Math Review for Algebra and Precalculus

Transcription

1 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Alger nd Prelulus Stnley Oken Deprtment of Mtemtis Te City College of CUNY Copyrigt Jnury 00

2 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Tle of Contents Prt I: Alger Notes for Mt 9 Introdution.... Bsi lger lws; order of opertions.... How lger works... Simplifying polynomil epressions 9. Funtions.... Wen to use prenteses.... Working wit frtions. Adding frtions...

3 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Prelulus nd Clulus Prt I: Alger Introdution Alger is te lnguge of lulus, nd lulus is needed for siene nd engineering. Wen you ttk rel-world prolem, you wnt to represent te prolem using lger epressions. Wen you red tenil ooks, you wnt to e omfortle deipering nd working wit tese epressions. Computers n t do eiter of tese tsks for you. Alger used in undergrdute mtemtis involves tree min tivities: rewriting epressions, solving equtions, nd solving inequlities. You need to perform tese somewt menil tivities quikly nd urtely. It s very diffiult to ieve tis gol unless you understnd ow lger works. Alger is symoli lnguge tt llows ommunition etween people wo don t know e oters spoken lnguge. Te grmmr of te lnguge involves tree min omponents: epressions, identities, nd equtions. An epression involves numers, vriles, prenteses, nd lger opertions. Bsi types of epressions re integers, vriles, monomils, polynomils, nd so fort. We ll del mostly wit epressions in one vrile, su s te polynomil. An identity etween two epressions, written wit n equls sign, is sttement tt e epression n e otined y rewriting te oter. A simple emple is. Wit rre eeptions, sustituting numers for vriles turns n identity into true sttement out numers. For emple, setting to yields. An importnt prt of lger is using identities to rewrite epressions. An eqution is lso sttement tt two epressions re equl. In most equtions, owever, equlity olds only for speifi vlues of te vrile. For emple, te sttement is true only wen is or. We sy tt te solutions of te eqution re nd. Plese rememer;: we rewrite epressions ut we solve equtions. In tis preliminry edition, setion edings su s AN re used for Alger Notes, Setion.

4 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : BASIC ALGEBRA LAWS; ORDER OF OPERATIONS.. Alger opertions nd nottion Let s egin wit two triky emples. Emple..: Rewrite s n epression wit no negtive powers. Rigt: Wrong: Emple..: Simplify te epression Rigt: Wrong: To understnd wt s going on, we need to review in some detil te five lger opertions: ddition, sutrtion, multiplition, division, nd eponentition. E of tese is lled inry opertion euse it is used to omine two epressions. Te tle elow lists nottion nd terminology for tese opertions. Te lst entry sows speil opertion lled negtion, wi opertes on one epression nd is n revition for multiplition y. Opertion Write Sy Desrie te nswer Addition plus Te sum of nd Sutrtion sutrt or minus Te differene of nd Multiplition Trditionl nottion times Te produt of nd Trditionl nottion * Clultor nottion Implied times sign y Implied times sign Division Seldom used / Clultor nottion Trditionl nottion Eponentition Trditionl nottion ^ Clultor nottion Negtion divided y four tirds over slng rised to te rd power Negtive or Minus Negtive of minus Te quotient of y Te rd power of is te se. is te eponent. Te negtion dditive opposite of is. It s it nnoying tt te minus sign is used for tree different purposes: nming negtive numer, sutrtion, nd negtion. Speifilly:

5 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. A minus sign efore numer s in is prt of te numer nme. A minus sign etween two epressions s in y mens sutrt. A minus sign tt is te leftmost symol in n epression s in, or one tt follows left prentesis s in, mens negtion: multiply y. To redue te numer of menings from tree to two, some ooks reserve te long ds for negtion or sutrtion ut write negtive numers wit sort ds e.g., s -. Te fewer symols n epression ontins, te esier it is to understnd. Te multiplition sign is left out s often s possile: etween digit nd prentesis, nd etween two prenteses, s sown in te emples elow. Multiplition sign omitted Multiplition sign inluded. Evluting Numeril Epressions Alger egins wit ritmeti nd te properties of wole numers: 0,,,,,,, Te most si property is te identity lw, wi sttes tt dding zero to ny numer or multiplying it y leve te numer unnged. One importnt property of ddition of wole numers is tt ddition is ommuttive: two numers n e dded in eiter order. Some emples of tis property: To tully stte te lw, we use letters nd to stnd for ny two wole numers. Commuttive lw for ddition:. Anoter importnt property of ddition of wole numers is tt ddition is ssoitive: wen you dd tree numers, te two possile metods of grouping yield te sme nswer. Some emples: To stte te ssoitive lw, we let,, nd stnd for ny tree wole numers. Assoitive lw for ddition: Te ssoitive lw llows us to write epressions su s nd 89 witout using prenteses.

6 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Togeter, te ssoitive nd ommuttive lws for ddition llow us to write sums witout prenteses nd to reorder te numers in te sum. Tus et. For emple, using ; ;, we ve. Similr lws pply to multiplition: Commuttive lw for multiplition: Assoitive lw for multiplition: Togeter, te ssoitive nd ommuttive lws for multiplition llow us to write produts witout prenteses nd to reorder te numers in te produt. Tus et. For emple,. Addition nd multiplition re onneted y te two versions of te distriutive lw, wi sttes tt wen sum is multiplied y numer, te multiplier n e distriuted mong te numers tt mke up te sum. Distriutive lw: nd Te si rules, or lws, of ritmeti nd lger re listed in te following tle, in wi,, represent ny wole numers, ddition is indited y, nd multiplition is indited y. Bsi lger lws for wole numers Nme of Lw Sttement of lw Emple wit ; ; ; Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 0 Assoitive lw for? 9 9 Identity lw for Commuttive lw for Assoitive lw for? Distriutive lw? 8 In te ove disussion, we sid we were working wit wole numers. As you lredy know, tere re more generl lsses of numers. Integers signed wole numers re,,,, 0,,,,

7 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Rtionl numers re frtions of integers:,,, Deiml numers re rtionl numers wit denomintors, 0, 00, 000, : Emples re., 0.0, nd Rel numers inlude rtionl numers nd oter numers su s e, π, tt nnot e epressed s frtions of integers. Oter lws nd definitions tt pply to te tegories of numers listed ove re: 0 : Te sum of numer nd its negtive is zero. Sutrtion is defined y. were,,, d re integers. Here nd d re not zero. d d is lled te reiprol of if 0. Te produt of non-zero rtionl numer nd its reiprol is. d Division is defined y if,, d re not zero. d To divide y non-zero frtion, multiply y its reiprol. Eerise..: Using te tle ove s model, plese fill in te following tles. Bsi lger lws for integers Nme of Lw Sttement of lw Emple wit ; ; ; Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw

8 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Bsi lger lws for rtionl numers Nme of Lw Sttement of lw Emple wit ; ; Identity lw for 0 Cek te emple y working out ot sides Commuttive lw for Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw Bsi lger lws for deiml numers Nme of Lw Sttement of lw Emple wit.;.;. Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw 8

9 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. PEMDAS nd Order of Opertions We just sw tt n e written witout prenteses, euse te ssoitive lw tells us tt. Similrly, tere re mny wys to insert prenteses in, ut te ssoitive lw gurntees tt te nswer will e te sme in tis se, 0 no mtter were te prenteses re inserted. Very often, owever, te oie of prenteses mtters. For emple, 0 ut. Wen you onstrut longer epressions, te sitution gets worse. For emple, wt does 8 men? 8 Some possiilities: [ 8 ] [ ] 8 [8 ] [8 ] 0 To resolve te miguity in su epressions, mt people ve speified order of opertions rules tt tell you wt opertion to do first in n epression tt doesn t ve prenteses. Anoter wy to sy tis is tt order of opertions rules sow were to insert prenteses in ritmeti epressions. You proly know te ronym PEMDAS Prenteses, Eponents, Multiplition nd Division, Addition nd Sutrtion for order of opertions in numeril or more generl epressions. PEMDAS is usully desried s step-y-step proedure for removing prenteses from numeril epression. However, it pplies to epressions wit vriles s well. One wy to eplin PEMDAS is s follows te word evlute mens find te vlue of : To evlute numeril epression tt doesn t involve prenteses: Step : Evlute ll Eponents powers, working from rigt to left. Note: In te following disussion, we will refer to eponents s powers; do not onfuse te word powers wit te P in PEMDAS, wi stnds for Prenteses. Step : Evlute ll Multiplitions nd Divisions, working from left to rigt. Step : Evlute ll Additions nd Sutrtions, working from left to rigt. To evlute numeril epression tt does involve prenteses: Step I: Find n innermost set of Prenteses: pir of prenteses tt don t ve oter prenteses etween tem. Evlute te epression etween tem using Steps,, ove. Step II: If ny prenteses remin, go to Step I. Oterwise you re done. 9

10 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. You don t need to egin wit n innermost set of prenteses: you n simplify prentesized epressions in ny order you like. For emple, to simplify te epression, you ould eiter egin wit te inner set of prenteses: or egin wit te outer set of prenteses, otining te sme nswer: 8. Furtermore, you n work on severl sets of prenteses simultneously. Tis is done in te rigt olumn of te following emple: te left olumn works on one set t time nd follows te PEMDAS rules stritly. 8 8/ / 0 8/ 8/ 8/ / / / / / / / Reding epressions wit minus signs: It s wort mentioning few triky emples involving minus signs. Te word minus nd te word negtive men pretty mu te sme ting, ltoug te prse minus numer is not used. Don t use minus to men sutrtion. Symols Words sutrt equls sutrt equls minus 0 Minus sutrt equls minus 0 Minus sutrt minus equls times te quntity: sutrt : equls times sutrt equls Here re some simple emples wit numeril epressions. In e emple, we egin y inserting prenteses round te opertion tt omes first s speified y te order of opertions rules. 0

