Matematika za prvi razred gmnazije

PA~J10HAJIHJ1

AJifEEAPCKM J13PA3J1

IIojaM H3paaa. ,Il;pBo Hapaaa (254)

A IIojaM 11apa3a; np11Ka3 CTPYKTYPe 113pa3a p;pBeToM; 113paqyttaBalbe Bpep;HOCTH 113pa3a 3a 3a)J;aTe Bpep;HOCTll npoMeHJbHBJ1X (254) IJ;eJIH aJireOapCKM H3pa3H (256) A MOHOMH; cpel'JeHH ObJIHK MOHOMa; CTeneH MOHOMa; ca811paibe CJIJ1qHJ1X MOHOMa; Qen11 anre8apcK11 113pa311 11 ocHOBHe onepaQ11je ca IbHMa; KBap;paT 811H0Ma 11 pa3JIHKa KBap;paTa; pacTaBJbalbe Ha q11HHOQe; pacTaBJbaine KBap;paTHHX 811H0Ma 11 TPHHOMa (256) p;a

TaqKy

CD.

:0 Ky8 811H0Ma; pa3JIHKa 11 3(511p Ky8oBa (263) B PaCTaBJbaine Ha q11HHOQe CJIO)l(eH11jmc 113pa3a 11 rrpHMeHe (264) IlOJIHHOMH ca je)J;HOM npoMeHJbHBOM (265) A IlojaM noJIHHOMa ca jep;HOM rrpoMeHJbHBOM; KopeH rroJIHHOMa; p;eJhelbe IIOJIHHOMa (265) B oe3y0Ba TeopeMa (269) H3,D; H H3C noJIHHOMa (270) A Op;pe1)11Balbe H3,[(-a 11 H3C-a non11H0Ma pacTaBJbalbeM Ha q11H110Qe (270) Pau;HOHaJIHH aJireoapcKM H3paau (272) A 08nacT p;eqmH11caHOCTH paQHOHaJIHor anre8apcKor 113pasa; ycJIOBHa eKBHBaJieHTHOCT; TpaHccpopMaQHje paQHOHaJIHJ1X anre8apcKMX 113pa3a (272) HeKe ocHOBHe Hejep;naKOCTH (276) A ,[(oKa311 HeKHX eJieMeHTpaHHX Hejep;HaKOCTH (276) JlHHeapHH H3pa3H H jep;Ha•mHe npasu:x (278) A JI11HeapH11 113pa311 11 JIHHeapHa 3aBHCHOCT BeJIJ1qJ1Ha; rpacpffqKO npe,n;cTaBJbalbe JIHHeapHe 3aBHCHOCTH; je,n;ttaqJ1He rrpaBHX (278) JlHm~apHe jep;HaqJt:He ca jep;HOM Heil03HaTOM (283) A EKBHBaJieHTHe TpaHccpopMaQMje 11 pernaBalbe JIHHeapHHX je,n;HaqMHa ca je,n;HOM Heno3HaToM; npHMeHe JIHHeapHHX je,n;Haq11Ha; penrnBaine je,n;HOCTaHHjHX je,n;Haq11Ha Koje ce CBO,n;e Ha JIHHeapHe (283)

B Je,n;Ha•mHe ca napaMeTpHMa (287) Jl1t:neapne nejep;naqHne ca jep;noM nenoanaToM (289)

A EKBHBaJieHTHe TpaHccpopMaQ11je 11 pernaBalbe mrneapHHX Heje,n;ttaqJ1Ha ca je,n;HOM Heno3HaToM; np11MeHe JIHHeapHHX Heje,n;ttaqJ.ftta; pernaBaine je,n;HoCTaBHHjHX Heje,n;Haq11Ha Koje ce CBo,n;e Ha J1J1Heaptte (289)

:0 Je,n;Haq11He ca anconyTHHM Bpe,n;HOCTMMa (294) B Heje,n;ttaqffffe ca anCOJIYTHHOM Bpe,n;ttoCTHMa; Heje,n;ttaqffffe ca rrapaMeTpHMa (295) C1t:cTeMH JIHHeapHHX jep;naqHHa (296) A EKB11BaJieHTHe TpaHccpopMaQ11je 11 pernaBalbe c11cTeMa mmeapHJa je,n;Haq11Ha ca ,n;Be 11 Bmne Herro3HaTmc (296) B C11cTeM11 JIHHeapHHX je,n;Haq11Ha ca rrapaMeTpHMa (299) JlHHeapHe YHKD;Hje (300)

