptformorstandardform).Note:Oftentimes,theseequationsmaylooksimilarbeforetheyarecompletelysimplified.Iftheyareinthesameform,youmaynoticethecoefficientsaredifferent,yetproportional.Thiscanbeasignthatdividingoneorbothequationsthroughbyacertainfactorwillthenrevealtheequationstobeidentical.ThisiswhyitissoimportanttotrytosimplifyanyequationbylookingtofactoroutaGCFandarrangingintostandardforminthebeginningofeveryproblem.Doingthisherewouldinstantlyrevealthatthesystemisdependent.3.Graph&Check–Graphbothlinesandlookatthegraph.Itshouldbeprettyobviousthatthelinesoverlap.Actually,itwilljustlooklikeoneline.

TRINOMIALS&QUADRATICSThewords“trinomials”and“quadratics”areoftenusedinterchangeablybecausetheyoverlap,bothincharacteristics,looksandapplication(particularlyduringfactoringandsolving).Despitetheirsimilarities,theyshouldnotbeseenascompletelysynonymousbydefinition.Becauseofthewaymanybooksandlessonsarearranged,sometimestheseareseenandusedtoodisconnectedlyorseparately.Thisisunderstandableaswell(whendonecorrectly),howeverthismayalsomisleadstudentstomisstheimportantconnectionandoverlapbetweenthem.Thissectionistohelpyouclearlyrelateanddifferentiatethesimilaritiesanddifferencesbetweenthem,bydefinitionanduse.

Atrinomialisanexpressioncontainingthreedifferentterms,oftenwithatleastonesquaredvariable.Isay“often”becausewhenyouareintroducedtofactoring(usingtheTrial&ErrororReverse-FOILmethod,forinstance),youarefactoringtrinomialsintobinomials.Bydefinition,trinomialscanbecomprisedofanythreetermstoanypower,buttrinomialsareveryoftendirectlyassociatedwithfactoringintotwobinomialsasthesegue-waytosolvingquadraticequations.Atrinomialsometimesoverlapsasaquadraticexpressionandmaybepartofaquadraticequation.Althoughanexpressionmaybebothatrinomialandaquadraticexpression,theyarenotsynonymousbydefinition.Atrinomialisaquadraticexpressionwhenthehighestpower(degree)ofanytermis2.Whentheydooverlap,theycanbesimplified(factored)theexactsameway.Also,notallquadraticexpressionsaretrinomials,asyouwillreadnext.

Aquadraticequationis:-Anequationcontainingasquaredvariable(likex2),yieldingamaximumoftwosolutions[butcouldcontainonesolution(thatoccurstwice),ornosolution].-Itmustcontainasquaredvariableandthusisconsidereda2nddegreeequation.-Althoughaquadraticequationcancontainatermofx(totheunwrittenpowerof1),itcannevercontainatermofapowerhigherthan2.-Also,aquadraticequation,whengraphed,alwaysmakesaparabola(aU-shapedcurve).Aquadraticequationappearsinthestandardform:ax2+bx+c=0

Thereareafewthingsyoushouldunderstandabouttheequationwrittenabove.Itmayalsobewrittenas:y=ax2+bx+c=0,inwhichy=0,asabove,or:f(x)=ax2+bx+c=0,becausequadratics(whichmakeparabolas)areconsideredtobe“functions.”Specifically,theyarefunctionsofx.(Idonotdelveinto“functions”inthisbook,butifyou’rewondering,anequationisconsideredtobea“function”ifitsgraphcrossesthey-axisonce.)Itiswrittenindescendingorderandstandardforminthiscase.Foraquadraticequation,“standardform”meansalltermsareononesideoftheequalsign,andsetequaltozero(ontheotherside).Descendingordermeansthetermsarearrangedfromthehighesttothelowestpower,fromlefttoright.Allquadraticsarenotalwaysoriginallypresentedindescendingorderorstandardform.Ifandwhentheyarenot,youshouldrearrangeeachoneintostandardformanddescendingorderbeforesimplifyingandsolving.Thelettersa,bandcarerepresentativeofnumbers,notvariables(moreonthatdownthepage).Also,“a”cannotbezero.If“a”iszero,itisnolongeraquadraticequation;itisthenalinearequation.

Aquadraticequationcontainsatrinomialexpressionwhena,bandcareallnon-zeronumbers.However,sometimes,eitherthecoefficientb,orconstantc,orboth,arezero(remember,“a”cannotbezero).Thisisworthhighlightingbecausethisiswherestudentsoftenrunintotrouble.Ibelievetheyrunintotroubleatfirstbecause,whenborciszero,theequationsjustlookdifferently,andusuallythesolvingmethodisdifferent.Forthatreason,thereasegmentdedicatedtothosespecificcases.Iwillshowwhattheylooklike,explaintheirgraphicalsignificance,howtosolvethem,andtheirexpectedsolutions.Thisiscontinuedin:QuadraticsWithZero.Butfirstyoushouldunderstandthesolutionstoquadraticequations.

WhatAre“Solutions”toQuadraticEquations?

Itisgoodyouknowwhatsolutionstoquadraticequationsare,togiveyouabetter,overallperspective.(Thevariableofaquadraticequationisusuallyx,butcanbeotherletters).Whenthevariableisx,thesolutionsare“x-intercepts,”whicharesimplythepointsonagraphwheretheparabolacrosses(orthesinglepointwhichtouches)thex-axis;thex-interceptsaredefinedthesame,nomatterwhattypeofequationorgraphtheycomefrom.Andx-interceptsarepoints(orderedpairs)atwhichy=0,whichiswhyyousetyourquadraticequationequaltozeroatfirst(inotherwords,makingsureitisinstandardform).Thisisalsowhy,ifyousolvebyfactoring,youseteachfactorequaltozero;or,ifyousolvebythequadraticformula,thisiswhytheformulaissetequaltozero.Youwilllearnaboutthismoreinthenextsection.

Allquadraticequationsproduceaparabolawhengraphed.Thesolutionsarethex-interceptsofthegraph.Asyouwillseeinthenextfewsections,thereareanumberofwaystosolvequadraticequations.Ifyousolvebyfactoring,youwillgetoneortwosolutions;thosesolutionswillbeeitherintegersorfractions.Ifyousolvebythequadraticformula,youranswersmaycomeouttobeintegers,fractionsorradicals(whichcouldbeconvertedtodecimalsforgraphing).Also,youmayfindthataquadraticequationhaseitherone,two,or“noreal”solutions.Hereisaquicksummaryofeachscenario:

Onesolutionmeansthattheparabolaonlytouchesthex-axisonce;itdoesnotcrossthex-axis.Youmaythinkofitas“sittingon”thex-axis.Anothermoretechnicalwaytosayitis,“Thex-axisistangenttothevertexoftheparabola.”

Thiswilloccurwhenthetrinomialfactorsintoabinomialsquared.Itisalsogoodtoknowthatabinomialsquaredcomesfroma“perfectsquaretrinomial.”

Twosolutionsmeanstheparabolacrossesthex-axistwice.Finally,youmayfind“norealsolutions”.Thismeansthattheparaboladoesnotcrossortouchthex-axis,butdon’tbefooled.Justbecauseitdoesn’tcrossortouchthex-axisdoesn’tmeanitdoesn’texist…itstillexists,andcanstillbegraphed.Thisconclusioncanonlybemadethroughuseofthequadraticformula.Ifaquadraticequationisprime(whichonlymeansitcan’tbefactored),thisisstillnotgroundsforsaying“nosolution”…itmayjustmeanthesolutionsareradicals.Butitmayalsomeanthereare“noreal”solutions.Thereasonthisisspecificallyansweredas“noreal”solution,insteadofnosolution,isbecausethisisoftentheresultofthesquarerootofa

negativenumber.Formoreonthat,see:TheSquareRootofNegativeOne,and:Primevs.NoSolution.

Theseconceptsandacloserlookatthesolvingmethodsbehindthemarediscussedinmoredetailinthenextfewsections.

SolvingQuadraticEquationsTrinomialsandquadraticequationscanbesolvedinthreegeneralways:1.Factor&Solve1b.TaketheSquareRootofBothSides2.UsetheQuadraticFormula3.Graph&CheckWhenusingfactoringtosolve,thereareactuallythreedifferentfactoringmethodsyoucanuse,soyoumightsaytherearefiveorsixtotalpossiblewaystosolvequadraticequations,(although,asyouwillsee,thefactoringmethod(s)won’talwayswork).Youalsosee“TakingtheSquareRootofBothSides”inthelistas“1b.”Thisisanalternativemethodtocertaincasesinwhichfactoringcanbeusedaswell.Thereisatypeofequationwhichcanbesolvedintwoways,eitherby“Factor&Solve”orby“TakingtheSquareRootofBothSides”.Thisisexplainedin:Whenbothb&care0:ax2=0.Ire-wrotethelistofwaystosolvequadraticequationsbelow,withthemorespecific,sub-methodsincluded,soyougetaconciselistofmethodsandchoicesyoucanuse.1.Factor&Solve:

Trial&Error/ReverseFOILMethodTheac/GroupingMethodTheCompletetheSquareMethod

1b.TaketheSquareRootofBothSides2.UsetheQuadraticFormula3.Graph&CheckInthefollowingsections,Iwillgooverthewhenasopposedtothehow(seeyourtextbookforthe“how”).Ihavegoodreasonsforthis.Thetextbooksusuallydoagoodjobofshowingyouhowtoimplementthemethods,andthestepsarenotreallythatcomplicated;alotofpracticeisthekeytobecominggoodatfactoringandsolvingquadraticequations.However,thebooksdon’tusuallyansweraquestionmanystudentshave,whichis,“whenisthebesttimetouseeachmethod?”I’mgoingtoanswerthatquestion,aswellasgivemoreofatop-downperspectiveonsolvingquadraticequations.

Whendealingwithquadratics,youshouldalsogetaccustomedtostartingthem(orpreparingthem)thesameway,nomatterwhichmethodyouusetosolvethem.Youshouldalways:

LookforaGCFtofactorout(thisisabigonestudentsoftenforgettodo),andArrangeintodescendingorderandstandardform.

Factor&SolveUsually,youshouldtrytofactorandsolveaquadraticequation(beforeusingthequadraticformula)becauseit’sfasterandinvolvesfewersteps(ifit’sabletobefactored).Factoringandsolvingcanleadyoutotheanswer(s),howeverifyoucan’tfactor,thisleavesyouranswerinconclusive.Ifaquadraticisprime(can’tbefactored),itdoesn’tnecessarilymeanthereisnosolution,butyoumustthenusethequadraticformulatocometothatconclusion(toeitherfindtheanswersorfindthatthereisnorealsolution).Butmanyquadraticequationscanbefactored.Therearethreegeneralwaystofactor,butmoreimportantly,therearebettertimestouseeachmethodandcluestodictatewhenthosetimesare.AlthoughIdonotteachyouhowtodoeachmethod(asIstated,yourtextbooksdoagoodjobatthat),Iwillhighlightthecluesandtellyouthebesttimetouseeachmethod.Beforefactoring,youmustgothroughaseriesofstepstosetupandprepareyourequation,nomatterwhichmethodoffactoringyouwilluse.Theseareveryimportant,andstudentsoftenforgetoneorallofthesebecauseyoudon’talwayshavetodothem:

Simplifyasmuchasyoucanbycombininglike-terms,ifnecessary.ArrangealltermsintoDescendingOrderaccordingas:ax2+bx+c=0.PutintoStandardFormbymovingalltermstooneside(theleft)andsettingthemequaltozero.LookforaGreatestCommonFactor.Sometimesyoucanfactorandsolvesuccessfullyifyouforgettodothis,butitwilloftenleaveallnumberslarger,andtheproblemmoretedious.IftheGCFisanumber,youcanfactoritout,thenremoveit(becauseifyoudividebothsidesbyit,thezerodividedbyitontheothersideeliminatesit,andthezeroremainszero).