11 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Powers efore ddition 0 Multiplition efore ddition 8 Powers efore multiplition 8 Do powers from rigt to left Add/sutrt from left to rigt 8 Multiply/divide from left to rigt. Te following emples omit prenteses implied y order of opertions. 0 Add/sutrt from left to rigt 8 8 Multiply/divide from left to rigt 9 / Multiply/divide from left to rigt 0 8 Do powers from rigt to left top to ottom 9. Multiply efore you dd or sutrt Multiply efore you dd or sutrt 9 Do powers efore nyting else. 9 Do powers efore nyting else. 9 Do powers efore nyting else. / / Do powers efore nyting else. Eerise..: Use PEMDAS to simplify e of te following numeril epressions. Strt y inserting prenteses tt re implied y te order of opertions. m 0 8 n 0 o d p 0 e /8 q 0 f r 0 g 0 8 s 0 / / t i u j / v k w 8 9 l

12 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : HOW ALGEBRA WORKS.. Implied multiplition signs in epressions involving vriles Te multiplition sign is left out s often s possile wenever it ppers etween two vriles, etween two prenteses, or etween ny two of te symols: digit, vrile, prentesis. One eeption: te epression, wi migt resonly e tken to men times, is not used; write or, preferly,. Epression wit multiplition signs omitted Signs inluded digit followed y vrile y vrile followed y vrile y vrile followed y left prentesis vrile following rigt prentesis y digit followed y left prentesis y y digit following left prentesis y left prentesis following rigt prentesis y eponent followed y ny se y. Using te distriutive lw to remove prenteses Erlier, we wrote te distriutive lw s Omitting te multiplition signs, we ve Finlly we n rewrite tis s sine multiplition preedes ddition in te order of opertions. Te distriutive lw omes up in mny situtions ut is triky to work wit wen minus signs pper. Te following re te ptterns tt ommonly our. Cution: is red minus sutrt nd is not relted to te produts on te previous line. Note tt. Removing prenteses following minus sign: Te following identity illustrtes very importnt rewriting opertion: removing prenteses tt follow minus sign.. d d

13 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Tis opertion sould e utomti: tink of multiplying te minus sign y e term inside te prenteses. Here re te detils, wi you sould understnd one, ten put side in fvor of te utomti proedure. d Tis is sutrtion prolem. d Apply te definition of sutrtion: u u v u v wit v d d Apply te definition of negtion: v v v wit v d d Use te distriutive lw to multiply y -: d d d d d Remove prenteses. Here is relted emple of te form d d d Tis is sutrtion prolem. d Apply te definition of sutrtion. d Apply te definition of negtion. d Apply te distriutive lw. d Remove prenteses. Eerise..: Fill in te tle elow. Version of distriutive lw Emple wit ; ; ; d??? d? e f g? ut i Wy? Cek te emple y working out ot sides? 8

14 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. j d d k d? l d? m d? n d d o d d Te preeding ws review of ritmeti. Now we move on to lger.. We do lger y rewriting epressions An epression involves numers, vriles, prenteses, nd lger opertions. Some si types of epressions seprted y semiolons re integers: ; 0; ; 9; 8 vriles: ; y; z; ; ; monomils in : 00 0; ; ; ; ; ; polynomils in : 0; ; ; ; ; rtionl polynomils in : ; ; 00 Alger nd first yer lulus fous on polynomils nd rtionl polynomils wit one or two vriles usully nd y, s well s on equtions nd inequlities involving su epressions. Te rdest prt of lger is rewriting epressions orretly. A typil lger prolem is sequene of rewriting opertions, wi n e written using two possile formts: Horizontl: Originl epression Epression Epression Finl Epression Vertil: Originl epression Epression Epression Finl Epression

15 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. It s mu esier to void mistkes wen you use vertil nottion, euse you re opying e epression from te one diretly ove it. You sould tink of tis formt s si prgrp of mtemtis, in wi e epression is sentene. In most ses te finl epression is otined y simplifying te originl epression to mke it esier to understnd. Simplifying numeril epression mens rewriting it s single numer. However, tere re two wys to simplify n epression wit vriles, y rewriting it s simplified sum or s simplified produt. For emple, one you review ow to multiply using FOIL, you ould use vertil formt to rewrite s simplified produt, wit no repeted ftors: or s simplified sum, wit no repeted terms:. Sustituting in epressions Simplifying epressions uses very si rewriting proess lled sustitution. In sketll gme, sustituting plyer A for plyer B mens repling plyer B y plyer A. Similrly, to sustitute epression A for epression B, reple epression B y epression A, wit one tenil requirement: Definition of sustitution: To sustitute epression S for vrile in epression E, erse every letter in E nd reple it y te epression S enlosed in prenteses. Plese engrve te following sttement in te innermost ore of your mtemtil suonsious: Wen you sustitute n epression for vrile, put prenteses round te epression. Here is n informl eerise: try to find tetook tt orretly desries sustitution. Emple..: Sustitute for in te epression Corret Metod: Erse every tere s just one in te epression nd reple it y.

16 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Corret Answer: Inorret Metod: Reple every y Inorret Answer: Two importnt oservtions: Prenteses mke differene, sine 8 weres 9 9 Te nswer is not simplified. It s orret, euse te question sid sustitute rter tn sustitute nd simplify. Sustitution nd simplifition re different proesses: Sustitution is purely menil erse nd reple opertion. Don t tink, just erse nd reple! Simplifition involves rewriting te epression y pplying lger rules. Tinking out lger wen you do sustitution very often leds to errors. Te tle elow lists ommon errors tt our rter frequently. To revel e error, we simplify te result of e orret sustitution. Question mrks pper in te lst olumn wen omitting prenteses produes result wit onseutive minus signs, nottion tt is usully onsidered illegl. Sustitute Corret Inorret for in for in for in for in for in? for in 9 for in 9 for in? for in for in

17 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te following tle gives emples of sustitution in epressions, one per row: Sustitute For In tis Corret Answer Inorret tis tis f f f f f f f f f Eerise..: Sustitute s indited ut do not simplify in e of te following epressions. for in for in for in d for in e for in f for in g for in for in i for nd for in j k l for nd for in for nd for in for nd for in Eerise..: Even toug we ven t disussed simplifition metods in detil, see if you n rewrite e nswer in Eerise.. s simplified sum. Here is n emple using te nswer to.. i, :

18 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Answer to.. i: Te lws of lger re listed elow. Run troug tem quikly, ten ontinue to te net setion nd refer k to tis list s neessry. Addition lws Identity: 0 Commuttive: Assoitive Bot re written s. Inverse: For e epression tere is n epression wit 0., lled te negtion or negtive of, is equl to te produt. Te ommuttive nd ssoitive lws llow us to ompute sum of tree or more epressions in ny order. Emple: d e e d. All su sums n e written simply s d e. Definition of sutrtion: Te differene of nd is written nd is defined y. Multiplition lws Identity: Commuttive. Assoitive Bot re written s. Inverse: Let e ny nonzero epression. Ten tere is n epression wit te property. In lultor nottion, we write /. 8

19 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te ommuttive nd ssoitive lws llow us to ompute produt of tree or more epressions in ny order. For emple, d e e d. All su produts n e written simply s de. Definition of division: If is nonzero ten te quotient of y, written In lultor nottion te quotient is / /. If 0, te quotient is undefined. Distriutive lw: te onnetion etween produts nd sums, is defined to e Tis lw is te sis for multiplition rules su s FOIL, wi sys tt d d d. Power lws rules for eponents: ; In lultor nottion: E ^ ^ ^ ^ ^ ^ ^. Te ove lws re lled ioms or definitions. All oter lger rules n e derived s teorems. In prtiulr, you migt wnt to prove te following: Teorem : Eponentil nottion 0 ;, et. d e Teorem : Sutrtion nd ddition re inverse opertions: Teorem : Division nd multiplition re inverse opertions. 9

20 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In lultor nottion: / / Teorem : Frtion rules: d d d d d In lultor nottion: / / / / d d / d / / d / d. Sustituting in Identities In Cpter, te disussion of lger rules su s te distriutive lw seemed to ssume tt te vriles used were stnd-ins for numers. Te ig piture of lger is tt te vriles in tose lws n stnd for ritrry epressions wi temselves involve vriles. Te min ide is s follows. Te lws of lger, listed in te lst setion, re si identities tt tt pply to ll epressions. Wen you sustitute epressions for vriles in lger lws, you get new identities. Just rememer to put prenteses round te sustituted epressions! Alger involves using wt students usully ll formuls, or rules: we ll ot of tese identities, euse tey re sttements tt two epressions re identil. Given ny epression E, te sttement E E is n ovious identity. Insted of sying tt F E is n identity, we will simply write F E. Wen you do lger, you sustitute for vriles in lger rules to get new identities. Fundmentl Sustitution Priniple: Wen you sustitute n epression for vrile in n identity, te result is n identity. In prtie, tis priniple is pplied s follows: Suppose is vrile, F is n epression you re working wit ontining vrile, nd E is n epression tt you know is equl to tt is, E. If you sustitute epression E for vrile in epression F to get new epression F*, ten F* F. Emple..: Sustitute for, for, nd 0 for in te distriutive lw:. You get new identity: 0 0 Now sustitute for in tis new identity to get tird identity: 0 0 0

21 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Agin we ution tt mny, mny lger errors our euse students forget te prenteses wen tey sustitute. Rememer: Wen you sustitute n epression for vrile, erse nd reple te vrile y te epression written inside prenteses. Eerise..: Sustitute s indited ut do not simplify for nd for in for in for in d for, for, nd for in e for in f for in g for in y y for in i for nd for in j for nd for in k for nd for in l for nd for in m for nd for in n for in f f o for in f p for in f z z Emples n, o, nd p my seem puzzling. Emple n is n identity if f is vrile. In emples o nd p, f is te nme of funtion nd te identity sown is speil kind of identity lled funtion definition, wi will e disussed in te following setion. Eerise..: Find n lger lw wit te following property: Omitting prenteses wen you sustitute for nd for yields flse sttement.. Order of opertions for generl epressions It s triky to work wit epressions tt involve ot vriles nd prenteses. In speil ses su s d e nd de, you n perform te opertions in ny