:0 OcHOBHe oco811He JIHHeapH11x cpyHKQHja; rpacpm~11 11 oco8MHe cp~rja ca anconyTHHM Bpe,n;HOCTHMa (300)

A

IIoj aM H3pa3a. ,IJ;pso H3pa3a IlpHMep I. TproB~ op; npm13Bo9aqa K)'IIYje po8y ,n;a 811 je rrpo,n;aBao y CBojoj rrpo,n;aBtt11u;11.

TproB~ patIJHa p;oforr Ta.KO mTo op; rrpop;ajtte Bpep;HOCTJ1 po8e op;y3Me cyMy HOBu;a

Kojy je yno)l(Jfo. IloTPe8Ho ynaraibe rrpep;cTaBlba 3811p tta8aBHe Bpe,n;ttocT11 po8e 11 IpOilIKOBa TpaHCIJOpTa. Ilocrynai< K3patIJHaBaiba p;o811TJ.1 rrpKKa3aH je Ha cnMu;11 neBo. 0BaKaB rrp11Ka3 Ha311Ba ce PJ>BO, 38or Bl13JeIDie CIDl'lliOCTJ1 ca p;pBeTOM. •

CBaK11 I! p;pBO Hei pa3nora IIOlbl1Ma KOHCTall je p;a ce ' KOJIJ1Hel

3a;:i;aTaK TproB~ je op; je,romr npoH3Bo9aqa KYJI110 op;pe9etty KOJI11q11tty HeKor rrpoH3Bo,n;a l1 3a lhY IIJiaTMO 4 230 p;MHapa. TpoIIIKOBJ1 TPaHCIIOpTa cy l13HOCHJIH 560 p;MHapa. YKyrrHa Bpep;HOCT oBe po8e y lberOBoj npop;aBHHIJ;M H3HOCJ1Jla je 5 700 p;IDiapa. KomneroBoM rpa9elhy: p = 5 700,

' x+6,2

n =4 230, t=560.

Be.nncy rrpHMeHJhHBOCT MaTeMaLHKe oMoryhaBa TO DITO ce OHa bairn: pema.BaIDe 1 BeoMa OillIITIIX rrpobJieMa KOjn oc5yxsarajy Bemlly HeKor H3pa3a, HaBO):IHneMo HX y 3arpai:iaMa nopei:i 03HaKe H3pa3a. TaKo, Ha npHMep, aKo je A H3pa3 (x + 6,2) · y + 5 -z, nHcaneMo A(x,y,z). 0BaKBO 03Ha'laBaH>e je noroA}fo Hy cny-qajeBKMa Ka):la )l{eJIHMO i:ia HCTaKHeMO Bpe):IHOCT H3pa3a A 3a HeKe KOmq>eTHe Bpe.!(HOCTH npoMeHJbHBHX. AKo je x =2, y =-1, z = 14, owosapajyny BpeAHocT H3pa3a 03Ha'IHheMo ca A(2, -1, 14). ,[(aKJie, A(2, -1, 14) =(2 +6,2) · (-1) + 5-14 =-17,2.

255~

-A

I.J;errH arrrebapcKH H3pa3H Bpcre mpaaa pa3mn) -lx' -

£_

.

CBaKJll u;eo arrre8apcKM J113pa3 ce MO:>Ke rrpwKa3aTJll Kao 3bwp Mel)ycobHO HeCJIJ!lqHJllX MOHOMa.

.259

3a fomo Koje H3pa3e A 11 B BIDKe jep;HaKOCTH:

KBaµ;paT OHHOMa

(A + B) 2 =A 2 + 2AB + B2 [ ¢opM)'Jla 3a KBap;paT bHHOMa], (A - B)(A + B) =A 2 - B2 [ ¢opM)'Jla 3a pa3lIHKY KBap;para].