However,iftheGCFisavariableorapowerof“x”,itwon’tbeeliminated,butitwillequalzeroasoneofyoursolutions.Factoringoutavariablemayallowyoutoproperlyfactorusingoneofthefactoringmethods(includingtofactoragain)thatyouotherwisewouldn’tbeabletodo.Actually,whentheGCFcontainsavariableorpowerofx,theproblemmightnothavebeenaquadratictobeginwith.Factoringthatoutmayleaveyouwithaquadraticthatyoucanthenfactorbyquadraticmethods.

Makesurethecoefficientoftheleadingterm(the“a”connectedtox2)ispositive.Youcan’tfactorifit’snegative(itcanbenegative,though,whenusingthequadraticformula).Ifitisnegative,treat-1asaGCFofeachterm.Byfactoringitout,youwillsimplychangethesignofeachterm,andthezeroontheothersideisn’taffected.(Optional)Youmaychoosetoeliminateallfractionsfirst,iftherearefractions.YouareusuallytaughtthatitisagoodruleofthumbtobeginanytypeofproblembyremovingfractionsbymultiplyingbytheLCD.Youdon’thavetothough,andsometimesyoucanevenfactorthemintobinomials,butyouwillaccountfortheminthefinalsolvingstepsifyoudon’tremovethemfirst.

Trial&Error/ReverseFOILMethod

Variousbookshavedifferentwaysofnamingfactoringmethods.IuseboththesenamesherebecauseIthinktheyaccuratelydescribetheprocessthey’reusedfor.Todothismethod,youmustsimplify(combinelike-terms),arrangealltermsintostandardform(movealltermsontooneside),andputintodescendingorder.Youthenpayattentiontothefactorsofthefirstandlasttermsofthetrinomial,writeoutthetwobinomials(or,onceyougetgoodenoughatit,keeptheminyourhead)thenFOILthesefactorstoseeifyouarrivebackattheoriginaltrinomial.Ifitworks,you’vefoundyourfactors;thenseteachbinomialequaltozeroandsolve.Keeptryingfactorsuntilyoufindtheonesthatwork(that’swhat’strialanderroraboutit).Also,Icallit“Reverse-FOIL”becauseyouarestartingwiththeproduct(theoriginaltrinomial),thencomingupwithfactorstotry,thenFOILingthemtocheck.ButasIstated,myintentionisnottofocusonthehow,butthewhen.Sowhenisthebesttimetousthismethod?

Itiscommontofirstapproachatrinomial/quadraticexpressionwiththismethodbecause,ifitcanbedone(easily),itcanbethequickestmethodwithfeweststeps.AnequationwithanexpressionofthistypehasthebestlikelihoodtobesolvedbystartingwiththeTrial&ErrorMethodwhen“a”and“c”arerelativelysmall.Iusetheword“small”loosely,andeveryone’sinterpretationof“smallnumbers”maybealittledifferent.Myuseof“smallnumber”couldbetakentwoways.Itcouldmeananumberwithfewfactors,oritcouldbetakenmoreliterally,meaningbetween1andabout15.Eitherway,thesmallerthenumber,thefewerfactorsitwilltendtohave.

Ideally,primenumbersornumberswithfewfactorswillyieldthefewestpossiblecombinationsoffactors.Thefewerfactorcombinationsthereare,thefewerthechoicestherearetotry(multiplyandtest).There’snotreallymuchelsetosayaboutthismethod.Asthe“a”and“c”numbersapproachlargervalues,therecouldbesomanypossiblecombinationsoffactorstotrialthatitcanbecomeverytimeconsuming.Whenthesizeandpossibilitiesoffactorsseemsoverwhelming,thisisagoodtimetodothe“ac/GroupingMethod.”

Theac/GroupingMethod

Youwilloftenfirstbeexposedtothe“GroupingMethod”whenyouarelearningfactoring(beforelearningtosolvequadraticequationsandequationswithtrinomials).TheGroupingMethodissimplyamethodoffactoringthatisintroducedtoteachyouhowtofactorfourterms[ifthereareenoughsimilarities(commonfactors)amongpairsofterms].Booksdon’toftencallthe“acMethod”the“ac/GroupingMethod;”theyusuallycallitoneortheother.Thisac/GroupingMethodiscomprisedoftwomainparts:

Firstusingtheaandctofindthecorrectfactors(bymultiplyingaandc,thenlookingatallthetwo-factorcombinationsofthatproduct,called“ac”),thenWritingoutthefourassociatedtermsandusingthegroupingmethodtofactor(intotwobinomials),

thensolving.Whenisthebesttimetousethismethod?Itisworthnotingthatthereare

oftentwogroupsofstudents:thosewhoprefertheTrial&Error/ReverseFOILMethod,andthosewhopreferthisac/GroupingMethod.ThereasonI’mtellingyouthisisbecauseyoucanalwaysskiptheTrial&ErrorMethodandstartbyusingthismethodeverytime(inotherwords,youdon’thavetotrytheTrial&ErrorMethodfirst,thengoontothismethodnextifTrial&Errordoesn’tworkout)ifyoulikethismethodbetter.Somestudentspreferthismethodbecauseitcanbequickeronaverage,becauseitremovesalotofguesswork(trialanderror)andthetimespentonalltheerrorfactorcombinations.

Regardless,agoodtimetousethismethodiswhentheaandcfactorsarelargeand/orhavemanyfactors.Togiveyouanideaofwhatmightbe“large”or“havingmanyfactors,”ifa=18andc=-24,therecouldbemanyfactorcombinationsfromthem.The“18”hasthreepairsoffactors{(1∙18),(2∙9),(3∙6)},andthe“-24”haseightpairsoffactors{(-1∙24),(1∙-24),(-2∙12),(2∙-12),(-3∙8),(3∙-8),(-4∙6),(4∙-6)}.Thenegativesigndoublesthenumberoffactorcombinations,becauseeitherfactorcouldbenegative.

Startbymultiplyingaandc;thisgivesyoutheproduct“ac,”(itwillbeanactualnumber).Youarethentolookateverypossibletwo-integer-factorcombinationoftheproductac.Youmayconsidersettingthisupinthefollowingway:maketwocolumns:onewiththeheading“factors,”andtheother“b.”Thepointistofindthenumbersthatwhenmultipliedgiveyoutheproductof“a”times“c”,andwhenaddedgiveyouthemiddletermoftheoriginaltrinomial,“b.”Whenthe“a”and“c”numbersofatrinomialarelargeorhavemanyfactor

possibilities,thismethodwillhelpyouquicklyfindthecombinationneededtocompletethe“factoringbygrouping”method.Again,don’tforgettosolveyourbinomialsonceyoufactorintothem.

CompletetheSquareCompletingtheSquareisdefinitelyinacategoryofitsown.Youmaynotevenconsideritfactoringbythesamedefinitionastheotherfactoringmethods,sinceitismore-solikeamanipulationtechnique,involvingfactoringasastep.Thismethodismainlyusedwhenconventionalfactoringdoesn’twork,becausethec-numbermaynotfactorintointegers,butitcanstillbemanipulated.It’simportanttorememberthattobegin,theleadingcoefficientmustalwaysbepositive1,sobeforeproceeding,ifthecoefficientisanythingotherthan1,divideeachtermbythe“a”coefficient,andthiswillmakethecoefficientoftheleadingterm“1”(andtheothertermswillchangeproportionally).Also,asyoumakeyournew“c-number,”don’tforgettoaddittotheotherside,tomaintaintheequality.Studentsoftenforgettodothetwothingsmentionedinthisparagraph.Thiscreatesa“perfectsquaretrinomial”(ononeside).Aperfectsquaretrinomialisaspecialcase,inwhichthecoefficientoftheleadingtermwillbeanunwritten“1,”(whichisaperfectsquare),thenewc-numberyoumadewillbea

perfectsquarenumber[thisnewc-numbermayalsobereferredtoas ,whichistheformulaforhowtomakeit],andthecoefficientbwillbeexactlytwotimesthesquarerootofthenewc-number.Accordingtothisspecialcase,“Aperfectsquaretrinomialfactorstoabinomialsquared.”Butthisslightlydeviatesfromaregularproblemwhereyou’regivenaperfectsquaretrinomialtosolve.Aperfectsquaretrinomialwillfactorintoabinomialsquared,andwhensetequaltozeroandsolvedwillgiveyouoneanswer.Thisisbecauseaperfectsquaretrinomialmakesaparabolawhichtouches(butdoesn’tcross)thex-axis,thusithasonex-intercept(solution).However,whenyoucompletethesquare,your(whatbecomesa)binomialsquaredequalsanumberontheotherside,andonceitissolved,willresultintwoanswers(notone).

TheQuadraticFormulaTheQuadraticFormulais:

inwhich“a,”“b”and“c”refertothenumbersfromthestandardformequation:ax2+bx+c=0.Thethingaboutthisformula/methodisthatitalwaysworks.Itwillworkwhenaquadraticequationcanorcan’tbefactored.Evenifthereisnosolution,theendpointofthismethodwillrevealthat.Abouthalfthetime,yoursolution(s)fromusingthequadraticequationwon’tbeintegersorevenrationalnumbers.Inthosecases,thebestwaytoexpressyouranswerswillbeinradicalform(asopposedtodecimalform).Ifyoueverwonderwhysimplifyingradicalsisdrilledintoyourminds,it’ssoyoucanuseandnavigatethroughtheQuadraticFormulafrombeginningtoend.Butgettingtotheveryend…theverylaststepiswherestudentscommonlymakeamistake.Simplyput,theyoftenforgettofinishit.Note:“Standardform”isslightlydifferentforquadraticequationsthanlinearequations.

ThePartEveryoneForgets(TheLastStepoftheQuadraticEquation)Sometimes,atthispoint,youransweriscompletelysimplified…butsometimesit’snot.Youshouldneverassumeitiscompletelysimplifieduntilyouattemptthislaststep.Consideryou’vegonethroughtheQuadraticFormulaandgetto

thislaststep:LookforaGCFinthenumerator(andfactoritout),andfactorthedenominator.Inthisexample,theGCFinthenumeratoris3,whichshouldbefactoredout.Inthedenominator,6factorsinto3and2.Nowitcanbeseenthatthenumerator

anddenominatorhaveacommonfactorof3:Cancelthecommonfactorof3outofthenumeratoranddenominator,thenre-writethesimplifiedanswer.Ifthereisaradicalinyouranswerandyoumustgraphit(remember,ananswerisanx-interceptpoint),convertyourradicaltoadecimalandreduceeverythingtoonenumber.Also,takeextracarenottodothelaststepimproperly,asmanyoftendo,asexplainedin:TheWrongWayToSimplifyaRationalExpression

Graph&CheckGraph&Checkismuchdifferentforquadraticequationsthanitisforlinearequations.Forexample,Graph&Checkforlinearequationsisusedtofindasolutionofasystemoftwolinearequations(thepointwheretwolinescross).Thecircumstances(solutions)aredifferentforquadraticequationsbecausequadraticsmakeparabolas.Thereforethewaytofindpointstobegraphedisdifferent,becauseyoucan’tjustfindandgraphanythreerandompointsforaquadraticwithaguaranteethattheywillberepresentativeofthecomplete(parabolic)shape,asyouwouldforalinearequation.Also,thecontextoftheword“solutions”isdifferentforlinear(systems)andquadraticequations.Solutionstoquadraticequationsarex-intercepts.[Justtobeclear,youcan(anddo)findthex-interceptofalinearequation(youjustdon’tcallitthe“solution”),andyoucouldalsoplottwoparabolasonthesamegraphandfindtheirpointsofpossibleintersection…butagain,thosepointsaren’ttheprimarycontextualuseof“solutions,”andit’ssomethingyouaren’tcommonlyaskedtodo].