22 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. order. In most ses, te order of opertions does mtter nd te sfest proedure is to insert prenteses tt indite te order of opertions. Emple..: Write s simplified sum. Notie tt te epression doesn t ve prenteses sowing weter te first opertion is or. However, ording to te order of opertions rules, te multiplition is done first. Now omes te triky prt: Sine 8, you migt egin y writing 8. Tt s not orret! Te order of opertions rule tells you were to insert prenteses: te miguous epression relly mens [ ]. Here we used squre rkets to sow te new prenteses tt were inserted to indite te order of opertions. Terefore, you sould proeed s follows: Te originl epression is Insert prenteses round te produt, sine multiplition omes efore sutrtion. [ ] Find te produt inside te prenteses. [ 8 ] Use te distriutive lw. 8 Collet like terms. 8. To sve step, figure out wi lger opertion needs to e done first nd ten ple prenteses round te result of te opertion, s follows: Te originl epression is Insert prenteses round te produt. [ 8 ] Use te distriutive lw. 8 Collet like terms. 8 Alwys keep in mind te following: lger works y rewriting epressions. You sould Know nd use PEMDAS to follow te order of opertions. Put prenteses round te result of e lger opertion. In some ses, te inserted prenteses migt e superfluous, ut it s etter to e sfe tn sorry! Keep in mind te order of opertions s you red troug te emples elow. For lrity, tey ll sow te inserted prenteses s squre rkets.

23 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In, prenteses re inserted ot efore nd fter te lger opertion is rried out. You n sfely omit te efore step s in. You my omit te fter step s well, s in, ut e reful! Oter prolems re written out in full detil. 8 8] [ ] [ 8 8] [ 8 0 ] [ ] [ ] [ ] [ ] [ ] [ ] [ [ ] / / / [ ] ] [ ] [ ] [ ] [ ] [ ] [ ] [ 8 9 ] [ ] [ 0 8

24 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. [ ] [ ] [ 8 ] 0 [ ] Eerise..: Rewrite e of te following s simplified sum. d e f g i j k l m n Eerise..: Rewrite e epression in one vrile s stndrd form polynomil, wit no repeted powers nd powers deresing from rigt to left. Rewrite e epression involving two vriles s simplified sum. Rememer to put prenteses round te result of e lger opertion. A] B] C] d e f g

25 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. i j k l y y y y y m n y y. Clssifition of Epressions Tis setion is kind of tenil referene to terminology found in mtemtis tets. It lists te kinds of epressions tt n e sustituted for vriles in te si identities disussed in te previous setion. Some of tis mteril s een overed efore: it s lwys good to review. An integer is wole numer, possily preeded y sign: 0; ± ; ± ; ±,,,. A rtionl numer is quotient of integers: ; ; 8 ; 8 A vrile is letter, usully tken from te Englis or Greek lpet. Te most frequently used vriles re, y, z, s nd t. A positive integer power of is one of te following:,,,,... Tese symols re revitions for repeted multiplition, s follows. 0 ; ;, wi mens times ;.. Negtive integer powers re lternte nottion for reiprols: is wy of writing tt voids using frtions. Anoter point of view: is wy of writing tt voids using negtive eponents. A monomil in is n integer, power of, or n integer times positive integer power of. 0

26 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te degree of monomil in is te power to wi is rised. It is undefined if te monomil is zero, or is zero if te monomil is n integer. For emple,,,,, re monomils wit degrees 0, 0,,,, nd respetively. Te degree of te zero monomil is undefined. Te oeffiient of monomil is defined seprtely for te tree ses just listed: Te oeffiient of n integer is te integer. Te oeffiient of power is. Te oeffiient of n integer times power of is te integer. Here re nine monomils: 0,,,,,,,, Teir oeffiients re: 0,,,,,,,,. A polynomil is monomil or sum or differene of monomils. Emples re ; ;. Te lst emple is lwys written. 0 0 A stndrd form polynomil is one in wi ll te monomils ve different powers nd te powers of te monomils derese s you red from left to rigt. If polynomil is in stndrd form, ten te leding term is te monomil wit te igest power of. te leding oeffiient is te oeffiient of te leding term. Every polynomil n e rewritten in stndrd form: reorder nd ten ollet terms y using te distriutive lw.. Emple: Te degree of polynomil is te igest power of te monomils in te polynomil otined y rewriting te originl polynomil in stndrd form. Emples: 0 s leding oeffiient 0. s leding oeffiient

27 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. s leding oeffiient. 8 s leding oeffiient. s degree. s degree. in stndrd form is nd so s degree. A moni polynomil is stndrd form polynomil wit leding oeffiient. Mke sure you know te ove definitions well. Te remining ones elow re more tenil: refer k to tem wen neessry. A polynomil epression is uilt out of polynomils y te opertions of dding, sutrting, multiplying nd rising to positive integer power. Every polynomil epression n e rewritten s polynomil y multiplying out to remove prenteses. For emple: A rtionl polynomil is frtion of polynomils, possily preeded y minus sign. Te frtion line mens tt te top numertor is divided y te ottom denomintor. A stndrd form rtionl polynomil is one wose numertor nd denomintor re in stndrd form nd ve positive leding oeffiients. Emple: Rewrite in stndrd form s follows: A polynomil is speil kind of rtionl funtion, sine p p. A rtionl polynomil epression is frtion of polynomil epressions. Emple:. Every rtionl polynomil epression n e rewritten s rtionl polynomil y multiplying out to remove prenteses.

28 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. A generl rtionl epression is uilt out of polynomil epressions y dding, sutrting, or multiplying two rtionl polynomil epressions, or y rising rtionl polynomil epression to positive or negtive integer power. Here is modest emple Every rtionl epression n e rewritten s rtionl polynomil y possily gret del of multiplying out nd working wit frtions to omine everyting into stndrd form rtionl polynomil. You migt try to do tt simplifition for te ove emple. You will quikly e disourged unless you ve ess to omputer lger system. In te ove emple, eponents re restrited to e integers. If tey re llowed to e rtionl numers, ten te epressions re lled lgeri epressions. You know, of ourse, tt rtionl powers n e written wit rdil signs: te si emple is tt / n e rewritten s. 8

29 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : SIMPLIFYING POLYNOMIAL EXPRESSIONS You re fmilir wit polynomil epressions in single vrile. Su epressions involve numers, te vrile, nd wole numer eponents. Division signs nd frtions re not permitted. Some emples of polynomil epressions, in list seprted y semiolons, re: 0; ; ; ; ; ; ; ; 8 ; In tis setion, we sow ow to use te lws of lger to rewrite ny polynomil epression s polynomil in stndrd wy.. Monomils nd Polynomils A monomil in is numer, wole numer power of, or non-zero numer times positive integer power of. Emples: ; ; ; We will utomtilly rewrite te monomil 0 k s k, sine 0 A polynomil in is sum of simplified monomils in. Emples: 0; ; ; ; ; ; A simplified polynomil in is sum of simplified monomils in in wi no two monomils ve te sme power. Emples: 0; ; ; A stndrd form polynomil in lso lled polynomil in stndrd form is simplified polynomil in in wi te powers of derese s you red from left to rigt. Emples: 0; ; ; 9; Almost everyting we ve studied so fr: te ssoitive, ommuttive, nd distriutive lws, s well s te order of opertions, will e used in te net setion.. Rewriting polynomil epressions in stndrd form. To multiply powers of, dd eponents 9

30 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple:. Use te ommuttive nd ssoitive lws for multiplition to rewrite produt of numers nd powers of s simplified monomil Emple: 8. Use te distriutive lw to omine like terms Emple: We ve desried polynomil s sum of monomils. Rememer tt omintion of dditions nd sutrtions is still sum. Tt s euse sutrtion is defined y. Here is n emple, sown wit n intermedite level of detil. Originl epression wit dditions nd sutrtions 9 Sutrtions rewritten s dditions 9 Reorder te dditions ommuttive lw 9 Rewrite some dditions s sutrtions 9. Use te ommuttive nd ssoitive lws for ddition to rewrite polynomil in stndrd form y prentesizing nd reordering terms, nd ten use te distriutive lw to omine like terms. Wen you rerrnge te terms of te polynomil, rememer tt te sign in front of e monomil is prt of te term, nd must e rried long wen te term is moved. Emple: All detils re sown. Try to solve te prolem y leving out s mny intermedite steps s possile! 9 [ ] [ ] [ ] 9 9 [ 9 ] More emples re sown elow. Emple sows detils of reordering, nd omits te prenteses used in te ove emple. Emple omits detils of reordering, nd tt s ow you sould eventully e le to write out su prolems. 0

31 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor Te following emples, preprtion for te net setion illustrte ow te distriutive lw is used to remove prenteses wen monomil is multiplied y polynomil Rewriting produt of sums s sum of produts E of te si lws of lger desried in te setion on numeril epressions pplies s well to epressions involving vriles. All of te lger lws tt you know re te result of using sustitution in te si lger lws. As n emple, te following mteril sows ow to use te distriutive lw to derive te FOIL lw. d d d Strt wit te distriutive lw Sustitute y for to otin y y y On te rigt side, pply te distriutive lw y y y Use ommuttive lw for to reorder te rigt side y y y

32 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te result is te FOIL lw. It my look more fmilir if you sustitute for, for y, for, d for to otin te identity d d d In turn, you n sustitute ny epressions wtsoever for te vriles,,, d to otin new identities tt re onsequenes of te FOIL lw. Any identity tells you two metods for rewriting epressions. In tis se: You n rewrite d s d d, or You n rewrite d d s d Te first rewriting opertion, lled multiplying out, rewrites produt of sums s sum of produts. Te seond rewriting opertion, lled ftoring, rewrites sum of produts s produt of sums. Te si ide of multiplying out is tt to multiply two sums of vriles, you multiply e vrile in te first sum y e vrile in te seond sum nd dd te resulting produts: y y y y z y z y z y z y z y z y z Te lst line is n importnt speil se of te FOIL lw, sine Te sme ides work if some dditions re repled y sutrtions. Rell tt nd. Ten y y y y y y y z y z y z We repet n erlier wrning: is red minus sutrt nd s noting to do wit te produts Te following identity differene of squres is of prtiulr importne: nd is usully presented s metod for ftoring te rigt side..