JI

pa3JIJIKa KBaµ;para

b

/ioKCJ.

a b

a

Obe jep;HaKOCTH cy je)J;HOCTaBHe norne~e 3aKoHa )J;HCTpHbyTHBHOCTH:

(A + B)2 =(A + B) · (A + B) =A2 +AB+ BA+ B2 =A2+2AB+B2 (A-B)(A +B) =A2 +AB-BA-B2 =A2-B2 •

a 3a ~•KeMO pacTaBHTJ1 Ha 'IIDU10Qe (o,q Kojmc HJ1 je,qaH HJ1je KOHCTaHTa J1 KOjJ1 cy CBJ1 CTerreHa MaH>er OA CTerreHa TOr l13pa3a). J1nycTpoBaheMO TO y Hape,qHOM IIpl1Mepy IIOKY1llaBajyh11 ,qa paCTaBJ1MO je,qaH caCBJ1M je,qHOCTaBaH Qeo amec5apcK11 113pa3.

IlpHMep 13. Mo:>KeMo JIH ,qa pacTaBHMO 113pa3 X2 + 1 Ha 'lHHHOQe? Axo c511 TO c511no Moryhe, 'll1HJ10QJ1 c511 Mopan11 c511T11 cTerreHa 1 11 y H>HMa c511 ce rrojaBJbHBana caMo rrpoMeHJbHBa x. ,/J;aKJie, 3a HeKe peaJIHe c5pojeBe a, ~, y, 0 J1MaJIJ1 c511CMO ,qa je + 1 = (ax+~)( y.x + 0), rrp11 'leMy je a:;eO 11 y:;eO.

r

Kao IlITO je II03HaTO, He IIOCTOjl1 peaJiaH cSpoj x TaKaB ,qa je X2+1 =0, OAHOCHO, je,qHa'lHHa X2 + 1 = 0 HeMa pellleH>a y cKyrry peaJIHmc cSpojeBa. MeQyrHM, axo c511 ropI:De pacTaBJbaH>e c511no Moryhe, OBY je,qHa'!Htty c511cMo Monrn 3aIIHCaTH y oc5m1ry (ax+~)(yx+o)

=0,

IlITO c511 3Ha'll1JIO ,qa je ax+~=O mrn yx+o=O,

OAHOCHO ,qa cy c5pojeBJ1 _ Q_ J1 -

a

,lJ;aKJie,

~ perneH>a je,qHa'lJ1He r

+ 1 = 0, a TO Jrnje .Moryhe.

3a CBaKH peanaH

5poj

x Ta'!Ha je Bejenuaxocr r ~ 0, Ila cnemtja,"IHO, He nocrojH peana.H bpoj TaKaB ;xa je r =- I, O.D;HOCHO, .ua je r +I =0.

& Jfapa.3 r + I MOXe ila ce pacraBu Ha 'IHHltone creneHa I axo H caMo a.Ko je.1Wa'Utifa r-+- l =0 HMa pemeH>a }" acyny pea.JIHHX 6~iesa.

y

X2 + 1 He MO:>KeMO paCTaBJ1TJ1 Ha 'IJ1HJ10Qe CTeIIeHa 1. •

26

• PacraBJbalbe IIBaJlPaTB11X OHBolila

ar-6rHar-y

IlpB11ep 14. IaTHH fornoMH 'Ilrja je jewma rrpoMeHJbHBa x jecy H3pa3H o8mrn:a ax2 + ~x HnH ~ JI! y HeKe peantte KOHCTaHTe pa3RJllqJl[Te 0,1.1; Hyne. IlpKMeTIL.\fO p;a ce H3pa3 ax2 + ~x MO)l(e pacTaBHTH Ha qJl[HMOQe 3a 8Hno Koje peantte

ax2 +y, rrpH qeMY cy a,

nKHOM

.:rp)Tor creneaa

Ha3HBa ce 1IBaJq>aillll fumoK.. TpimoM .:rpyror creneaa Ha3HBa ce ~parKH

Yp11HOlllL

8pojeBe a H ~: ax2 +~x=x(ax+~). Jtfapa3 ax2 +y ce MO)l(e pacTaBHTH Ha qJl!HHOQe je,l.l;MHO aKo cy a Hy 8pojeBH pa3nJ11qJ11rnr 3HaKa Hy TOM CJI)"lajy KOpHCTHMO cpopAfYRY 3a pa3nHKY KBa,1.1;paTa.

r -9 ={x-3){x+3)

.x2-3 = (x-{3)(x+f3)