Herearetheminimalpointsyouneedtographaquadraticequation:ThevertexThex-intercept(s),AKAthe“solutions”They-interceptAnyadditionalpoints

Let’slookatthesepointsingreaterdepth.Regardingthey-intercept,everyquadratic(parabola)hasone.Ifyouhavetheequationwrittenindescendingorderandstandardform,it’sthenumberwhichis“c”fromax2+bx+c=0.Everyparabolawillcrossthey-axis(once),regardlessofifitcrossesthex-axis.Thevertexmustalwaysbefound,asthisisthe(eithermaximumorminimum)pointwherethegraphshiftsfromthepositivetonegativedirection(orviceversa).Thisisitsinflectionpoint.

Asforthex-intercepts,youmayfind:twox-intercepts,iftheparabolacrossesthex-axis,onex-intercept,ifthe(vertexofthe)parabolatouchesbutdoesn’tcrossthex-axis(asisthecaseforperfectsquaretrinomials),ornox-intercepts,iftheparabolaneithertouchesnorcrossesthex-axis.Forthis,youwillget“nosolution”,buttheparabolamaystillexist.

Thisiswhere“anyadditionalpoints”comesin.Ifyoufoundthevertex,they-intercept,andthex-intercepts,youcansuccessfullysketchadecentrepresentationoftheparabola,bygraphingthepointsanddrawingthelinethroughthepointsinasmooth,curvedway(notinarigidwayasifyouwereplaying“connectthedots”).Findingmorepointswilljusthelpyoumakeamoreaccurateandcompletecurve.

QuadraticswithZeroThissectionisdedicatedtoshowingyouthatthefollowingcasesarestillconsideredquadratics(andthereforealsoseconddegreeequations.Actually,quadraticsinwhichborcarezeroarecalled“incompletequadratics”).Thecoefficient“a”mustbeanumberotherthanzero,otherwise,theequationwouldnolongerbeaquadratic.Also,keepinmindthatwhenb=0,the“bx”termequalszeroandwillnotbewritten.Likewise,whenc=0,itwon’tbewritten.However,azerowillappear(only)ifitisaloneononesideofthe=sign.Eventhoughoneortwotermswithinaquadraticcanbezero,youmayhavetosolvethemdifferentlythanifalltermsarenon-zeros.Here,welookatthosecases.

Whencis0:ax2+bx=0Whenc=0,ax2+bx+0=0whichwillbeshownas:ax2+bx=0Afewexamplesare:A)4x2+2x=0,B)3x2+x=0,C)-x2+5x=0,orD)7x2–3x=0.Ineithercase,youwillalwaysexpecttwosolutions:x=0,andx=another#.Note:Incaseswhere“cis0,”then“x=0”isalwaysoneofthesolutions.Fromagraphicalperspective,anytimecis0inaquadraticequation,theresultingparabolawillalwayscrossthroughtheorigin(0,0),aswellasanotherpointalongthex-axis.Asinanyquadraticequation,“c”representsthey-intercept,whichinthiscaseis0.Inthiscase,theorigin(0,0)isboththey-interceptandoneofthex-intercepts.Thesearethestepstosolving:

FactorouttheGCF…whichwillinclude“x”andpossiblyanumberaswell.Settheoutsidefactor(s)equaltozero,andinthiscase,this“x”automaticallyequalszero.Setwhatisinsidetheparenthesesequalto0andsolveforx.

Usingthefirstexample:4x2+2x=0,factoroutxand2(theGCFis2x)frombothterms:2x(2x+1)=0,settheoutsidefactorsequaltozero:2x=0,dividebothsidesby2,andthusx=0.Setwhat’sinsidetheparenthesesequaltozeroandsolveforx:(2x+1) � 2x+1=0.Subtract1frombothsides:2x+1–1=0–1Giving:2x=-1.

Dividebothsidesbythecoefficient2,andx=

Thesolutionsarex=0andx= .

WhenBothb&care0:ax2=0Whenbothbandc=0,ax2+0x+0=0,whichwillbeshownas:ax2=0.Someexamplesare:2x2=0,x2=0,or-3x2=0.Inallcases,theonlysolutionisx=0(becauseifyoudividebothsidesbythecoefficientinfrontofx2,thentakethesquarerootofbothsides,youwillget“0”).Graphically,thiswillproduceaparabolawhosevertexistheorigin(0,0),withtheline“x=0”astheverticallineofsymmetry.Thereisonlyonesolutionbecause(thevertexof)theparabolaofthistypetouchesbutdoesnotcrossthex-axis.Here,theorigin(0,0)isboththey-interceptand(theonly)x-intercept.

Whenbis0:ax2+c=0Whenonlyb=0,ax2+0x+c=0whichwillbeshownandseenas:ax2+c=0Someexamplesare:E)2x2–2=0,F)9x2–4=0,G)x2–36=0,H)4x2+25=0,orI)3x2–5=0.Ofthegivenexamples,allexcept“4x2+25=0”areconsideredtobe:“Thedifferenceoftwosquares.”ExampleHisdiscussedafewpageslater.

“TheDifferenceofTwoSquares”Wewilllookateachexample,specifically,butbeforethat,it’simportantyouseethetwowaysinwhichproblemslikethesecanbesolved,soyoucannoticethepatternintheexamplestofollow.Eitherapproachisstartedinthesameway.First,lookforaGCF.IfthereisaGCF,factoritout,thenproceedtodividebothsidesbyit.Sincezeroisontheright,any(non-zero)numberyoudividebyitwillequalzero.Atthispoint,therearetwowaysyoucanproceedtosolve.Onewayisbymovingtheconstanttotheothersideoftheequalsign,thentakingthesquarerootofbothsides.Theotherisbyfactoringintobinomials.Wewilllookateachmethodinmoredetail.Eitherchoiceiscompletelyvalid.It’sreallyuptoyoutodecidewhichmethodyouprefer.Toreiterate,thetwowaysare:A1.Factorintoconjugatepairbinomials,orB1.Takethesquarerootofbothsides.

Thereisatimethat“takingthesquarerootofbothsides”willbepreferable,asIwillshowinthefollowingexamples.ExampleE:2x2–2=0ThisisaclassicexampleoffactoringouttheGCFfirst,whichhereis2.2(x2–1)=0Dividebothsidesby2,whichgivesyou(x2–1)=0Atthispoint,youcangoforwardwitheithermethod.I’mgoingtodemonstrateboth,toprovethateachisvalidandyieldsthesameoutcome.Butfirst,I’mgoingtoshow“factoringintoconjugatepairbinomials.”As“x2–1”isthedifferenceoftwosquares,itcaneasilybefactoredinto(x–1)(x+1)=0Seteachfactorinparenthesesequaltozero,andsolveforx.x–1=0x+1=0x-1+1=0+1x+1–1=0-1x=1x=-1sox=+/-1Thereisaveryimportantlessoninthisexample,whichisthat“thedifferenceoftwosquarescanbefactoredintoconjugatepairbinomials.”Itshouldalsobeexpectedthat“thedifferenceoftwosquares”willalwaysyieldtwooppositesolutions(howeverthereisonetechnicalexceptiontothis,explainedin:Clarification:WhentheSolutionis0).

ConjugatePairBinomialsIt’sgoodtobefamiliarwithconjugatepairbinomials,visually,bydefinition,byname,andbycommonuse.Conjugatepairbinomialsaretheresultoffactoring“thedifferenceoftwosquares.”Theyareknownasconjugatepairbinomialsbecause…

theyareconjugateinthattheyarejoinedandconnectedinsomeway,theycomeinpairs(asconjugatesdo),andtheyarebinomials(eachsetofparenthesescontainstwoterms).

Theyappearastwoparentheseswiththesamefirstandsecondterms,butoppositesignsinbetweenthem.Whenconjugatepairsaremultiplied,theresultis“thedifferenceoftwosquares.”Theadvantageofidentifyingandfactoringthedifferenceoftwosquaresintoconjugatepairbinomialsisthatitisquickandinvolvesfewsteps.Alsotheuseofconjugatepairbinomialsareanessentialpartof“rationalizingthedenominator”whenthedenominatorcontainsabinomialwithatleastoneradical.Thistopicisonethatisusuallycoveredneartheendofthesemester,oftendisplacedfromthelessonswhichintroducefactoringquadraticequationsand“specialcases.”Duringthistimedisplacement,studentssometimesforgettoseetheconnectionofthisconcept.Additionally,uponlearningit,studentsoftendonotrealizetherelevanceforwhichitwillbeneededlater.Moreontheprocedureformultiplyingconjugatepairbinomialsandgraph-relatedinformationiscoveredintheSpecialCasesubsection:TheDifferenceofTwoSquares.Theotherwaytosolveforx,startingfrom“x2–1”isbytakingthesquarerootofbothsides…

TakingtheSquareRootofBothSidesWestartbackwith“x2–1”fromexampleEtoseethatitcanalsobesolvedbytakingthesquarerootofbothsides.Itissetequaltozero.x2–1=0Inthiscase,weproceedbymovingtheconstant(here,-1)totheotherside:x2–1+1=0+1,makingitx2=1.Takethesquarerootofbothsides:

Rememberthattakingthesquarerootofanumbergivesthepositiveandnegativerootnumber(becauseifyousquareapositiveornegativenumber,youalwaysgetapositiveresult),soitcanbesaidthatxequalsplusorminus1,writtenasx=+/-1.

Thenexttwoexamplesdemonstrate“factoringthedifferenceoftwosquaresintoconjugatepairbinomials.”ExampleF:9x2–4=0Factored:(3x–2)(3x+2)=0Noticethat3xisthesquarerootof9x2and2isthesquarerootof4.Seteachsetofparenthesesequaltozeroandsolveforx.3x–2=03x+2=03x–2+2=0+23x+2–2=0–23x=23x=-2dividebothsidesby3dividebothsidesby3

x=

ExampleG:x2–36=0Factored:(x–6)(x+6)=0x–6=0x+6=0x–6+6=0+6x+6–6=0–6x=6and-6

ExampleI:3x2–5=0Thisisaninterestingexample,oneofwhichyouaresuretoencounter.It’simportanttorememberthatyoucantakethesquarerootofany(positive)numberorterm,butonlywhenyoutakethesquarerootofaperfectsquarewillyourresultbeintegers(non-radicalornon-decimalnumbers).Inthethreeexamplesbeforethisone,thetermswereperfectsquaresandthereforecouldbefactored(intobinomials),howeverinthisexample,the3andthe5arenotperfectsquares(however,thex2stillis)…sotheycan’tbefactoredusingintegers.Therefore,whenoneorbothofthetermsinvolvedarenotperfectsquares,itisoftenpreferredtoapproachsolvingbymovingtheconstanttotheothersideoftheequationandtakingthesquarerootofbothsides,asseeninthefollowingsteps.3x2–5+5=0+53x2=5

Dividebothsidesby3,giving:

x2= ,takethesquarerootofbothsides:

and

x=+/-

TheSumofTwoSquaresInanyquadraticequationinwhichb=0,anytimethec-numberisadded,thepolynomialintheequationisconsideredprime,andhasnorealsolution.(Pleaseseethetwonotesbelow).Youshouldrecognizeequationssuchastheseas:“thesumoftwosquares.”andyoushouldremember:“Thesumoftwosquaresisprime.”Whenfacedwithaproblemsuchasthis,theacceptableanswerresponsesare:“prime,”and“norealnumbersolutions.”*PleaseNote:Ifyoulookatthestandardformofaquadraticequation:ax2+bx+c=0,orinthiscase:ax2+c=0,youprobablynoticethatcisadded,whichmayleadyoutothinkthateveryequationsuchasthisisprime,butthisisnotenoughinformationtomakethisjudgement.Tobeclear,theformax2+c=0iscorrect,butwhethertheequationisprimealldependsonthesignoftheactualnumberthatisrepresentedbyc.Inotherwords,ifthenumberplugged-inforcisnegative,youhavethedifferenceoftwosquares,whichisnotprime.Or,ifthenumberforcispositiveyouhavethesumoftwosquares,whichisprime.Thereistechnicallyoneexceptiontothis,explainedin:Clarification:WhentheSolutionis0.**AlsoPleaseNote:Thefirstsentenceofthissectionstates,“…theequationisconsideredprime,andhasnosolution.”Thisstatementmustbeproperlyunderstood.Althoughthestatementmayseemtoinsinuatethat“prime”issynonymouswith“nosolution,”bydefinition,thisisnottrue.Sincethe“sumoftwosquares”issocommon,youcanpredictitsoutcomeof“prime”and“nosolution”automatically,butthefactthattheyhaveboththeseoutcomesismerelycoincidental.(Thisexplanationiscontinuedin:Primevs.NoSolution).Withthatinmind,rememberthat“thesumoftwosquares”canstillbegraphedandwillproduceaparabola.