33 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emples inlude 9y y y Emple..: Use similr rule to rewrite polynomil. s stndrd form Solution: lerly follows te pttern of te following identity, in wi we will sustitute for ; for ; for ; d e d e d e for d; nd for e : 9 Eerise..: Convert e produt of polynomils to simplified sum: first multiply out nd ten reorder nd ollet terms. Te lst two epressions were generted y sustitution in te distriutive lw nd sould yield te sme stndrd form polynomil. ] ] ] d] e] f] g] ] i] j] k] l] m] n] o]

34 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. p] q] r] s] 0 t] 0. Ftoring: rewriting sum s produt In te previous setion, we used te distriutive lw to rewrite produts s sums, proess lled multiplying out. Now we reverse diretion nd use te distriutive lw to rewrite sum s produt. Wen you ftor ver n epression, you rewrite it s produt of simpler epressions, wi re lled ftors noun of te originl epression. Te distriutive lw sys tt te produt of nd is identil to te sum of nd. In oter words, you n ftor s follows Sustituting for, for, nd for, gives Multiply on te left side nd remove etr prenteses: However, we ould ftor te rigt side furter y pulling out more times: We wnt to utomte te proess of ftoring sums su s. To do tis, look for te gretest ommon ftor of nd, nmely te lrgest epression in tis se, te igest power of wi is ftor of ot. In te epression, we see tt nd. We pull out from nd otin. Note tt tis is wt you get from te distriutive lw wen you sustitute for, for, nd for. Tenil note: in te following definitions, ll epressions re ssumed to e rtionl polynomil epressions wit integer oeffiients. Definitions involving ftoring: ll epressions re polynomil epressions Epression is ftor of epression E if tere is tird epression su tt E. In tis se, is lso ftor of E. Epression E is prime if its only ftors re,, E, nd E. Emples re, 9,, nd.

35 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Epression is te gretest ommon ftor GCF of epressions E nd F if tere re ftors e of E nd f of F su tt o E e, F f, nd o e nd f don t ve ommon ftor. In su se we n ftor E F s e f, ut e f n t e ftored ny more, t lest y using tis metod. Emple..: Ftor y pulling out te GCF of te terms. Solution: Te terms of te ve no ommon ftor, ut you proly relize tt we n ftor y te FOIL metod. Here s te lnguge tt would e used. Emple..: Ftor ompletely. Solution: Plese mster te emples in te following tle of ftoring prolems of inresing diffiulty. It is followed y tle of relted eerises. Emple.. Rewrite e sum of terms s produt y ftoring out te GCF of its terms. Ftor te epression: 8 Solution sometimes uses sustitution, sown in te olumn to te rigt 8 Finised:,, ve no ommon ftor. Comments goes into te oeffiients,8, goes into 0, 9 nd 0. Ten goes into,9, Pull out te lowest -power, wi is te GCF 9 8 Pull out te GCF of te oeffiients, 9, 8 nd lso pull out lowest -power.

36 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Pull out lowest -power. Ten pull out lowest -power. [ [ 8 ] [9 ] ] Sustitution used ws Prolem rises in Cl s derivtive of Eerise..: Ftor e of te following sums y pulling out te GCF of te terms, or write DNF does not ftor using GCF metod. Ten rewrite e ftor s simplified sum d e 0 0 f g i j k l m n o 0 Ftoring using FOIL is n importnt tenique tt is disussed in Cpter 9. Te identities in te following eerise re importnt emples of te FOIL metod. Eerise..: Rewrite e epression s produt of sums y using one of te following identities

37 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. ; ;. 9 d e y f y g y 9y i j 0 k l y y m y y n y y o 9 p 8 8. Rewriting epressions wit rdils A squre root n e written in two wys, nd. Te defining property of tese epressions is tt, wi ould lso e written in te form. Be wre tt tere is sutlety wen we del wit numers rter tn epressions: te usul onvention is tt ut not, even toug. In te following identities involving nottion for eponents, te letters m nd n n represent ny epression. In lmost ll te emples we ll work wit, m nd n re integers or rtionl numers.

38 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Lws of eponents mn n m m m n m n m Rdil nottion: m m Emples: / / / / / 9 For te time eing, use te following si fts : Eerise..: rewrite te following produts s simplified sums in wi powers of derese s you red from left to rigt. Te first one is done for you d e f g Te following emples sow ow sum of terms involving rdils n e rewritten s produt y ftoring out te lowest power of. Do not ssume tt one form is simpler tn te oter. Plese fous on te instrutions in te left olumn: tey speify ow te nswer sould e written. Be wre tt te emples 8

39 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Pull out lowest -power in Your nswer sould ontin rdil signs, not frtionl powers / / / / / / / Pull out lowest -power in / Write te nswer wit frtionl powers ut no rdil signs. / / / / / / / Pull out lowest -power nd rewrite using only positive powers. Rewrite s single frtion witout using frtionl powers. Esy wy: to rewrite ny non-frtion s frtion wit denomintor e, use te identity e. Ten 9

40 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Rewrite s single frtion witout using frtionl powers.. Hrd wy: Eerise..: rewrite e epression s indited. As one frtion, witout frtionl powers of Witout using frtions oter tn / By ftoring out, wit no frtionl powers in te nswer d By ftor out te lowest power of e Witout negtive powers f Witout negtive powers 0

41 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Rel life funtions CHAPTER : FUNCTIONS Very often, in rel-life situtions, one mesurement depends on noter. Suppose we wnt to use feet of fene to surround retngulr field. Tus te perimeter of te retngle will e feet. Wt re possile dimensions lengt nd widt for te field? Sine te perimeter is twie te widt plus twie te lengt, it follows tt L W, were L nd W re te lengt nd widt of te field. It follows tt L W nd terefore L W. Of ourse, it s importnt to note tt ot te lengt nd te widt re etween 0 nd. In prtiulr, we require 0 < W <, nd te following tle sows few oies of su W. Possile lengt nd widt of retngle wit perimeter Widt W Lengt L W Cek: L W Te tle sows tt if we know W, ten we know L. We sy tt te retngle s lengt depends upon its widt, or tt its lengt is funtion of its widt. Here s summry of wt we did. Emple..: A retngle s widt W feet nd perimeter feet. Wt is te lengt of te retngle? Solution: Let L e te lengt. Solve te eqution L W to otin L W. Answer: Te lengt of te retngle is W. Emple..: I ve ord feet long. I ut off dmged piee tt is D feet long. How long is te remining good piee? Here we will use very si priniple: Te wole is te sum of its prts. In tis se: Wen you ut ord into piees, te lengt of te ord is otined y dding te lengts of te piees

42 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Solution: Cll te lengt of te good piee G. Ten G D. Solve tis eqution to get G D. Answer: Te lengt of te left over piee is D feet.. Funtion definitions: sustituting for te rgument vrile Mny tetooks sow piture of funtion mine, gdget tt swllows numer lled te input nd spits out numer lled te output. Tt s resonle desription of rel-life funtion. In tis setion, we generlize tt desription y llowing not only numers, ut lso generl epressions wit vriles, s inputs nd outputs of funtions. Te equtions G D nd L W from te previous setion epress te sme ide. A new mesurement of interest to us G or L depends on given originl mesurement D or W. In ot ses te reltionsip n e epressed y lling te originl mesurement. Ten we n write Cll te originl mesurement : Arevite tis s New mesurement originl mesurement. New mesurement New On te lst line, te rigt side is n ordinry epression involving vrile. However, te left side is not n ordinry epression. It looks odd euse it s n Englis word New in it. Write N insted of New. Now we ve: N Te left side looks like N times. But N is not vrile, nd tere is no multiplition involved. Te entire sttement N is speil kind of identity lled funtion definition. It sys tt new mesurement, revited y N, depends on n originl mesurement, revited y.. Here s wt you need to know: Te letter N is te nme of funtion. N s noting wtsoever to do wit N times. is te rgument vrile. Bot sides of te funtion definition re referred to s te vlue of N t. Te rigt side of te funtion definition n e ny epression wtsoever: it my or my not inlude n instne of te vrile. Te funtion definition N is speil kind of identity Te Fundmentl Sustitution Priniple remins true: If E is ny epression, ten sustituting E for in te identity N produes new identity N E E, wi n e rewritten s N E E

43 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. We ve een working wit mny different kinds of epressions. E nd every epression turns into single numer wen numers re sustituted for vriles. However: Tere s no su ting s n epression tt produes more tn one output for given input. In prtiulr, tings su s ± nd ± re not epressions nd my not pper on te rigt side of funtion definition. Emple..: Sustitute for in te funtion definition f to find f f g f g Solutions: f f f f g g g f f g g g Te lst emple is interesting, euse te sustituted epression g ould e eiter funtion vlue or simply te produt g times. It doesn t mtter: in eiter se, te sustitution priniple gurntees tt f g g g is n identity. A funtion is kind of lger lw tt s vlid only for te durtion of prtiulr mtemtil disussion, in ontrst to lger lws su s te ommuttive lw, wi is true lwys. Te euty of funtion definition is tt sustituting ny epression wtsoever for te rgument vrile yields new lger lw. A funtion definition is templte for n infinite numer of lger lws desriing te vlues of te funtion t vrious input epressions. For emple, sustituting for in te funtion definition f yields f. Te doule prenteses to te left of te equls sign re unneessry: we write f. However, te prenteses on te rigt of te equls sign re solutely essentil, sine nd re very different epressions!