4.x2- l = {2x-1){2x + 1)

2.x2 - 1 = (-fi.x - 1) (-fi.x + 1)

Y OIIIIITC Ha qJ1HJ1 j e cTeqe1 peIIIaBai HaBO,l.l;ll qJ,fHHOl.l

-18.x2 + 2 =-2(9.x2- l) =-2(3x-1)(3x + 1) A.Ko cy a

JI!

y 8pojeBH HCTOr 3HaKa, OH,l.l;a ce

ax2 + y He MO)l(e pacTaBHTH Ha qJl!HHOQe. •

Ilp1D1ep 15. PacraB;balb(! KBallpaTHOTYpHHOMa ar-~r~y

flpHJrnKOM pacraalbalba

K:Bal!p3THOrrpHHOMa ar-~r+y,a>0Ha

a

r 111m

Ha q11Hl10Qe. Koje heMo cpopMyne rrp11Melb11BaT11 3aBJ1Cl1 o,n; crryqaja ,n;o crryqaja, rra je CTeqeHO l1CKYCTBO rrp11 pacTaBJbalby l13pa3a Ha qJ1HJ10Qe rnaBHJ1 OCJIOH~ rrpH peIIIaBalby OBaKBMX 3a,n;aTaKa.

eanHe IJIJ1qJ1TOf

HaBo,n;11Mo joIII HeKe cpopMyne Koje Mory 811T11 Kop11cHe rrpl1JIJ1KoM paCTaBJbruna Ha qJ1HJ10Qe. 3a 811no Koje l13pa3e A 11 B B3.)Ke je,n;HaKOCTl1:

(A + B) 3=A 3+ 3A 2B + 3AB 2 + B3 [cpopMJna 3a Ky8 811H0Ma], A3 -B3 =(A -B)(A 2 +AB+ B2 ) [ cpopMyna 3a pa3Jil1KY KyboBa], A 3+ B3=(A + B) (A 2-AB + B3) [cpopMyna 3a 3b11p KyboBa).

Teopnta

IOQe. • 3a~aTa

,lJ;OKa)lKe rrpe,L\CTaBHTI1 Kao 38Kp Mel)yco8tto HeCJIJ.1qHJ1X MOHOMa. Cneu;HjaJIHo, y cnyqajy nomrnoMa JiMaMO cneJJ;ehe TBpl)elhe.

ett,

ou;e

µ

e

roj

j:\a j e

p

IIOJIKHOM ca

rrpoMeHJbHBOM

x. KKeMo ):\a je noJIHHOM P(x) cTerretta n J1 mnneMo deg (P) = n, • MOHOM a nXn Ha3J1BaMO BOp;enHM MOHOMOM TIOJIJ1HOMa P(x) , • Koecl>Mu;MjeHT an Ha3HBaMO Bop;ehHM KoeHI:t;HjeHTOM TIOJIMHOMa P(x), Koecl>Hu;HjeHT a 0 Ha3MBaMo c1100op;HHM 11JiaHoM IIOJIMHOMa P(x) . Cneu;MjaJIHO, aKo je P(x) = a 0 , a 0 ;t: 0, Tj. aKO je P(x) KOHCTaHTa pa3JIWIBTa OA ttyne, OHJJ;a je deg (P) = 0. )J;aKJie, peaJIHe 8pojese pa3JIWIHTe OA ttyJie cMaTpaMo noJIJ1HOMHMa HYJITOr CTenetta. AKo je P(x) = 0, Tj. aKo je P TaK03BaHJ1 HYJITH noJIKHoM, oH,qa Iberos CTeneH HMje Aecl>MHHCaH, Tj. ttyJITOM TIOJIMHOMY He np:mrncyjeMO Hl1KaKaB OApel)elllf cTeneH.

03HaKa deg ,n;ona3M 113 ettmecKor je311Ka Kao CKpahett11u;a 3a peq

degree = cmeueH. 3, -5, 1, 14, - 7 cy KoecpMll;J1j eHTH IIOJIMHOMa

....