Nowlet’slookattwomoreexamples,JandK.Noticeeachexampleissimilartoanexamplefrombefore,buttheybothhaveanegativesignintheleadingcoefficient.Whatwe’regoingtodoissimplifyeachexamplefirst,andthendecideifwhatremainsisprimeorfactorable.Tosimplify,wemustdoanumberofthings:

IdentifyandfactoroutaGCF(ifthereisone)DividebothsidesthebytheGCF(ifthereisone)

MovetheconstanttotheothersideoftheequalsignEnsurethecoefficientinfrontofx2ispositive1

Thismayhavebeentakencareofinapreviousstep,otherwise…Youcandoitnowbydividingbothsidesbythecoefficientinfrontofx2

LookatExampleJ:-4x2+25=0Let’ssimplifybygoingthroughtheprocedurejustpreviouslymentioned:

IsthereaGCF?No.…SothereisnoGCFtodividebothsidesby.Movetheconstant,25,totheothersidebysubtractingitfrombothsides:

-4x2+25-25=0–25

-4x2=–25

Ensurethecoefficientinfrontofx2ispositive1.Inthiscase,wewilldividebothsidesby-4,whichcancelsoutbothnegativesigns,giving:

x2=

Takethesquarerootofbothsides…whichwecando,becausethesignof ispositive.

Thesolutionsare:x=+/- .

Next,let’slookatExampleK:-3x2–5=0Andfollowthestepstosimplify,aswedidinthelastexample.

IsthereaGCF?Inthiscase,yes.Itis“-1”;factorthatout:-(3x2+5)=0

DividebothsidesbytheGCF“-1”,whichmakesit:3x2+5=0

Ifwepausehere,weseethatwehavethesumoftwosquares,whichisenoughinformationforustostopsolving,andanswerwith:“prime;noreal-numbersolutions”Note:Ifyou’rewonderingif3and5aresquarenumbers,youcouldsay“sort-of”becausetheyarethesquaresofthesquarerootof3andthesquarerootof5,respectively;however3and5arenotperfectsquares.Anypositivenumberisasquareofanothernumber,butaperfectsquareistheresultofsquaringaninteger.Let’sproceedwiththeremainingtwostepsanywaytoseewhathappensifwecontinuetosimplifyandsolve.

Movetheconstant5totheothersidebysubtracting5frombothsides,giving

3x2=-5

Dividebothsidesby3toensurethecoefficientinfrontofx2is3,giving

x2=-5

Atthispoint,whenyougototakethesquarerootofbothsides,youshouldrealizethatyoucan’ttakethesquarerootofanegativenumberandgetareal-numbersolution.Thisfurtherprovesthat“-3x2–5”hasno(real)solution.

SpecialWordsforSpecialCases

Thetextbooksaregenerallygoodathighlightingthespecialcase(quadratics)intheirownsection,andteachinghowtosolvethem.Andstudentsaregenerallygoodatfactoringandsolvingthemwhentheyareintheirownisolatedareasandwhentheyknowwhattypethey’redealingwith.Butoncethespecialcasesaremixedintogeneraltypesofproblems,studentssometimesforgetthesignalsinidentifyingthem.

Identifyingthemisthefirstcrucialstep.Thissectioncontainsafewsentencesandkeywordsthatwillhelpyouidentifythetypeofspecialcase,andtellyouwhatoutcometoexpectintermsoffactoring,graphing,andtheshortcutformultiplying.Ifyoumemorizethesewords,itwillhelpyoufigureoutthefactorsandanswersmorequickly.Somepartsmayseemredundantfromsomeearlierpartsofthebook,butthat’sokay,therepetitionisgoodforyou.Icovertwomainspecialcasesinthecomingpages.Pleasenotethatthereisanothercommonspecialcaseinvolving“thesumanddifferenceoftwocubes,”whichIdon’tcoverinthisbook(becausemyprimaryfocusislinearandquadraticequations,andsquaresandsquareroots).

PerfectSquareTrinomialRememberthat:“Aperfectsquaretrinomialfactorsintoabinomialsquared.”Likewise,“Abinomialsquared,whenmultipliedout,givesaperfectsquaretrinomial.”Youwillnoticeinaperfectsquaretrinomialthat:

thefirstterm(ax2)andthelastterm(c)areperfectsquares,thecoefficientbwillbetheproductof

(thesquarerootofthefirstterm,ax2),(thesquarerootofthelastterm,c),and2;

thesigninfrontofcwillalwaysbepositive.Itisalwayspositivebecausethelastnumberinthebinomialissquared.

Itwillresemble:(Perfectsquarenumber)x2+ x+perfectsquarenumberSuchas:4x2-12x+9=(2x–3)2withtheperfectsquaretrinomialontheleft,equaltoitsfactoredbinomialsquaredontheright;andanotherexample:x2+8x+16=(x+4)2Therearetwowaystomultiplyabinomialsquared.OnewayisbyexpandingitintotwobinomialsandmultiplyingbytheFOILmethod.Thiswayisfine,butyouarehighlyencouragedtousetheshortcut.Thisistheshortcutformultiplyingabinomialsquared,inwords.Iwillrefertothetermsofthebinomialassuch:(firstterm+lastterm).

1. Squarethefirstterm.2. Leavespace(forwhatwillbeinstruction#4).3. Squarethelastterm.4. Multiplythefirsttermtimesthelastterm,thendoubleit,andwriteit

inthespaceyouleftininstruction#2.*Acommonmistakemadeispeopletreatthismethodliketheshortcutformultiplyingconjugatepairbinomials,andtheyforgettodostep4,sotheyarethenmissingthemiddle(bx)term.

Doyouseetheconnectionofhowtheprocedureforthisspecialcaseisappliedwhencompletingthesquare?Whengraphed,aperfectsquaretrinomialisaparabolawhichtouches(butdoesnotcross)thex-axis.Thevertexisaty=0andthesolutiontox,and(saidtobe)tangenttothex-axis.

TheDifferenceofTwoSquaresRememberthat:“Thedifferenceoftwosquaresfactorsintoconjugatepairbinomials.”Likewise,“Conjugatepairbinomials,whenmultiplied,givethedifferenceoftwosquares.”Thedifferenceoftwosquaresisaspecificcaseofwhenbis0;ax2+c=0,and“c”isanegativenumber.When“c”isapositivenumber,youhave“thesumoftwosquares.”Remember:“Thesumoftwosquaresisprime.”Therearetwowaystomultiplyconjugatepairbinomials.YoucanmultiplythemusingtheFOILmethod,butyouareencouragedtousethespecialcaseshortcutmethod.Gettingusedtotheshortcutwillbeusefulwhenyoulearntorationalizedenominatorswithradicalsandbinomials(notcoveredinthisbook).Plus,theshortcutisfaster.Thisistheshortcutmethodinwords:

1. Squarethefirstterm,2. Writeaminussign,3. Squarethelastterm.

Therewillbenomiddle(bx)termbecauseifyouFOILtheconjugatepairbinomials,theproductoftheOutertermsplustheproductoftheInnertermscanceleachotherouttozero.Thisiswhatmakesthedifferenceoftwosquaresaspecialcase.Whengraphed,thismakesaparabolathat:

Hastwox-interceptsbecauseitcrossesthex-axisintwoplaces,hasavertexatx=0,[atpoint(x,c)],andthelineofsymmetryisthey-axis(theline:x=0).

Thereisonetechnicalexceptiontothestatementthat“thedifferenceoftwosquaresyieldstwosolutions,”whichisexplainedin:Clarification:WhentheSolutionis0.

Primevs.NoSolutionAsinthesection:TheSumofTwoSquares,whenthequadraticexpression[inanequationinwhichb(only)iszero]isprime,theresultis“no(real)solution.”Bydefinition,“prime”shouldnotbethoughtassynonymouswith“nosolution.”Thefollowinginstancescanoccur:

thepolynomialcanbefactoredanddoesyield(real)solutions;thepolynomialcan’tbefactoredandyieldsno(real)solutions;andthepolynomialcan’tbefactoredbutdoesyield(real)solutions;however,apolynomialwhichcanbefactoredwillalwaysyieldoneormore(real)solutions.

Keeptheirdefinitionsinmind.“Prime”means“can’tbefurtherfactored(intofactorsotherthanitselfand1).”And“No(Real)Solution”withregardstoapolynomialmeansthattheresultinggraphhasnox-intercepts(doesnotcrossortouchthex-axis).Agoodexampleiswhenyouhaveaquadraticequationcontainingatrinomialexpression,inotherwords,therearenon-zeronumbersinfora,bandc.However,asdiscussedearlierin:SolvingQuadraticEquations,thetrinomialmaynotbeabletobefactoredusinganycombinationofintegerfactorsintheTrial&Error/ReverseFOILMethod,ortheac/GroupingMethod;inthiscase,itwouldbeconsidered“prime.”YouarethentousetheQuadraticFormula.YoumaynotyetknowiftheQuadraticFormulawillyield“real”solutionsornot(butthat’swhyyoupluginthea,bandcvaluesandsolve).IftheQuadraticFormuladoesyieldrealsolutions,thesolutionswillcontainradicalsofnot-perfectsquares(whichcouldbeconvertedtodecimals).Thereisalsoachancethatnorealsolutionswillresultfromthatoriginalprimepolynomialduetoanegativenumberinthesimplifiedradical.

Clarification:WhentheSolutionis0Before,Istatedthatthedifferenceoftwosquareswillyieldtwosolutions,butthereisoneexceptiontothis.Anyquadraticequationinwhichbandcarezero,asseenin“WhenBothb&care0,”canalsotechnicallyqualifyasthedifferenceoftwosquares,suchas“4x2–0=0”,orthesumoftwosquares,asin“9x2+0=0”,becausezeroisa(perfect)squarenumber.Incaseslikethis,

thereisnottwosolutions,norisitprimewithnosolutions…Ithasonlyonesolution,thatbeing“x=0”.

Thatalsomakesthisanexceptiontothestatement:“Thesumoftwosquaresisprimeandyieldsnorealnumbersolutions.”

RATIONALEXPRESSIONSByitstechnicaldefinition,arationalexpressionisafractionthatcontainspolynomials.Butsince,tobeapolynomial,itmustcontainatleastonevariable(otherwise,ifitjustcontainedconstantsornumbers,itwouldbeconsideredaregularfraction),mydefinitionis:Arationalexpressionisafractioncontaining(oneormore)variables.Noticetherootword“ratio.”Sometimes,youwillbeaskedtosimplifyarationalexpressionandsometimesyouwillhavetosolveanequationcontainingrationalexpressions.Whendealingwithrationalexpressions,youmustknowhowtoproperlysimplifythem.