44 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Any oter epression, su s f eomes g eomes f g n e sustituted for. Ten. Te proess just demonstrted is lled rgument sustitution. It s esy nd fun to do it wit word proessor: just pste te sustituted epression into te funtion definition, reple e ourrene of te rgument vrile, nd insert prenteses! Study refully te tle elow. It ontins some surprises. In prtiulr, notie te lst olumn, in wi every funtion vlue tt ppers s prt of lrger epression is enlosed in squre rkets. Resons will e given in te net setion. Given f defined elow: f f q q Ten f nd f nd f f z sin z sin sin sin f r sin z sin z sin z sin z f nd finlly f f [ ] [ ] [ ] [ ] [sin ] [sin ] [sin z ] [sin z] [ ] [ ]. Sustituting epressions for funtion vlues Te lst setion empsized tt wen you sustitute n epression for te rgument vrile in funtion definition, te sustituted epression must e enlosed in prenteses. For emple: given te funtion definition f, ten f Wen funtion vlue is used in n epression, you lso need to use prenteses wenever you sustitute for te funtion vlue. Tt s euse te symol f eves like vrile, s sown in te following emple. Agin define f.

45 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple..: Sustitute for u in te epression. u u Answer: u u Emple..: Define nd sustitute for f f in te epression. f f Solution: Tret etly like te vrile u in te previous emple. f Answer: f f Emple..: Define nd sustitute in te epression f f f Answer: nd so f f f ] [ ] [ In tis emple, te first set of inserted prenteses ere written for lrity s rkets is not needed, ut omitting te seond set is serious error, s seen from te following Eerise.. Simplify ot te given nswer to.. nd te inorret nswer otined y omitting rkets.. Funtion evlution emples Te following emples sow ow to n epression for te rgument of funtion. Tey lso sow ow to simplify te result. Te simplifition proess depends on following te order of opertions refully. Given, find e funtion vlue nd rewrite it s simplified sum. f 8 9 f z z z f ] [ ] [ f ] [9 ] [ f In te following emples, inserted prenteses will e written s res { }. Given simplify f f f

46 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. } { ]} [ { } { ]} [ { } { } { } { } { f f f f Tis nswer is simplified sum. Tis nswer is simplified produt. Given simplify f f f Step : Simplify f f : Step : Multiply y nd simplify: f f f f } { } { ]} [ { ]} [ { ]} [ { ]} [ { } { } { } { } { f f Given, simplify f f f f f ] [ ] [ ] [ ] [ ] [

47 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In e eerise elow, find nd try to simplify e of te speified funtion vlues. Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f. Composition of funtions Emple..: A rel life emple wit numer gme. I give you numer. You ve one ne to doule te numer nd one ne to dd ten to te numer. Wi proedure gives te igger nswer: dding 0, ten douling, or douling, ten dding 0? Solution: Cll te strting numer. In tis prolem, tere re two funtions. Te douling funtion is d. Te dding 0 funtion is 0 Strt wit. Douling first gives d. Adding 0 to te result gives d d 0 0 Tis strtegy strts wit, yields 0, nd n e tougt of s funtion: f d 0

48 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Strt wit. Adding 0 first gives 0 Douling te result gives d d Tis strtegy strts wit, yields 0, nd n e tougt of s funtion: g d 0 Te result g of dding first, ten douling, is lwys 0 more tn te result f of douling first, ten dding. Te funtions g d nd f d re lulted y putting togeter, or omposing, te funtions nd d. We ve just seen tt te order of omposition mtters. In oter words, omposition of funtions is not ommuttive. We just ssigned somewt rndom nme g to te omposite funtion formed y first pplying, ten pplying d. Te offiil nme of tis funtion is d. It is usully enlosed in prenteses wen we write te funtion definition: d d. Similrly, omposing in te reverse order is defined y d d Tke note: Wen omposite funtion is nmed using te omposition opertor, te rigtmost funtion is evluted first. For emple: to find f g, strt wit, net find g, nd lstly find f g Emple..: Buying lumer. I m uying wood tt osts dollrs per foot. A foot long ord s dmged prt wose lengt is feet. I m llowed to ut off te dmged prt nd py only for te good prt tt I tke. Te rge for utting te wood is 0 dollrs. How mu do I py? In tis prolem tere re tree importnt mesurements nd two funtionl reltionsips. Te lengt of te good prt equls minus te lengt of te dmged prt. Te ost of te good prt is times te lengt of te good prt. For emple, if te lengt of te good prt is 8 feet, it osts 8 dollrs. Te rge for utting te wood is n dditionl 0 dollrs nd terefore I py 8 0 dollrs. In generl, if te lengt of te good prt is G, ten I py G 0 G 0 dollrs. Tere re two wys to proeed, e relted to wy of prsing te prolem. Metod. Let e te lengt of te dmged piee. Find formul for C, te mount I must py. Solution : Nme te quntities s follows. Lengt is mesured in feet. Let D e te lengt of te dmged prt. Let G e te lengt of te good prt. Ten G D s efore. 8

49 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Let C e te ost of te good prt. Ten C G 0 Sustitute G in te identity C G 0 to otin C 0. Answer : If te lengt of te dmged piee is, ten I py C 0 dollrs. G Summry : C G 0 C 0 Tis metod ws strigtforwrd. Te following one seems it more omplited. We use it for resons tt will eome ler in Clulus I. Metod : Let e te lengt of te dmged piee. Find te definition of funtion of tt gives te ost of te good piee. Solution : Te lengt of te good prt is given y te funtion definition g. Cll te lengt of te good prt u: ten te ost of te good prt is given y te funtion definition u u 0. Comine tese funtion definitions: te ost of te good prt is g 0 Answer : Te ost of te good piee is g 0 dollrs. Summry : g u u 0 g g 0 Te metod of omining funtions sown ove is lled omposition: te word ompose mens put togeter. Te ide is simple: you ve definitions for funtions f f nd g, for emple g It migt ve een nier to write te definition of g using letter oter tn, ut te letter used in te funtion definition is ritrry nd tere s noting wrong wit using te sme letter in two different funtion definitions. To define omposite funtions, we sustitute s in te following emple. f Emple..: Given, find f g nd g f. g Solution: 9

50 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. g f g f g g f g Tis ompletes te sustitution. In e se ontinue y simplifying. [ 0 9 ] [ g f ] f f f g As efore, we see tt te order of omposition mtters. A funtion my e omposed wit itself. For te funtion g we ve g g g g g Eerise.. Given f nd g find nd simplify g f f f ; f f ; d f g ; e g g Eerise.. Given f nd g find nd simplify g f f f ; f f ; d f g ; e g g. Pieewise defined funtions A very generous nk offers te following del: Deposit your money wit us t te eginning of ny yer. At te end of te yer we will deposit interest in your ount s follows: if you ve less tn 80 dollrs in your ount, we ll dd 80 dollrs to your ount.. if you ve 80 dollrs or more, we ll doule your ount. Emple..: Assume your ount strts out wit dollrs. Give funtion definition for te mount in your ount fter one yer. If your ount strts out wit 0 dollrs, ow mu is your ount fter one yer? fter two yers? fter tree yers? Prt sks ow to write te symoli definition of pieewise defined funtion. 0

51 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Prt involves finding te vlue of te funtion f nd its omposites f f f for te speifi input 0. f f nd Solution If you strt te yer wit dollrs, let f e te mount fter one yer. Aording to te nk s rules: If < 80, ten f 80. If 80, ten f Tese two sentenes nswer prt. Tey re usully omined nd written in mt ooks s Answer If your ount t te eginning of yer s dollrs, ten t te end of tt 80 if < 80 yer, your ount is given y te funtion f if 80 We sy tt te funtion just defined is pieewise defined funtion euse te epression tt will e sustituted for te input vrile depends on wi piee of te numer line te numer lies on. Solution If you strt wit dollrs ten te mount in your ount in dollrs is f fter one yer ; f f fter two yers; f f f fter tree yers, nd so fort. Strt wit 0. Note tt 0 < if < 80 Wen you sustitute 0 for in te funtion definition f if if 0 < 80 nd omit unneessry prenteses, te result is f 0 0 if 0 80 Sine 0 < 80, it follows tt te top line of te definition sould e used. Terefore f is te mount fter one yer if 00 < 80 Te mount fter two yers is f f 0 f if In tis se te seond line pplies nd so te mount fter two yers is f f From tis point on, te ount doules e yer. Answer Te ount s f 0 00 dollrs fter one yer; f f 0 f dollrs fter two yers; nd f f f 0 f f 00 f dollrs fter tree yers. A pieewise defined funtion is lwys rel-life funtion in te sense tt te input vrile is ssumed to e rel numer. Certinly te rigt side of te funtion definition ove is not n epression of te sort we ve een disussing. However, it stisfies te ruil property of funtion: tere s etly one output for given input numer.