ProcedureforSimplifyingRationalExpressionsItisimportantandhelpfulthatyoucanclearlyseeeachnumeratoranddenominatorasseparatepiecesbecausetheywillneedtobesimplifiedindividually,first,beforecontinuing.0.Forthisreason,Irecommendputtingparenthesesaroundnumeratorsanddenominators.(ThisreinforcesthefirstruleofOrderofOperations,beingthatnumeratorsanddenominatorsareinfact“groups”butarerarelywrittenwithparentheses.Puttingparenthesesaroundthemmakesthemlookmorelikegroups,andwillremindyoutotreatthemassuch).1.Factoreachnumeratoranddenominatorseparately,completelyfactorthepolynomials.2.Lookforfactorsthatarealikeinthenumeratoranddenominator,thencancelthemoutto1(correctly,byavoiding:TheWrongWaytoSimplifyaRationalExpression).

ProcedureforAdding&SubtractingRationalExpressionsThisfollowstheprocedureforadding&subtractingfractions,onlynow,variablesareinvolved.Youadd&subtractrationalexpressionsthesamewayasfractionsbyfindingandusingtheLCD.Remember,thegoalforadding&subtractingrationalexpressionsistoproperlycombinethemsothatonerationalexpressionremains.Followthisprocedure(thefirststeps,0a–0c,aremorepreparatorysteps):0a.Putparenthesesaroundallnumerators&denominators.0b.Lookforanyminussignsinfrontofafraction.Ifthereisaminussigninfrontofafraction,youmustdistributeitthroughitsassociatednumerator.Todothis,replacetheminuswithaplusandchangeeverysigninitsassociatednumerator.Thisisoftenanoverlookedstepwhichresultsin+/-signerrorslateron.*Note:Abigerrorstudentsmakehereisapplyingthenegativesigntoonlythefirstterminthenumerator,insteadofapplying(distributing)ittoeveryfactorinthenumerator.0c.Putalltermsinnumerators&denominatorsindescendingorder.1.Factorallthenumerators&denominatorsseparately.2.LookatallfactorsofthedenominatorsanddeterminetheLCD.

WritetheLCDofftothesidesoyoucanrefertoit,andwritetheLCDinthedenominatortotherightofthe=sign(asofnow,thenumeratorisblank;youwillfillinthenumeratorinstep5).LeavetheLCDasunmultipliedfactors(whichwillmakesimplifyingeasierinthelatersteps).

3.Lookateachdenominator(ofeachfraction)anddeterminewhatfactorsaremissing(Icallthesethe“missing-factors”)ifany,tocompletetheLCDineach.4.Multiplythenumerator&denominatorofeachfractionbyitsmissing-factor(s)5.Multiply&distributefactorsinthenumeratorandwritetheproductsinthenumeratorabovetheLCDwrittenontherightsideofthe=sign.Don’tforgettotransferthepropersigns.6.Simplifythetermsinthenewnumerator.6a.Combinelike-terms.6b.LookforaGCF,andwhetherthereisoneornot,trytofactorcompletely.7.Simplify:Canceloutanycommonfactorsinthenumerator&denominator.

Makesureyouavoidthewrongwaytosimplifyarationalexpression.

SimplifyingaComplexRationalExpressionSimplifyingacomplexrationalexpressionissomethingyouwillsurelyberequiredtodoonafinalexam.Acomplexfractionsimplymeansfractionswithinafraction.Sincerationalexpressionsarefractionscontainingvariables,acomplexrationalexpressionmeansrationalexpressionswithinrationalexpressions…orsimply:fractions-containing-variableswithinfractions-containing-variables.

Therearetwowaystosimplifycomplexrationalexpressions:1.The“All-LCDMethod.”Multiplyallmini-fractionsbytheLCDofallmini-fractions.Or2.Simplifytheoverallnumerator&overalldenominatorfirst(byapplyingtherulesforaddition&subtractionoffractions),separately,intoonefractioneach,thendividethetop-fractionbythebottom-fraction(see:DividingFractions).Eithermethod,ifusedcorrectly,willyieldthesameresult.

Whenshouldyouuseeachmethod?Method1,the“All-LCD-Method”avoidsneedingtoadd&subtractfractionsandyouonlyneedtofindoneLCD.Thisistypicallyeasierandmorepreferred.Whenindoubt,defaulttousingthismethod.Althoughthismethodismorestraightforward,itisalsotediousandmaymakeyourpapermessy.Forthatreason,mistakesareoftenmadeintheprocessfornotbeingabletoreadyourownwriting,writingtoosmall,ornotleavingenoughroom.Whendoingtheseproblemsbythismethod,besuretoleaveplentyofroomonthepaperandwriteclearly.Method2,“SimplifyingtheOverallNumeratorandOverallDenominatorSeparately”istypicallyusedwhentheaddition&subtractionoffractionsintheoverallnumeratorandoveralldenominatorwillbeaquickandeasyprocedure.Thismethodisalsousuallyselectedwhenthevariablesintheoverallnumeratoraredistinctlydifferentthanthoseintheoveralldenominator.ThereasonforthisisbecausefindingandusingtheAll-LCD-Methodmayintroducevariablesintooppositepartsofthefractionthatwillrequireextraandtedioussteps(suchasfactoring)togettotheend.Ultimately,thechoiceofmethodismoredependentonthestudent’spreference,asbothmethodsaretediouswithanumberofintermediatesteps,butwillstillyieldthesameoutcome.Becauseofhowtedioustheyare,youmaychoosetostickwiththeoneyougravitatetowardsandworkongettinggoodatit.Thetwoprocedureswillbegivennext,atfirstinaverydetailedform,andtheninacondensedform,soyoucanreferbacktoeitherversion.Thesemayhelpyoudecideonyourpreference.

All-LCDMethod(detailedversion):0a.Iftherearewholenumbersorpolynomialsthatarenotfractions,it’sagoodideatoputthemover1,tomakethemfractions.0b.Irecommendputtingallpolynomialnumerators&denominatorsinparentheses.Booksoftenleavethemwithoutparentheses,butusingthemmakesiteasiertoviewandusethepolynomialsorfactorstocontinue.1.Completelyfactorallnumeratorsanddenominators,ifpossible.2.FindtheLCDofallthemini-fractionsinvolved.3.MultiplytheLCDbythenumeratorsofallmini-fractions.(SeeNote4).4.Simplifyallmini-fractionsbyapplyingtherulesofmultiplication&divisionofbaseswithexponents.Inthisstep,alldenominatorsofmini-fractionsshouldcancelout.ThefactorsoftheLCDareintendedtodirectlycancelout(with)everyentiredenominatorofallmini-fractions.However,thismaystillleavetheremaining,un-cancelledfactorsfromtheLCD,ifany,inthenumeratorsofthemini-fractions,andthisisexpected.Therationalexpressionisnowsimple,notcomplex,asthereisnowonlyonenumeratorandonedenominator.

4a.Simplifythe(new)numerator.Multiply,distributeandcombinelike-terms,wherepossible.

4b.Simplifythe(new)denominator.Multiply,distributeandcombinelike-terms,wherepossible.4c.Arrangealltermsintodescendingorder.5.Completelyfactorthenumeratoranddenominator,separately.ThatmeansfactoringoutaGCFfirst,ifthereisone,and/orfactoringthepolynomial,ifpossible,(intosmallerpolynomials).6.Canceloutanycommonfactorsinthenumerator&denominator.*Note1:Sometimesfactorswillcanceloutinthelaststepandsometimesnonewill.Bereadyforeitherscenario.

**Note2:Atthispoint,itisuptoyourprofessorifhe/shewantsyoutomultiplythefactorsinthenumerator&denominator(individually)foryourfinalreportedanswer.Ipersonallypreferthemtobeleftinfactoredform.***Note3:Don’tcommitthefrequentlymademistake:TheWrongWaytoSimplifyaRationalExpression.

SimplifyOverallNumerator&OverallDenominatorSeparatelyMethod(detailedversion)0a.Iftherearewholenumbersorpolynomialsthatarenotfractions,setthemover1,tomakethemfractions.0b.Irecommendputtingallpolynomialnumerators&denominatorsinparentheses.Thismakesiteasiertoview&usethesepolynomialsorfactorstocontinue.1.Completelyfactorallnumeratorsanddenominators,ifpossible.2.Next,youwillbesimplifyingtheoverallnumerator&overalldenominatorseparately,usingtheirownLCDs.2a.FindtheLCDofthefractionsintheoverall-numerator2b.FindtheLCDofthefractionsintheoverall-denominator.3.Intheoverallnumeratoranddenominator,separately,usetheprocedureforadding&subtractingrationalexpressions.Atthispoint,youstillhaveacomplexrationalexpression,butnowwithonlyone(unsimplified)mini-fractionineachtheoverall-numeratorandoverall-denominator.4a.Leavealldenominatorsofmini-fractionsasunmultipliedfactors.4b.Simplifyallnumeratorsofmini-fractionsbycombininglike-terms.4c.Factorthenumeratorsofthetwomini-fractions,separately.Bythispoint,thereshouldstillbeonefractioneachintheoverall-numeratorandtheoverall-denominator,butnow,eachmini-fractionissimplified.Youarenowreadyto…5.Dividethetopmini-fractionbythebottommini-fractionusingtherulefordividingfractions.Younowhavea(simple)rationalexpression(onenumeratorandonedenominator).Theymayormaynotalreadybesimplified.6.Simplify:Canceloutcommonfactorsfromthenumerator&denominator.

All-LCDMethod(shortversion)0.Putwholefactorsover1andputalloverallnumerators&denominatorsinparentheses.1.Factorallnumerators&denominators.2.FindtheLCDofallmini-fractions.3.Multiplyallmini-fractionsbytheLCD.4.Simplify:canceloutalldenominatorsofmini-fractionswithassociatedcommonfactors.*Therationalexpressionisnowsimple.4a.Multiply,distribute,combinelike-termsinnumerator.4b.Multiply,distribute,combinelike-termsindenominator.4c.Putalltermsindescendingorder.5.Completelyfactorthenumeratoranddenominator,separately.6.Canceloutanycommonfactorsfromnumerator&denominator.

SimplifyOverallNumerator&OverallDenominatorSeparatelyMethod(shortversion)0.Putwholefactorsover1andputalloverallnumerators&denominatorsinparentheses.1.Factorallnumerators&denominators.2a.FindtheLCDofthefractionsintheoverall-numerator.2b.FindtheLCDofthefractionsintheoverall-denominator.3a.Convertandaddfractionsinoverall-numerator.3b.Convertandaddfractionsinoverall-denominator.4a.Leavealldenominatorsofmini-fractionsasunmultipliedfactors.4b.Simplifyallnumeratorsofmini-fractions.4c.Factorbothnumeratorsofthetworemainingmini-fractions.5.Dividethetopfractionbythebottomfraction.6.Simplifytheoneremainingfraction.

Note4:UsingtheLCDissimilar,yetdistinctlydifferentintheAll-LCD-Methodthanforaddition/subtractionofrationalexpressionsintheoverall-numerator&overall-denominatormethod.Inaddition/subtractionofrationalexpressions,themissingfactorsoftheLCDaremultipliedtimesthenumeratoranddenominatorofeachfractiontoconvertthefractionsintolike-fractions,whereasforsimplificationofcomplexrationalexpressions,thewholeLCDismultipliedtimesthenumeratoronlyofeachmini-fraction.

AnnotatedExample1UsingtheAll-LCDMethodWriteouttheproblemleavingplentyofroomonbothsidesandbelow:

0. Put4over1,andputallpolynomialsinparentheses:

1. Thenumeratorsarealreadysimplifiedandcannotbefactored.Alldenominatorsexcept“x2–4”cannotbefactored.Noticethat“x2–4”isthedifferenceoftwosquares.Factoritintoconjugatepairbinomials:

2. FindtheLCDofallminifractionsandwriteittotheside.LCD=x(x–2)(x+2)

3. MultiplyallminifractionsbytheLCD:

Crossoutthecommonfactors:

Andremovethecrossed-outcommonfactors.Noticethatalldenominatorsofthemini-fractionswillbeeliminated(also,removethedenominator“1”)andtheexpressionwillchangefromcomplextosimple.Itwill

looklike:

4. Simplifyboththenumeratoranddenominator,separately,bymultiplying,distributing,thencombininglike-terms

4a.Thesestepsshowthemultiplication:

4b.Thisstepshowscombininglike-termsandarrangingintodescendingorder:

5. Completelyfactorthenumeratoranddenominator,separately.Inthenumerator,theGCFis-3x.Thedenominatorisatrinomial,sotrytofactoritintotwobinomials.

Thedenominatorcannotbefurtherfactored.

Note:thenumeratorcouldhaveautomaticallybeenfactoredtothisfromstep4aeitherbyusing“(x2–4)”asaGCF,orbycombininglike-terms,butthiswillnotalwaysbeanoption.6. Lookforanycommonfactorsinthenumeratoranddenominator.In

thiscase,therearenone,sothelaststepisthemostsimplifiedform.

AnnotatedExample2UsingtheOverallNumerator&DenominatorMethodWriteouttheproblemandleaveplentyofroomonthesidesofeachterm:

0. Put4over1.Inthiscase,therearenopolynomialstoputparenthesis

aroundinanymini-fraction:

1. Sincetherearenopolynomialsinthenumerators,theycan’tbe

factored.Thedenominatorsinthetopfractionscan’tbefactored,howeverthedenominatorsofthebottomfractionscanbefactored(intoexponentialforminanticipationoffindingtheLCD):

2. FindtheLCDs:

1. FindtheLCDofthetopfractions.Itis“x”.2. FindtheLCDofthebottomfractions.Itis“8”.

3. Convertfractionsintolike-fractions,thenadd:

1. Intheoverall-numerator,then2. Intheoverall-denominator.

4. 1. Leavedenominatorfactorsunmultiplied.

2. Simplifyallnumeratorsofminifactors.Inthiscase,theyarealreadysimplified.

3. Factorthenumeratorsofbothmini-fractions.Thenumeratorofthebottomfractioncan’tbefactored.

5. Dividethetopfractionbythebottomfractionbyinvertingand

multiplying:

Therearenocommonfactorstocancelout,sothesimplifiedformis:

TheWrongWaytoSimplifyaRationalExpressionThissectionhighlightsaseriousmistakethatstudentsmakeallthetime.Itinvolvesthelaststepofsimplifyingarationalexpression.Oddlyenough,studentsoftenperformthemoredifficultpartoftheproblemcorrectlybeforegettingtothisstep,whichiswhyIbelievestudentscommitthismistakemoreoutoflazinessthanignorance.Regardlessofwhy,itmustbeprevented,especiallybecausethisisoftenthelaststepinaproblem(andifyouhaveaninstructorthatdoesn’tgivepartialcredit,thisstepcouldmakeorbreakaproblem).Hereareexamplesofthewrongandrightwaytosimplifyarationalexpression.Thestep(s)I’mhighlightinginthissectionarethesameseeninsteps5&6ofTheAll-LCDMethodforSimplifyingRationalExpressions.Whatyouneedtorealizeis:youcan’tfactoroutaterminthenumeratorwithaterminthedenominatorwhen(andbecause)termsareseparatedby“+”and“-“signs.Youcanonlycancelfactorsinthenumeratorwithfactorsinthedenominator…andfactorsaremultiplied,notaddedorsubtracted,together.Thewrongthingtodoistoinstantlycanceloutafactorinthenumeratorwithatermorfactorinthedenominator,withoutfirstfactoringthenumerator(eitherfactoringtheGCFoutorfactoringitintosmallerpolynomials),andtakingintoconsiderationthesignificanceoftheplusorminussignontopbetweenthetopterms.Let’sstartwiththisexample,whichcontainsabinomialinthenumeratoranda

monomialinthedenominator:

Thefollowingisthewrongway:inwhichoneattemptstofactor3xoutofthe12x2(to4x)andthe3x(to1).

Alternatively,thefollowingisalsothewrongway:inwhichoneattemptstofactor3outof-6(to-2)andoutof3x(to1,leavingx).Noticehow,ineachwrongwayexample,thetermsincorrectlycancelledouthaveplusorminussignsinfrontorbehindthem.Thisisthekeysign(nopunintended)thatshouldtellyounottocanceloutterms.

TheCorrectWay:YouneedtolookforaGCFinthenumerator,whichinthiscaseis6,andthenfactoritout.Atthispoint,thecommonfactorof3canbecancelledoutofthe

numerator(6)anddenominator(3x),asshown:Theexpressionaboveontherightcanbeconsideredthemostsimplifiedform.Comparethisanswertothewronganswersfrombefore.AsImentionedinthenotesattheendoftheAll-LCDMethodforSimplifyingRationalExpressions,thefinalanswercanbeshownlikethis,orbydistributing(multiplying)thefactorof2throughthe(2x2–1)inthenumerator.Sincesimplificationofteninvolvescompletefactorizationandnotthereverse(multiplying),Ibelievethisformisthemostsimplified.Ifyouchoosetomultiplythrough(perhapsatthesuggestionofyourinstructors–youshouldalwaysreporttheanswertheway

theypreferit,sincethey’regradingyou),itwillappearlikethis:or,ifyoubreakitapartintoseparatefractions:

whichwillthensimplifyto:

Again,findoutfromyourinstructorshowtheywantyoutoreportyouranswer.

Let’slookatanotherexample,onewithatrinomialinthenumeratoranda

binomialinthedenominator:

TheWrongWay:Therearemanywrongwaystoapproachasuchproblem.Onewrongwaymightbetoattempttocanceloutxfrom3x2(to3x)inthenumeratorandfrom2x(to2)inthedenominator.Anotherwrongthingtodowouldbetofactor3outof-15x(to–5x)inthenumeratorandoutof-6(to-2)inthedenominator.Ifthoseerroneouscancellationswereperformed,itwouldwronglygive:

…whichwouldconcludetobeundefined.

TheCorrectWay:Goingbacktotheoriginalexample,factortheGCF(whichis3)outofthenumerator.Then,factortheGCF(whichis2)outofthedenominator,which

wouldmake:Next,gobacktothenumeratorandseeifthetrinomialinsideparenthesescanbefactored,whichitcanbe,intothetwobinomials,seenbelow.Itisrevealedthatthecommonfactorinthenumeratoranddenominatortobecancelledoutis(x–

3),shownbelow:After(x–3)iscancelledout,thefinalsimplifiedformis:

Comparethistothewronganswershownabove.

ExtraneousSolutionsItisimportantyouknowwhatextraneoussolutionsare,whentolookforthem,andhowtodealwiththem,becausetheyaretrickyanddeceptivethings.Extraneoussolutions(alsocommonlyknownasextraneousroots)appeartobesolutionstoaproblemyoujustsolved,butactuallyaren’t.Theytendtocomefromthefollowingtwoplaces:

A(variableinthe)denominator,andA(variableinsidea)radical.

Basedonthelocationofvariablesinanequation,thesecanbethoughtofassolution-exceptions,forthefollowingreasons:

Anyfractionwhosedenominatoriszeroisundefined.Also,anytimearadicand(ofanyevenroot)isnegative,theresultisnotreal(stillcountsasundefined).

Youcanfindextraneoussolutionsinoneoftwoways.

Oneisbycheckingallanswersafteryou’vesolvedfortheunknowns.Anotherisbyfindingit(orthem)first,beforesolvingtheequation.

Irecommendthisway,asexplainedinthenextsection;it’seasier,andthisway,ifyouforgettodothecheckstepattheendofaproblem,asmanypeopledo,itwon’tmatter.ThemethodforfindingextraneousrootsinradicalsisshownintheRadicals,Roots&Powerssection.

ProcedureforSolvingEquationswithRationalExpressions&ExtraneousSolutionsA.Findextraneoussolutions(solution-exceptions)first[findallpossiblevaluesofxthatwouldmakethedenominator(anydenominatorintheproblem)=0].Whenyougettotheendoftheproblem,compareyoursolutionstotheexceptions,andeliminatetheextraneoussolutionsfromyouranswers.Todothis:A0.Writeoutthedenominatorsonly(separately,ifmorethanonefraction).A1.Factoreachdenominator.A2.Setalldenominatorfactors=0andsolveforx(orwhateverthevariableis).A3.Putaslashthoughthe=sign,toremindyourselfthatxdoesnotequalthenumber(s)justdetermined.SavetheseofftothesidetorefertothemattheendofPartB.B.SolvingtheProblem:B0.Writeoutthewholeproblem.WritethedenominatorsasthefactorsyoudeterminedthroughfactoringfromstepA1.B1.DeterminetheLCD.B2.MultiplytheLCDtimeseach(numeratoronlyofeach)fractionandnon-fraction-term(onbothsidesoftheequation).Thiswilleliminatealldenominators(andthusallfractions).B3.Simplify(combinelike-terms)andsolveusingtheProcedureforSolvingaSimpleAlgebraicEquationwithOneVariable).B4.CompareyouranswerstothosefoundinstepA3andcrossoutanyextraneoussolutions.

CrossMultiplicationCrossMultiplicationistheactofmultiplyingthenumeratorofonefractiontimesthedenominatorofthefractionontheothersideoftheequalsign,andviceversa.Crossmultiplicationiscommonlyusedwhendoingproblemsinvolvingproportions,specificallywhenthereisonefraction(only)oneachsideoftheequalsign.Whenshouldyouuseit?Youshoulduseitwhentryingtosolveforavariableinthedenominatorandwhenthereisonlyonefractiononeachsideoftheequalsign.Youcanonlycrossmultiplyifthereisonlyonefractiononeachsideoftheequalsignoryousimplycan’tdoit.However…

Ifyouhavemorethanonefractiononeithersideoftheequalsign,youcaneither:

Moveoneofthefractionstotheotherside(youcandothisifyouhavetwofractionsonthesamesideequaltozeroontheotherside),or:FindtheLCDofallfractions&multiplyallfractionsbytheLCD.Thiswilltheneliminatealldenominatorsandyouwillnolongerhavetodocross—multiplication.

Don’tbefooled.Forcrossmultiplicationtooccur,theremustbeonefractiononeachsideoftheequalsign,however,thenumerators&denominatorsthemselvescanbepolynomials(iftheyare,multiplyaccordingly).Also,youcaneasilyconvertawholenumberorpolynomialintoafractionbyputtingitover“1”.Amistakestudentscommonlymakeistryingtocrossmultiplyfractionsthatareonthesamesideoftheequalsign.Crossmultiplicationcanonlybeperformedacrossequalsigns.Seeintheexamplebelowhow:

iscross-multipliedtobecome(2)(x)=(5)(3)

whichbecomes2x=15andcanbesolvedbydividingbothsidesbythecoefficient2:

,andthus or7.5Cross-multiplicationisnotthesameasmultiplyingfractions(onthesamesideoftheequalsign).Whenfractionsareonthesameside,multiplythenumeratorsbynumeratorsandthedenominatorsbydenominators(see:MultiplyingFractions).Also,Cross-Multiplicationisdifferentthan“Cross-Cancelling.”

Cross-Multiplicationvs.CrossCancellingItisimportanttousethesemethodsattheappropriatetimesandtousetheterminologycorrectly,astheyarecompletelydifferent.Cross-multiplicationisdonewhenyouhaveonefractionsetequalanotherfraction,oneofwhichcontainsanunknownvariable.Thisisoftenseenwhendoingworkwithproportionsandsometimespercentproblems.Crossmultiplicationisandcanonlybeperformedacrossan“=”signbymultiplyingthenumeratoroftheleftfractionbythedenominatoroftherightfraction,andsettingthatproductequaltotheproductofmultiplyingthedenominatoronthelefttimesthenumeratorontheright.Thisisexplainedinmoredetailintheprevioussection.Crosscancelingisasimplificationtechnique.Itistheprocessofsimplifyingandreducingfractionsbycancelingoutcommonfactorsinthenumeratorofonefractionwiththedenominatorofitselforanotherfractionitismultipliedby.