52 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. You sould wt out for possile mistkes in pieewise funtion definitions. If you nge te inequlity sign in te top line of te funtion from < to, ten you 80 if 80 would write f Tis is still legl funtion definition, sine te top if 80 line nd te ottom line ot give 0 wen you sustitute 80 for. Te funtion so defined is etly te sme s te one ove. 80 if 80 However, g is not legl funtion definition, sine wen 80 is if 80 sustituted for, te top line eomes 0 ut te ottom line gives 0. Wtever tis ting is, it s not funtion definition, sine te single input 80 produes two different outputs. f Eerise.. Given g f g e f g g f g g f g g f g g d f f f f i j k find e of te following: i j k f g u g f u g g u m n o l f f u p f g w f g g w g g g g g g g if < 0 Eerise.. Given f if 0, find e of te following: if > 0 f f 0 f f f f e f g f f f f f f f f f 0 d f f f 0 f f f 0. Wrnings out funtion nottion If f is funtion, te symol f is neiter numer nor vrile. Te symol f s noting to do wit f times. Computer siene nd logi ooks void tis prolem y using squre rkets elusively to enlose funtion rguments, ut we will follow te ustom of mt ooks nd use round prenteses. If f were vrile, we ould use te distriutive lw to write f y f f y. Tis tempting sttement is responsile for mny errors in students lger work in pre-lulus nd lulus. Te sttement is usully ut not lwys flse: te

53 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. eeptions re funtions defined y lws of te form emples were it s true: f k. Here re some o Let f. Ten y y. o Let f. Ten y y o Let f. Ten y y Tis n e rewritten y y. o Let f. Ten y y. Tis n e rewritten z z z z z y y. z z z Te lst two lines re emples of priniple you lredy know: A frtion n e rewritten s sum of frtions provided its numertor is sum. Te orresponding sttement for denomintors is flse: see te tird line of te following tle, wi ontins mistkes frequently found on students ems. Every error in te tle is sed on te flse elief tt funtion nme n e treted s vrile. Tt in turn leds to te misoneption tt f y f f y, wi in turn is responsile for most of te errors sown in te tle elow.

54 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. You will mke your mt teers very ppy if you void te errors in te tle elow. Emple..: Some ommon funtion errors to void Funtion Error! Error! NO! NO! Corret sttement f y y y y f y y y y f y y y does not simplify. f y y y y y k k k k k f does not simplify. y y y f k k k k k does not simplify. y y y f os os y os os y os y os os y sin sin y f sin sin y sin sin y sin y sin os y os sin y f log log y log log y log y does not simplify. y y y y e f e e e e e e f k k y k y k y k ky f k k Hrd to mess tis one up y k k y k

55 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise.. Fill in te following tle. First sustitute in te funtion definition, ten simplify te result Funtion f f y Sustitute to find te funtion vlue. Ten simplify your nswer or write DNS does not simplify d f f f f y f y f f e f f f f g f f u v f f u v i f f f f f j f k f f f l f f f sin m f f z n f u f f

56 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : WHEN TO USE PARENTHESES As you my suspet, every step in n lger prolem n e tougt of s sustitution opertion. Mny students get into troule euse tey don t use prenteses orretly wen tey sustitute. Te following summry of previous setions sows wen prenteses re needed in sustitutions tt re done in prelulus. Wen you sustitute, put prenteses Around te result of n lger opertion. Around n epression tt is sustituted for o vrile in lrger epression; o vrile in n lger lw; o funtion rgument; o funtion vlue. Prenteses re sometimes optionl, of ourse. However, it s very diffiult to give omplete lws for wen prenteses re needed. Wen in dout, use tem! Tis setion presumes tt te reder knows te si FOIL identity, disussed in more detil in Setion, for multiplying epressions: d d d.. Put prenteses round te result of e lger opertion. Emple: Rewrite s simplified sum. Solution: Do te multiplition first. Put prenteses round te produt. We ll write tese inserted prenteses s rkets [ ] for lrity. [ Tere re two opertions, sutrtion nd multiplition. ] Multiply using FOIL nd put te produt inside prenteses. Remove prenteses see elow Collet terms. Inserting te prenteses ws solutely ruil. Oterwise, you would ve written Tere re two opertions, sutrtion nd multiplition. FOIL is used orretly, ut omitting te rkets is ftl error. Full redit off! Even toug te order of opertions is orret te result is wrong! In te orret version, we performed very triky opertion

57 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Removing prenteses following minus sign It s importnt to understnd wy we wrote. Tis opertion sould e utomti: tink of multiplying te minus sign y e term inside te prenteses. After wile, you sould e le to remove te prenteses witout sowing intermedite detils. For now, ere re te detils for tis emple: note tt is n emple of te identity d d Tis is sutrtion prolem. Definition of sutrtion:. Negtion: Do te multiplition. Remove prenteses. Comine terms. Here is relted emple of te form d d Tis is sutrtion prolem. Definition of sutrtion: Do te multiplition. Remove prenteses. Comine terms. Here s noter emple, in wi you follow te lws for order of opertions y evluting power efore doing sutrtion Corret: Corret nd sorter: ] [ ] [ ] [

58 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Put prenteses round ny epression tt you sustitute for vrile in noter epression. Wen you do tis, te Fundmentl Sustitution Priniple gurntees tt te new epression is equivlent identil to te originl. For emple, if ten.. Put prenteses round ny epression tt you sustitute for vrile in n identity or lger lw. An identity is sttement tt two epressions re equl. However, it sys mu more: sustituting ny epression for every instne of te vrile produes noter identity. For emple, te ommuttive lw sys tt AB BA. Sustituting ny epressions for A nd B will produe true sttement, provided you enlose tose epression in prenteses. Here re tree emples. If you omit prenteses in eiter of te first two emples elow, you re mking serious error. In tose emples, we use lnguge su s Let B s sorter wy of sying: sustitute for B. Let A nd B. Ten. Let A y nd B u v. Ten y u v u v y Sustitute A for A nd A for B. Ten A A A A In te lst emple we didn t sy Let A A. Wy? Perps te most importnt lger lw is te nelltion lw, wi n e written Wrning #: Mny students misuse te nelltion lw. Don t nel in sitution were te nelltion lw doesn t pply. For emple, te ttempted nelltion yz yz is wrong, euse isn t ftor of te numertor. yz yz Wrning #: Wen you simplify epressions, you my e using n lger lw or identity witout relizing it. For emple, suppose you wnt to perform te esy frtion sutrtion. 8

59 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. If you relize tt te relevnt identity is, you will sustitute for, for, nd for, using prenteses nd orretly otining weres if you forget to put prenteses round te wen you sustitute it for, your nswer will e!. Put prenteses round ny epression tt is sustituted for funtion rgument. Suppose te funtion definition is f Sustitute for : f is te vlue of f t. Sustitute z for : f z z is te vlue of f t z. Sustitute for : f is te vlue of f t. Sustitute for : f is te vlue of f t.. Put prenteses round n epression tt is sustituted for funtion vlue. Tis is ompletely nlogous to putting prenteses round n epression tt s sustituted for vrile. For emple, suppose f nd we wnt to simplify te epression f. You need two sets of prenteses: one wen you sustitute for to get f, nd noter to rewrite f s [ ]. One wy to proeed is sown elow. If you re omfortle omitting te seond line, do so. Corret f [ f ] [ [ 9] [ 8] ] Inorret f f 9 9 9

60 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. One more time:. Deide wt do net y using te order of opertions rules. Put te result of e lger opertion in prenteses.. Put prenteses round n epression tt is sustituted for ny of te following: vrile in noter epression; vrile in n lger lw or identity; funtion rgument; funtion vlue. Eerise..: Perform te indited sustitution in te given epression, identity, or funtion. Use f ; g 9 u for w in u for w in y w u for w nd u for in w d u for nd v for in e u for w in w f u for nd v for in g u for nd v for nd w for in u for nd v for nd w for in i u for nd v nd for nd w for in j for in f k for in f l Find f f m Find f f n Find g g o Find g g p Find f g q Find g g r Find f f Eerise.. Rewrite e result of te lst eerise witout prenteses s simplified sum. 0

61 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : WORKING WITH FRACTIONS Your initil gol in tis pter is to rewrite e of te following emples s ompletely redued frtion y neling ommon ftors of numertor nd denomintor. Two emples re done for you. If you re omfortle doing tese prolems, skip to setion e d de e d de d e u v u u u uvw d d d d d d d 8 y y 8 w w w w 8 w w w w

62 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Identities involving frtions Working wit frtions n e triky wen te numertor nd denomintor involve vriles. First we list importnt identities you sould know. All of tem re si lger rules listed erlier or n e derived from tose rules y using sustitution. Frtion sis: Frtion ddition: sme denomintors generl rule: ; d d d Frtion sutrtion: sme denomintors generl rule: ; d d d Frtion multiplition: Frtion division: d d d d Frtion negtion: Frtion nelltion Implied prenteses in vertil frtion nottion: E E Let E nd F e ny epressions. Ten. F F Wrning: Te orresponding sttement is flse for frtions written in lultor nottion. Wen you use lultor nottion, you usully need to ple prenteses round te denomintor. For emple pq pq /rs, wi is not te sme s pq / rs. rs Eerise.. Use order of opertions to sow tt pq / rs pqs / r.

63 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Multiplying nd dividing frtions Frtion multiplition is esy to desrie:, lso written d d Often we omit te times sign nd write d d d d Tis is esy to rememer ut lmost too simple: to multiply frtions, you just onnet te frtion rs. Anoter wy to write te multiplition lw is d Here we use te stndrd onvention tt te pir of onseutive prenteses relly mens. Wrning: Te similr looking formul is 00% wrong! d d Te following emples eplin ow to sustitute using te si neling lw. Emple.. Use te Fundmentl Sustitution Priniple FSP in te following emples. Wen you sustitute for vrile, use prenteses! Use te frtion multiplition lw to rewrite Solution : Sustitute for, for, for, nd for in d to get FSP requires prenteses round nd, ut we omit tem euse PEMDAS sys tt ll epressions ontin implied prenteses round powers. w u w u w Find Here prenteses ren t needed: w v w v wv Furtermore, te nswer n e rewritten y using te ommuttive lw togeter wit te frtion multiplition lw written in reverse: u w u w u w wu v wv wv w u w v u u v v Tis is relly n emple of nelltion, wi will e disussed in te net setion. d d

64 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. To divide y frtion, multiply y its reiprol. Te quotient of u divided y v is u epressed in tree wys: u / v u v. Te definition of division sys tt v u u were, lled te inverse of v, stisfies v v v v v v v d d d d In prtiulr, te inverse of frtion is its reiprol, sine d d d d Tus we ve two si division lws: Epression divided y frtion: Frtion divided y frtion: u / d / / d u d d d u d ud d Te lrge prenteses re not neessry, ut mke it esier to see wt s going on. Te dots tt indite multiplition re optionl. In te lst se te nswer is written s produt divided y produt. Simplifying su epressions is te gol of te net setion.. Simplifying frtions: te nelltion lw Perps te most importnt frtion identity is te nelltion lw: Sustituting ny epressions wtsoever for e of te vriles,, yields n identity. For emple, sustitute for ; uvw for, nd yz for to otin yz uvw yz uvw, usully written s yz uvwyz uvw. Most students lern to nel tis emple y putting sls troug te identil ftors yz on top nd ottom of te frtion. Tt s fine, ut tese notes will omit te sls euse it s esy to get into d its nd misuse te sls wen neling isn t legl. Te nelltion lw is n identity. Like oter identities, it n e written using ny uw v vriles. For emple, sustituting u for, v for, nd w for in gives. vw w From now on we will restte lger lws wit wtever vriles re onvenient.

65 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te nelltion lw gives rise to very si identities using powers of vriles. Te following emples sow ow to nel su powers. Te si ingredient in most prolems is te following rule. Rule for neling powers: If numertor nd denomintor ot ontin powers of te sme vrile, ten ftor te iger power nd nel te lower power. For emple, if ppers in te numertor nd ppers in te denomintor or vie 8 8 vers, rwrite nd nel te lower power from numertor nd denomintor. In symols: 8. 8 y y y Offiilly, wt ppened is tt we sustituted u for for u; for v; nd y for w in te nelltion lw tis detil wen you nel: just write uv v to otin. Don t tink of ll uw w y y y. 8 y y Te importnt ting is to rememer tt nelltion mens: rewrite te frtion y neling rossing out, omitting ommon ftor from numertor nd denomintor. If numertor or denomintor onsists entirely of single vrile rised to power, e reful. Wen te power in te denomintor is iger, e sure not to nel te numertor:, NOT wi is wrong or wi is nonsense. 8 Similrly 8. y y y Wen te power in te numertor is iger, te nswer need not e written s frtion: 8. Wen you do more omplited emple, proeed s follows. Proedure for reduing frtion in wi numertor nd denomintor re produts of powers. For e se tt ppers in ot numertor nd denomintor, sutrt te lower eponent from ot eponents. Te result will e to: reple te lower power y nd to reple te iger power s eponent y te differene of te eponents.

66 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Here s n esy emple: You sould try to do tis in one step nd write just 8 0. Te following tle of dditionl emples inludes more detil tn is neessry. It is inluded for referene only, to sow ow neling frtions is just one more pplition of te Fundmentl Sustitution Priniple. Mke sure tt you n quikly write e rigt nswer witout writing down te detils of te sustitution. Emple.. Simplify te epression elow Solution on te top line is otined y sustituting in te identity on te ottom line. Sustitution used? w v uw uv w v u?? n m n m > n m n m n n n m n m u u v uv v u n n m? n m n m < m n m n m m n m mn n m n m 0 See lst emple

67 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. / / / doesn t redue Eerise.. Redue te following frtions witout inditing te sustitution tt you used. Note tt power of vrile tt ppers in numertor or denomintor, ut not ot, is left undistured d d d Emple:. Try to skip te middle step! 0 e e e 8 0 d e 8 0 d e u u v v 0u w w v w u v w 0 Eerise.. Fill in te tle t te eginning of Cpter.. Ftor numertor nd denomintor efore you nel Students ourt disster wen tey misuse te nelltion lw. You need to know te following mteril in order to understnd wen neling is, nd is not, legl. Proedure for simplifying frtion: Ftor te numertor nd denomintor. Cnel ommon ftors of numertor nd denomintor.

68 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te previous setion overed only te seond step of te ove proedure, euse numertors nd denomintors were given in ftored form. Definitions involving ftoring. Let, E, F, nd G e polynomil epressions. E is ftor of G mens tt G n e written s produt EF, were F is noter epression. E is ommon ftor of F nd G provided E is ftor of F nd E is ftor of G. E is prime if its only ftors re, -, E, nd E. A redued frtion is frtion wose numertor nd denomintor re polynomil epressions wit no ommon ftor oter tn - nd. A frtion is in stndrd form if it is redued nd if its numertor nd denomintor re e written s n integer multiplied y produt of powers of prime ftors. Simplify frtion mens: redue nd rewrite te frtion in stndrd form. Emple.. is ftor of yz euse yz yz. y is ommon ftor of y nd 9y euse y y nd 9 y yy.,, nd re ll prime epressions. is prime numer. y is redued frtion. Wen you simplify you get y, wi is frtion in stndrd form. Te nelltion lw llows neling ommon ftor from numertor nd denomintor of frtion. Noting else n e nelled. For emple, in te frtion y y is ftor of te numertor euse te numertor is equl to ; is ftor of te denomintor euse te denomintor is equl to y ; is ommon ftor of numertor nd denomintor euse it is ftor of e. Terefore we n sustitute for, y for, nd for in frtion s. y y to rewrite te given 8

69 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Don t nel if te nelltion lw doesn t pply. For emple, te ttempted yz yz nelltion yz yz is wrong, euse isn t ftor of te numertor. Oter frtions tt n t e simplified inlude: y y y d One of te resons tt students mke errors wen nelling is tt tey don t try to ftor efore tey nel. As result, tey nel piee of te numertor tt isn t ftor. Here re some more possile errors: os y os. y sin Worst of ll is si. just kidding!. n If you ren t sure weter you re llowed to nel, ply it sfe nd write down epliitly ot te nelltion lw nd te tree epressions tt re eing sustituted for,, nd. Study refully te tle elow. Emple..?? y? y y? y Doesn t simplify: numertor nd denomintor do not ve ommon ftor. Doesn t simplify: numertor nd denomintor do not ve ommon ftor. y y y y 9

70 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. 8 ] [ ] [ Used sustitution First step is multiplying y to get rid of te frtion in te numertor. Tis prolem is wrm-up for te net prolem. 0

71 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Now ftor. [ ] [ ] Don' t multiply! Sustitute. for nd for in te lst emple. Tis prolem rises in first semester lulus wen you find te derivtive of Eerise..: Simplify te follow frtions or write DNS Does Not Simplify. Strt y mking sure tt te numertor nd denomintor re ftored ompletely. d e f g i yz yw yz y w d d y z y z

72 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. j k l m n o p uvw uvw d y 8 8. More on ftoring. Te only ftoring topi disussed so fr ws pulling out te GCF of sum of terms. Two oter importnt topis re ftoring integers nd ftoring qudrti polynomils. Ftoring qudrti polynomil: Ftoring te polynomil wit wole numer oeffiients,, nd is esiest wen. Ten te polynomil is lled moni. We wnt to rewrite it s follows: r s. Notie tt r s s r rs s r rs r s rs In summry, we wnt to find integers r nd s su tt r s rs Certinly tis will e te se if r s nd rs. In oter words, to ftor s r s find wole numers r nd s wit sum nd produt. To find r nd s, first write down pirs of integers wose produt is. Ten look for pir tt dds to. If tere is no su pir, ten doesn t ftor.

73 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Here s sort ut: if you find pir wit r s insted of,, just reverse te signs of r nd s: ten r s. To ftor n ritrry qudrti epression, first pull out te GCF of its oeffiients. You don t need to find te GCF epliitly: just pull out ommon ftors until you n t ontinue. For emple, 0 0 Te lst polynomil is moni nd ftors esily. Terefore 0 If te leding oeffiient of is not, te tril nd error proedure sown ove is more omplited nd more nnoying. A systemti proedure tt voids tril nd error is disussed in Cpter 9. Emple.. Ftoring moni qudrti polynomils Ftor given Possile pirs r,s Suessful pir wit r s elow, wit solution sown tt multiply to, ;,;, -,-; -,-; -,- -,; -,; -, , wi is Multiply y - to get te suessful pir - - doesn t ftor,,, None of tese pirs dd to.,-,, -,,-,,- Ftoring wole numers. Wen te leding oeffiient is -, ftor it out efore you egin. Ten - - A systemti metod for reduing frtions of integers is to first ftor numertor nd denomintor s produts of powers of prime integers, or primes, for sort. Tese re positive integers m wit etly two positive integer ftors: m nd. Note tt s only one ftor nd is terefore not prime. Te first few primes re,,,,,. To ftor n integer, ftor out s mny times s you n, ten ftor out s mny times s you n, nd ontinue up te list of primes. Stop wen te squre of te prime eing tested is lrger tn te numer you re trying to ftor. For emple to test weter 9 is prime, first try,,, nd : none re ftors of 9. Te net prime to try is, ut its squre is, wi is greter tn 9, so stop trying. It follows tt 9 is prime.

74 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te most systemti wy of reduing frtion is to ftor ot numertor nd denomintor using te metod just sown, nd ten neling ommon ftors. Emple Anoter wy: Now ontinue s in te ove emple. Ftoring out signs. Use te fundmentl identity [ ] Tus ut Rewrite ftors s stndrd form polynomils efore you try to nel. Emple..