Seeintheexamplebelowhow

whenfactored,whichreducesto

because

oneofthetop2scross-cancelswiththe2inthebottomoftheotherfraction,andthe5inthetopcross-cancelswiththe5inthebottomoftheotherfraction,

whichthenequals afteryoumultiplythefractions.

RADICALS,ROOTS&POWERSNote:Beforebeginning,Imuststress(again)thatradicalsmayresemblelongdivision,buttheyarecompletelydifferent.Studentssometimesmixthemupandtrytoapplylongdivisiontoradicals,butdealingwithradicalsisnotdivision.Payspecialattentiontotreatradicalsastheirownuniquefunction.Whydoyouneedtounderstandradicals?OneofthemainneedsandusesisforsolvingquadraticequationsusingtheQuadraticFormula.TheyarealsocommonlyappliedinproblemsusingthePythagoreanTheorem(whichisnotcoveredinthisbook).Whyisusingradicalseasy?Becausetheyarenothingmorethansimplerearrangementsoffactors.Youjusthavetoknowwhatyou’relookingfor.Iwilltellyouwhattolookforinthecomingpages.Whenitcomestodealingwithradicals,themainobjectiveistosimplifythem.Tosimplifythem,youmustbeabletotogglebetweendifferentversionsofthem,andrearrangethem.Todothat,youmusthaveagoodhandleonperfectsquaresandfactoring.

PerfectSquares&AssociatedSquareRootsYoucantakethesquarerootofany(positive)numberorterm,buttheresultmaynotbeaninteger.Perfectsquaresarenumberswhosesquarerootsareintegers(non-decimalnumbers).Asetofverycommonperfectsquaresarelistedafterthefollowingexplanation.Thelistofradicalsabouttobelistedfollowtherelationshipseenbelow:

.

Inwords:Whentherootofaradicalisthesameasthepowerofthebaseoftheradicand,theradicalsimplifiestothatbase(which,here,is“x”).Also,ifaradicalisraisedtothesamenumberastheroot,italsosimplifiestoequalthebase(again,here,is“x”).Theimportantthingtorealizeisthatbothsimplifytothesamebase,“x,”inthiscase.Thisequalityalsoexemplifiesanotherproperty,whichisthattheexponentcanbemovedfromtheradicandtooutsidetheradical,andviceversa.Thisisanecessarymanipulationtechnique.Observethisexampleusing7asthesamerootandpower:

Inwords:Theseventhrootofbasextothepowerofsevenequalsx;and:Theseventhrootofxinparentheses,raisedtothepowerofseven(outsidetheparentheses)alsoequalsx.Noticeinbothversions,therootisthesameastheexponent,andbothversionsequal“x”.Youwillnoticethistrendinthelistofperfectsquaresandsquareroots,inthenextpages.

Thisalsofollowsinthenextexamplewheretherootandpowerareboth2.Youmaynoticethereisno“2”writtenastheroot,butdon’tletthatdeceiveyou.Forsquareroots,theroot2isusuallynotwritten,butit’salsonotwrongifyouwriteitin.

Inwords:Thesquarerootofaradicandwhosebaseissquaredequalsthatbase.Equivalently:Thesquareofthesquarerootofsomeradicandequalsthatradicand.Notice:whetherthesquareisinsidetheradicaloroutsidetheradical,theresultisthesame.Beforeworkingwith(simplifying)radicals,itisimportanttoknowsomeofthecommonperfectsquares.Inthenextlist,theleftcolumnshowsthesquares,andtotherightshowstheassociatedsquareroots.Havingthesememorizedwillhelpyousimplifyradicalsmorequicklyandeasily.Thesearelistedasaneasyreference,butalsosoyoucanseethepatternsasdiscussedabove.

ListofPerfectSquares&AssociatedSquareRoots02=0; =012=1; =+/-122=4; =+/-232=9; =+/-342=16; =+/-452=25; =+/-562=36; =+/-672=49; =+/-782=64; =+/-892=81; =+/-9102=100; =+/-10112=121; =+/-11.Yougettheidea…122=144132=169142=196152=225

162=256172=289182=324192=361202=400

CommonPerfectCubes&AssociatedCubeRootsThefollowingisalistofcommonperfectcubesandtheirassociatedcuberoots.Theyarehereforyoutoreference,butyoushouldalsomemorizethemandnoticethepatternsofmovingtheexponentinandoutoftheradicalasdiscussedafewpagesago.03=0; =013=1; =1-13=-1; =-123=8; =233=27; =343=64; =453=125; =563=21673=34383=51293=729103=1000

OtherPowers&Relationshipsof2,3,4&5Thisisanextrasectiontoshowothercommonpowersandexponentialrelationshipsforbase2throughbase5,atpowersof4and5.Noticeherehowthesenumberscanbefactoredandrearrangedusingtherulesofmultiplyingbaseswithexponentsandtakingpowersofpowers.24=(22)(22)= =(4)(4)=1634=(32)(32)= =(9)(9)=8144=(42)(42)= =(16)(16)=25654=(52)(52)= =(25)(25)=62525=(22)(23)= = =3235=(32)(33)= = =24345=(42)(43)= = =102455=(52)(53)= = =3125Theselistsarehereforyourreference.Keepthesepowerandrootrelationshipsinmindforthenextsection,astheyplayahelpfulroleinmanipulatingandsimplifyingradicals.

Manipulating&SimplifyingRadicalsThereasonit’simportanttobeabletorecalltheperfectsquares,perfectcubes,andotherperfectpowersisbecausetheyareessentialinsimplifyingradicals,whichiswhyIlistedmanyofthemintheprevioussection.Onemajorreasonforsimplifyingradicalsistofindwhichradicalsarelike-terms,sotheymaybecombinedasyouwouldcombinelike-termswithvariables(andthereareotherreasons,too).Itisimportanttoknowthatsimplifyingradicalsisdifferentthansimplifyingothertermsorexpressionsthatdon’thaveradicals,soyoucan’texpecttousethesamestrategy.Themaindifferenceisthatwhensimplifyingnon-radicaltermsorexpressions,youusuallyresorttofactoringintoprimefactorsand/orfindingaGCFtofactorout.Simplifyingradicalsactuallymeansfactoringandreorganizingfactorsoftheradicand,buttheradicandisnotnecessarilyprimefactored.

Tosimplifyradicals,youmustfactortheradicandintotwotypesoffactors:perfect-power—factorsandnon-perfect-powerfactors.

Andthereisaverylogicalreasonforthis.Theradicaloftheperfect-poweristobetaken,andthen(itsroot)willbemovedoutsidetheradical(andtreatedlikeacoefficientthatismultipliedbytheremainingradical).Thenon-perfect-powerfactorwillsimplyremainundertheradicalbecausethatisitsmostsimplifiedform.ObservethismethodinthenextsectionCommonRadicalFingerprints.Lookattheexampleforthesquarerootof12.Noticethatitsfactor“4”isaperfectsquarefactor,and“3”isnot,soyouseparatethemintofactors(4)(3).Now,since4isaperfectsquare,takethesquarerootofit.Sincethesquarerootof4is2,the2getsmovedtotheoutsideoftheradicalasacoefficient,andthesquarerootof3remainsintheradical,leavingyouwith(asyouwouldsay)“two(times)thesquarerootofthree.”Thereasonradicalsaresimplifiedthiswayissotheycanbemanipulatedintolike-termsthatcanbecombined(asin“combinelike-terms”).Forradicals,“like-terms”aretermsinwhichboththerootandradicandareexactlythesame.Whenthesecriteriaaremet,like-radicalsarecombinedviatheircoefficientsthesamewayaslike-termswithvariables.AsinthelastsectioninwhichIshowalistofcommonrootsandpowers,thereareothersthatarestillcommon,but“notperfect”…“notperfect”inthesensethattheradicalcannotbereducedtoaninteger.Thesearesocommon,thatIcallthem“fingerprints,”becauseafterencounteringthemenough,youmaymemorizethem,savingyouthestepofhavingtomanuallyfactorandsimplifythemeverytime.

Thefollowinglistaccomplishesthreepurposes.1. Itsimplyshowscommon“non-perfect”radicals,and2. itshowstheintermediatestepswheretheradicandsarefactoredinto

“perfect-powers”(inthiscase,they’reperfectsquares),and“non-perfectpowers”(whichinthiscasearenon-perfectsquares).

3. Italsodemonstratesthe“ProductRuleofRadicals.”Ialsodecidedtoincludeafewcommonradicalswhicharealreadyintheirmostreducedform,justtoputthemintoperspective.

ListofCommonRadicalFingerprints

Itisimportanttonotethatwhendoingsquareroots(oranyevenroots)onacalculator,mostcalculatorswillonlyreportthepositiveroot,soitisuptoyoutoalsowritethenegative.

ExtraneousRootsinRadicalEquationsThetopicofextraneousroots(a.k.a.extraneoussolutions)hasbeenexplainedpreviously,aswellashowtoidentifythemwhentheyareinadenominator.Buttheycanalsobeinanequationwithradicals.Whentheyare,youmustcheckyouranswerbysubstitutingbackintotheoriginalequationandsimplifying.Thisismentionedinanupcomingsection:CheckingYourAnswers.

FMMs(FREQUENTLYMADEMISTAKES)Thissectionofthebookisonelikenoother.Ibetyouwillnotfindonelikethisinatraditionaltextbook(atleastIhaven’tyet).Thissectionisstrictlydedicatedtohighlightingthemostcommonand“frequentlymademistakes”bystudents.Byhoninginonthesecommonmistakes,Ihopeyouwillbeabletoquicklyrecognizeandavoidthem.Thissectionisalsodifferentthantheothersectionsinthisbookinthewayitissetup.Ifit’satopicthathasn’tbeencoveredinthebookyet,Iwillgiveititsownnewsection.Ifit’samistakeI’vealreadyexplainedinaprevioussection,Iwilllistitwithabriefintroductionandprovidethehyperlinktodirectyoubacktothatrespectivesectionofthebook.

TheTwoMeaningsof“CancellingOut”“Cancellingout”canmeantwodifferentthings:

1. Cancellingouttozero,or2. Cancellingouttoone.

Studentsandinstructorsoftenjustsay“cancellingout,”whichisabitambiguousandcancauseconfusing.Youmustbeabletoproperlydifferentiatewhichcontextisbeingusedandwheneachishappening.Termsarecancelledtozerowhenoppositetermsareadded(meaningaddingandsubtractingthesameterm.Thisisoftenseenwhenyouareaddingorsubtractingthesametermtoeachsideoftheequationinordertomoveatermtotheoppositeside.Itisalsoseenduring“combinelike-terms,”whentermshappentobeoppositesofeachother.

Termscanceloutto1when:Anumberortermisdividedbyitself,orAfractionismultipliedbyitsreciprocal.

Thisisoftenseenduring:Reducingfractions;Multiplyingfractions;Thefinalstepofsolvingasimplealgebraicequationofonevariable,whereyoudividebothsidesbythecoefficientinfrontofthevariableyou’resolvingfor;andFactoringaGCFoutofaseriesofterms.

Studentsusuallydon’thaveproblemsrememberingthataddingoppositescancelsthemtozero.Butsometimesthemistakeiswhenstudentsthinkcancelingalwaysresultstozero.Youmustnotforgetthatwhenatermisdividedbyitself,itequals“1,”asshowninPropertyCrisesofZeros,OnesandNegatives.Thistypicallyinvolvesfractions(eitherduringreducingindividualfractionsormultiplicationordivisionoffractions).ButthemostcommontimeitisforgotteniswhenyoufindaGCFinaseriesoftermsandthenfactorthatGCFout(bydividingeachtermbyit),leavingtheGCFoutfront,multipliedbytheparenthesescontainingtheremainingfactorsofeachterm.