75 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise.. Redue te following frtions ompletely: ftor numertor nd denomintor, ten nel ommon ftors d 8 e f 8 g i j k l m n y 9 y y y 9 o 8 p

76 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : ADDING RATIONAL EXPRESSIONS. Sustituting in te frtion ddition lw y y Adding frtions wit te sme denomintor is esy, sine. Wrning: Te following is mjor error:. y y For frtions wit different denomintors, te si frtion ddition lw is d. d d Emple.. Use te frtion ddition lw s single frtion. d to rewrite d d Solution: You wnt to mt wit te ddition lw d To do so, sustitute for ; for ; for ; nd for d: d d Clerly tere re unneessry prenteses on te rigt nd side.wen you sustitute n epression E for vrile u, you don t lwys need to ple prenteses round E. Here re few situtions were te prenteses n nd sould e omitted. E is lso vrile; E is prentesized epression; E is power ut u is not rised to power. In te ove emple, it follows tt, wi ws sustituted for d, does not need to e pled in prenteses. We ould ve written Tis is orret nswer to te question, wi did not sk tt te result e simplified. Eerise..: Prove te ddition lw for tree frtions: e df f de d f df

77 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise..: Rewrite te following epressions y sustituting in te d identity. Rewrite te nswer s frtion in stndrd form. d d ] ] Hint: ] d] e] f] pq p g] pr q ] i] y j] k] d l] m] n] o] p] q]

78 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Adding frtions:find ommon denomintor, uild, dd, redue First we justify te si frtion ddition lw. y y. Strt wit te left nd epression. Rewrite te epression y multiplying e frtion y. Multiplition omes efore ddition. Use te ommuttive lw to rewrite te seond denomintor. Add frtions wit te sme denomintor. y y y y y In prtie, te frtion ddition lw n e ineffiient. For emple: Tis nswer is orret ut needs to e redued to. A etter metod for dding frtions is to egin y finding teir lest 000 ommon denomintor, s will e eplined in te net setion. Our gol is to eplin in detil te following proedure for dding frtions. Proedure for dding frtions : CD, uild, dd, redue Find ommon denomintor CD for te frtions. Build te frtions so tt tey ve tt ommon denomintor. Add te uilt-up frtions. Redue te result.. Finding te lest ommon denomintor To dd frtions, you need to rewrite tem wit te sme ommon denomintor. Te produt of teir denomintors n e used, ut tt produt n e very omplited. For emple, if you use te produt of denomintors s te ommon denomintor wen you find, you would ve to rewrite ot frtions wit ommon denomintor

79 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor Tt would e wste of time, sine you n esily rewrite te first frtion s. Tus, n esy lultion It s often est to find te smllest or simplest ommon denomintor of two or more frtions. Tt denomintor is lled te lest ommon denomintor LCD of te frtions. Te LCD n lso e desried s te smllest or simplest epression tt s e denomintor s ftor. Here s it more detil: Definitions involving LCD nd LCM. Assume tt E,, nd re polynomils E is multiple of mens tt tere is polynomil wit E. E is ommon multiple of nd mens tt E is multiple of nd E is multiple of. E is lest ommon multiple of epressions nd mens o E is ommon multiple of nd nd o Any oter ommon multiple of nd is multiple of E. Te lest ommon denomintor LCD of frtions is te LCM of teir denomintors. We ll find LCD s of frtions in te net setion. y Emple.. is ommon multiple of y nd y, sine y y y y y nd so y. Te LCM of y nd y is y. Note tt y is multiple of te LCM. Emple.. Find LCM of,, Find LCM of,, LCM is LCM is Te LCM of distint vriles is teir produt. Te LCM of powers of distint vriles is te produt of tose powers. In te ove emples, te LCM of epressions wit no ommon ftors is te produt of te epressions.. However, wen te epressions do ve ommon ftors, te LCM will e simpler tn te produt of te epressions. Emple.. Find LCM of nd LCM is sine Sine is ommon ftor of nd, it souldn t e repeted in te LCM. 9

80 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Find LCM of nd LCM is Sine is ommon ftor of nd, it isn t repeted in te LCM. Find LCM of nd LCM is Te LCM is te produt of te igest power of vriles tt pper in ny of te epressions. In tis emple: igest power of is, igest power of is, igest power of is. Te LCM is te produt Te rule stted in te lst o nd in more detil t te ottom of tis pge pplies not only to vriles, ut to prime ftors of te epressions. Rell tt n epression E is prime if its only ftors re ± nd ± E. Here re some emples listing prime ftors of epressions. Emple.. nd re te prime ftors of, wi n e written s. Oter ftors,, 9, 8,, re not prime.,, nd re te prime ftors of, wi n e written s. Tere re no oter prime ftors.,,, nd y re te prime ftors of y y.,, nd re te prime ftors of.,,, nd re te prime ftors of,, nd re te prime ftors of To find te LCM of two or more epressions: Step : Ftor e epression s produt of powers of primes. Step : For e prime tt ppers in ny of te ftoriztions, write down te igest power of tt prime in ny of te ftoriztions. Step : Te LCM is te produt of te powers written down in Step. 80

81 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple.. Find LCM of te following epressions: Find LCM of, 0,, Find LCM of y z 0yz y 9 z Find LCM of y Find LCM of Epressions rewritten s LCM of te epressions produts of prime powers Higest power of is 0 Higest power of is Higest power of is LCM produt of igest powers 90 y z y z Use te previous emple to see tt te LCM of te 0yz yz oeffiients is Higest power of is y z y z Epressions ftor s: y Epressions ftor s: 0 Higest power of y is y Higest power of z is z 9 LCM is y z 90 y z Higest power of is ; igest power of is ; igest power of is. LCM is y LCM is te LCM of te oeffiients, 0,, multiplied y te LCM of te vrile ftors of te epressions. Use results of te lst two emples to see tt te LCM is 90 y 9 Eerise.. Find te LCM for e of te following list of epressions:,,8,,, d, 0, 0 d, 0, e,, f,, 8

82 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. g 8,,. Building Frtions Tis seond step in te frtion ddition proedure is lled uild. Te ide is to multiply e frtion y in order to rewrite it wit osen ommon denomintor. Emple.. Rewrite s frtion wit denomintor 0. Solution:????? 0? 0? 0/ In oter words, to rewrite s frtion wit denomintor 0, multiply top nd ottom y 0/, wi is. Tus 0 In tis emple it ws esy to rewrite te new denomintor 0 s times te originl denomintor. Tt step n e rd in some ses. Study te emples elow. Emple.. Rewrite s frtion 0 wit denomintor 0? 0 0? 0? 0? 0/ ? 0 You ould lso find? y ftoring 0 nd 0 s follows 0 0 0?? is formed s te produt of ftors of 0 tt re missing from te ftors of 0. See net emple. 8

83 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple.. Rewrite s frtion wit denomintor To solve?, set? equl to te ftors of oter tn. Rewrite s frtion wit denomintor Rewrite s frtion wit denomintor.???????? is te originl numertor times te produt of ll ftors of te uilt-up denomintor tt re missing from te originl denomintor. Te point ere is tt nd so. Anoter metod is sown in te seond prt of Emple.. immeditely elow. Emple.. Rewrite yz s frtion wit denomintor yz yz Rewrite s frtions wit denomintor. k k k k / / / / / Te originl epression yz is not frtion. In tis esy se, te uilt-up frtion s numertor is yz times te desired denomintor. First tree emples: use sme proedure s in previous emple. Lst emple: solve /? y setting? 8

84 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple..? y? z? u? y? y y z z u u y y Importnt simple emples. Te first tree rise in te ddition prolem y z y z y z. Adding frtions: find LCD, uild, dd, redue We ve so fr ompleted te first two steps of te following generl proedure for dding frtions. Find ommon denomintor CD for te frtions Build up e frtion so tt it s tt ommon denomintor Add te uilt-up frtions Redue te result y ftoring numertor nd denomintor nd neling. In most ses, te work is esiest if you find te lest ommon denomintor in te first step. Terefore we summrize tis proedure s: LCD, uild, dd, redue. Rememer tt te LCD of te frtions is te LCM of teir denomintors. In te emples elow, we stte te prolem nd find te LCD in te left olumn, ten uild frtions nd dd in te seond olumn. 8

85 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple.. Find 0 0 LCD LCD Emple.. Find Find LCD: 0. 0 LCD is Answer is redued frtion sine 9 9 nd 90 ve no ommon prime ftors. In e of te net two emples, notie t te line mrked!!! ow importnt it is to ple prenteses round te numertor of frtion tt is sutrted from noter frtion. At tt step you re relly using te formul nd so you need to use prenteses wen you sustitute for te vriles! 8

86 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple.. Find 8 8 LCD is!!! Emple.. Find LCD ] [ ] ][ [ LCD!!! 8

87 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise..: Write e of te following sums s simplified frtion y z y z d e f g y y z i j k l m 8

88 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise..: Use two different metods to rewrite e of te following produts or powers s single frtion nd ompre te results of te two metods. Metod : o Multiply out y using FOIL. o Add te resulting frtions to get single frtion. o Simplify te resulting frtion Metod : o Rewrite e ftor s frtion o Multiply frtions o Simplify te resulting frtion. d e f g i j 88

89 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Simplifying Nested Frtions On osion you will enounter frtion wose numertor nd/or denomintor ontins frtion. Tese re often lled nested frtions. Some emples re nd. In e se, you strt wit ig frtion wose numertor or denomintor ontins one or more little frtions, wi we ll te little frtions in te nest. To mke nested frtion simpler, rell tt te nelltion lw, like every identity, is relly pir of rewriting rules. Wen you nel, you rewrite te frtion in simpler form s. Wen you uild frtions s in setion., you strt wit nd rewrite it s. Let s ll tis seond rewriting metod neling in reverse. It relly sys: You n rewrite frtion y multiplying ot numertor nd denomintor y te sme epression. It turns out tt neling in reverse n e used to get rid of te little frtions tt pper in te numertor or denomintor of n epression wit nested frtions. Proedure for simplifying nested frtions. Multiply numertor nd denomintor y te LCD of te little frtions in te nest. Emple.. Rewrite s redued frtion. Solution: Multiply numertor nd denomintor y, wi is te desired LCD. 8 89

90 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple.. Rewrite s redued frtion. Solution: Multiply ot numertor nd denomintor y.. Te frtion doesn t redue, sine doesn t ftor. Emple.. Rewrite s redued frtion. Solution: Multiply top nd ottom y te LCD of te frtions nd. In tis se te LCD is te produt. Ten ] [ ] [ you n nel. t multiply out!ftor te top nd ottom to see if Don' 90

91 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Eerise.. Rewrite e nested frtion s frtion in stndrd form. y y d d e f 9 g 9