Forexample,inthefollowingexpression:18x3–6x2+3xtheGCFis3x.Tosimplifythis,youwouldfactor3xoutofeachterm.Theintermediatestep(whichyouwouldn’talwaysshow)showseachterm

dividedby3x:Studentsoftenincorrectlyanswerthisas:

.Whenaskedaboutit,theywillrespondthat“threexoverthreexcancelsout,”whichistrue,butitcancelsto“1,”notzero,sothe1mustbeshown,asinthecorrectanswershownhere:Whileonthesubjectof“cancellingout,”thisplaysaroleinmultiplicationanddivisionoffractionsbymeansof“crosscancelling.”Anothercommonmistakeorareaofconfusioniswhenstudentsmixupcrosscancellingwithcrossmultiplication.Thisisexplainedin:CrossMultiplyingvs.CrossCancelling.Followthislinktoseetheproperwaytocrossmultiply.

CheckingYourAnswers

Thelaststepofanyproblem-solvingprocedureistocheckyouranswer.Specifically,thatmeanstakingthevalueoftheunknownyoudetermined,substitutingitbackintotheoriginalequation,simplifying,andreviewingtheoutcome.If,aftersimplifying,theleftsideequalstherightside,thisaffirmsthatyouransweriscorrect.However,ifitdoesn’t,therearethreepossiblereasonswhy.

1. Youmadeanerrordoingtheproblem.2. You’veidentifiedanextraneoussolution.3. Youmadeanerrorinyourmathinthecheckstep.

Oneclearrealityisthatstudentseitherforget,orjusthatedoingthecheck

step.IfIhadtoguess,it’sbecauseitcostsextratimeandworkspacetodo,andstudentsjustwanttobedonewithaproblem,especiallyproblemsthatarelongtobeginwith.Buttoattaincompleteanswers,you’reexpectedtocheckyouranswers,andtherearetimeswhencheckingwillhelpyoudiscoveranerrororanomaly.Thiscouldsaveyouvaluablepointsonatest.

First,itwillsimplydrawyourattentiontoanerroryoumadeduringproblemsolving.Ifyoucan’tfindyourmistakebyreviewingyourwork,considerstartingitoverwithoutlookingatthelastwayyoudidit.Consequently,youcanalsomakeanerrorduringthecheckstepswhichmayleadyoutothinkyoumadeanerrorintheoriginalsteps,butdidn’t.Eitherway,youranswersshouldcheckout.

There’salsoanothermajorreasonananswermightnotcheckout,andthatisduetoanextraneoussolution.Extraneoussolutionsoftenoccurwhenavariableisinthedenominatorinanequation(orinsidearadical).

MiscellaneousMistakes

Whenmultiplyingfactorsofacommonbasewithexponents,sometimesstudentsmistakenlymultiplytheexponents.Whenfactorsofacommonbasearemultiplied,theirexponentsareadded.

Sometimesstudentsarerequiredtodistributeanexponentthroughatermofmultiple(variable)baseswithexponents.Thisistakingthepowerofeachbasetothepowerbeingdistributed.Thereisoftenacoefficientattachedtothevariables,andwhenthereis,studentsoftenforgettoapplythepower(fromoutsidetheparentheses)tothecoefficient.Thereasonmightbebecausestudentsareusedtotakingthepowerofthepowerofeachvariablebase,andtheyjustforgetaboutthecoefficient.Whendistributinganexponentthroughagroupofbaseswithacoefficient,don’tforgettoapplytheexponenttothecoefficient.

Whengivenanequationwithatrinomial,oraquadraticequation,sometimesstudentswillsuccessfullyfactorit,butthenforgettodothelaststep,whichistosolve.

Studentscommonlymakethemistakeofusing“zero”and“noslope”or“undefined”interchangeably,buttheyhavecompletelydifferentmeanings.

ClickthelinkforcommonmistakesstudentsmakeduringtheSubstitutionMethod.

ClickthelinkforcommonmistakesstudentsmakeduringtheAddition/EliminationMethodforsolvingasystemoftwolinearequations.

Studentsoftenmakeamistakewhenanegativesignisinfrontofafractionbynotproperlydistributingthenegativesignthrough,changingthesignofeachtermintheseries.

Equationsandexpressionsareintendedtobesimplifiedcompletely.Oftentimes,studentsdomostoftheproblemcorrectly,butmakeoneoftwovitalmistakesthatcouldmakeorbreakananswer(especiallywheninstructorsdon’tgivepartialcredit).Sometimesstudentsgetnearthe

end,butsimplyforgettosimplifytheanswer.Or,sometimesstudentsattempttosimplify,butdoitwrong.Learntoavoid:TheWrongWaytoSimplifyaRationalExpression.

WhenstudentsusetheQuadraticFormula,theyoftenforgettosimplifythelaststep.Thisisexplainedin:ThePartEveryoneForgets:TheLastStepoftheQuadraticEquation

Whenapplyingthe“specialcase”shortcutmethodtomultiplyingoutabinomialsquared,studentsoftenmakethecommonmistakeofusingtheshortcutmethodformultiplyingconjugatepairbinomials.Thismistakeresultsinthemissing“bx”term.

Whenanegativesignisinfrontofarationalexpression(afractionwithapolynomialinthenumerator)studentsveryoftenforgettodistributethatnegativesignthroughalltermsinthenumerator.Thisthenincorrectlyassociatesthenegativetoonlythefirstterminthenumerator,leavingthetermstofollowwithoppositesignsthanwhattheyshouldbe.

RadicalsAreNotLongDivision.There’snotreallymuchtosayaboutthisotherthanthatthesymbolsandsetupofradicalsandlongdivisionaresimilarlooking,buttheyarecompletelydifferentoperations.AnytimeI’veeverencounteredastudentattemptingtoapplylongdivisiontoaradicalmayhavebeentheirdesperateattempttodosomethingwhentheyhadnoideahowtoapproachradicals(mostlikelyduetolackofpreparation).Longdivisionisaprocesstofindouthowmanytimesthedivisorgoesintothedividend,andtheansweristhequotient.Butradicalsareusedtoanswer:Whatnumber,whichwhentakentothepowershownastheroot,equalstheradicand?Theradicandwon’talwaysbeaperfectpowernumber,andinthatcase,assumingyoudon’tuseacalculator,youwillbreakitdownandsimplifyitusingtheruleofmultiplicationofroots,asbrieflyshowninCommonRadicalFingerprints.

ScientificNotationonYourCalculator

Scientificnotationisastandardizedwayofreportingnumbersthatareeitherverybigorverysmall,withmanyzerosand/ordecimalplaces.Itisawaytoexpressnumbersintoamanageableformat,andisoftenusedinscienceandstatistics.Scientificnotationisthealternatewayofwritinganumberfromitsexpandedform.AlthoughIdonotcoverscientificnotationinthisbook,Iwanttoaddressthemistakestudentsfrequentlymakewhenputtingscientificnotationintoacalculator.Themistakeissomevariationofnotknowinghowtoproperlyputitintothecalculator.

Sincetherearegenerallytwotypesofcalculators(scientificandgraphing)withthescientificnotationfunction,eachtypeandbrandvariesinwhatbuttonstheyhavetoaccomplishthisfunction,soit’sagoodideatobepreparedforeachpossibility.Thereisalsoacompletelywrongwaytoinputscientificnotation,whichresultsinthenumberbeingoffbyanorderofmagnitude(afactorof10,orinotherwords,offbyonezero).Typically,onallcalculators,youstartoffthesame,bytypinginthebasenumber.Next,youmusthittheexponentbutton,butnotthesameexponentbuttonyouwouldusefornormalexponents.Thebuttonyouwantmaylooklikeanyofthefollowing:[EE][exp][EXP][x10](meaning“timestentothe…”)[10x](meaning“meaningbasetentothepowerof”)[antiLOG](oftena2ndfunctionto[LOG])[e](nottobeconfusedwith[ex],whichstandsfor“thenumberetothex,”alsoknownas“inverseLN,”whichis“inversenaturallog”).Ifoneofthefunctionsshownaboveisa2ndfunction,meaningthesymbolisshowninanothercolor,aboveaprimarybutton,youmusthitorholdabuttonsuchas:“2nd,”“Shift,”or“Alt,”oftenlocatedatthetopleftcornerofthecalculator,thenhitthebuttonasitisshown(fromthechoiceslistedabove).Youmighthavetolookaroundforit;itdoesn’talwaysjumpoutatyouatfirst.

Toreiterate,youwouldfirsttypeyourbasenumber,thenscientificnotationbutton(shownabove),thentheexponent(ofthe10).

Hereistheplacestudentsoftenmakeamistake…bymanuallytypingout:[thebase#][x][10][EXP][theexponent]...Inwords,thatwouldsay,“basenumbertimesten,timestentothepowerofsomenumber.”Inotherwords,thiscausesaredundantmultipleof10,whichwillresultinyournumberbeingoffbyafactoroften.Topreventthis,youmustuseeitheroneortheother(eitherthe[EXP,thentheexponent],or[x10^theexponent]).Considertheexampleofconverting9,400,000toscientificnotation,whichwouldbe9.4x106.YouwouldtypeEither:[9.4][x][10][^][6]Inwords:Ninepointfourtimestentothesixth,usingtheexponentfeature,notthescientificnotationfeature.Or:[9.4][EXP][6]Thisisthepreferredwaytoinputnumbersinscientificnotation.Inwords,thisreadsthesameasabove(“Ninepointfourtimestentothesixth”),butthebuttonsareclearlydifferent.Inthisversion,thescientificnotationbuttonisused,notmanuallytypingtheten,thecarrot,andtheexponentsix.Irecommendgettingusetothe[EXP]button.

WhatDoes“Error”onaCalculatorMean?Oftentimes,studentswillputanoperationintothecalculatorandgettheresponse:“Error.”Somemisinterpretwhatthatmeans.Sometimesstudentsinterpretthatas“thestudentmadeanerror,”butthismanynotbethecase.Whentheoutputonthecalculatoris“Error,”itcouldmeanoneofthefollowingthings:

1. “Error”isthecorrectandexpectedresponse.Whatwemightcall“Undefined”or“NoSolution”toanarithmeticoperation,thecalculatorwillreportas“Error.”Forsuchexamplesinvolvingdivisionandradicals,see:PropertyCrisesofZeros,OnesandNegatives).

2. Sometimes,however,“Error”meansyoumadeamistakein-putting

yourintendedoperation.Inthatcase,youshouldcheckwhatyoutypedandlookforanerrorinthatrespect.Forinstance,youmayhaveaccidentallytypedtwodecimalsinanumber.Ifyoucheck,anddon’tfindaninputmistake,thenthereisagoodchance“error”isthecorrectresponseforareason.

CLOSINGMymissionistohelpaveragepeoplebreakthroughmathbarriers,whateverthesourceofthebarriersmaybe.Youlearnbasedonhowyouaretaught.Ilearnhowtoteachbasedonthetrials,errorsandpatternsofwhatandhowstudentslearn.Itisaconstantlearningcurvetryingtoperfecthowtopredicthowstudentsprocessthelessons;thelearningisatwo-waystreet.Iamextremelyinterestedinyourfeedback.Pleasetellmewhatworked,whatyouliked,whatyoudidn’tlike,whatwasconfusing,andwhatyou’dliketosee(moreof).Ifthisbookhelpedyou,Iaskthatyoupleasesupportmymissionbytellingyourfriendsandfamilyaboutitandleavingareview.Iwishyouthebestofluckinallyoudo.Pleasesendyourfeedbacktomypersonalemailaddress:bullockgr@gmail.com.Also,feelfreetofollow@GregBullockand@AlgebraInWordsonTwitter.