6·1

t4

Je creneH nomrnol'

3x 4

IlpHMep I. 38Mp J1 npOM3BO):\ ):\Ba TIOJIMHOMa TaKol)e cy TIOJIJ1HOMJ1. HeKa je P(x) = 2x3 + 3x-6, Q(x) = 2x3 + 7x2 - 3 H R(x) = -2x3 + 1. IlpwMeTHMO ):\a je deg (P) =deg ( Q) =deg (R) = 3. TaAaje P(x) +Q(x) =4x3+7x2 +3x-911 deg (P+Q) =3, Q(x) +R(x) =7x2-211 deg (Q+R) =2 < 3, w P(x) + R(x) = 3x- 5 11 deg (P + R) = 1 < 3. KaKo je P(x) · Q(x)=(2x3+3x-6) · (2x3 +7x2 -3) = 4.x6 + l 4x5 -6x3 + 6.x" + 2lx3 -9x-12x3-42.x2+18 =4.x6+14x5 +6.x" +3x3 -42x2 -9x+ 18, HMaMo Aa je deg (P · Q) = 6. Hwje TeIIIKO w 8e3 AHpeKTHOr MHO:>Kelba 3aKJbyqHTM Aa je deg (Q · R) =deg (P · R) =6. •

Sx3 +r + l 4x(- 7

je BOl:(etrn: MOHO

3 je BCJ::eL - KO

qi.

CJI060l:(HH 'C

l>poj max {a,b} je;J,HaK je Beile.Y on. OpojeBa a 11 b. Ha npm&ep, max {2,3} =3, max .·.i,o} =4, max {3,3J =3, H CL A1ra;10rno ce ::ieHIDmre min {a,b}. reope.Ma

_\Ko je 1 iIOJIIDIO :Oe3 ;J>tpeKTHor cafotpa&a 11 MHO)Kelba IIomrnoMa P(x) H Q(x), o,n;pe,n;H deg (P + Q) H

deg iP · Q) aKO je: Pix) =-x3+ 2.x2-14.x+5 H Q(x) =-x2-x- l; - P(x) =-24.x8 + 3,7£ -150.x2+15 H Q(x) = l 7x!'- l,2x + 58; - P(~)=2x13 +r+x+l HQ(x)=-2x 13 +5x7 -x2.

eyne, aJ ~eJUfX l ;:i:eJblIB I

AKo IIOJ111HOMH P(x) H Q(x) mtey H)'JITH IIOJIHHOMH, ott,n;a je deg (P+Q):::;; max {deg (P), deg (Q)} aKO p + Q HHje ttynrn IlOJIHHOM J1

IIp

deg (P · Q) =deg (P) +deg (Q).

ilomm IlpJL"1eTIL"10 ,n;a je BO,n;etrn Koeqnu."'11jeHT IIpOH3BO,n;a ,n;Ba IIOJIJ1HOMa IIpOH3BO,ll; BO,n;enHX Koeqnru;11jeHaTa THX IIOJilffiOMa.

Ilo::iceha.Mo ;ia cy $a

(na crremrja;mo ;3a no;nmo, ra) eKBJIBCL'IeHTHa ai.:o cy

IIomm -Ji IIOC =-1

Ka,ua ce rrpoMeHJbHBoj x ,n;o,n;eJ1J1 tteKa KOHKpeTHa Bpe,n;uocT, Bpe,n;ttocT IIOJIHHOMa P(x) ce pa'f}'Ha Ha yo8wiajeHH Ha'IHH.

H3pa.3a H

H>JIXOBe spen;HOCTH je:maxe 3a CBaKJt H30op spe:mocn1

rqxuteJLnJIBJfX Koje ce y H>Jn.ta IIOjaBJbyjy.

Ba&Ha reope.r.1a

IlpHMep2.

3a CBaKH IlOJIHHOM

AKo je P(x) =-3x3 +r-4, oH,n;a je P(2,l)=-3 · (2,1)3+(2,1) 2 -4=-27,373, P(-2,1) =-3 · (-2,1)3 + (-2,1)2-4 =28,193, P(l)=-3 · 13 +1 2 -4=-6, P(O) =-3 · 03 +02-4 =-4.

+

P(x) Bpe,n;HOCT P(O) je je,n;HaKa c11060,n;HOM q11atty, a P( 1) 3fo1py KoeqmQHjettaTa noJIHHOMa P.

,UBa IIOJUfHOMa (ca J1CTOM npoMeHJbHBOM) cy eKBHBaneHTHa aKO H caMO aKO HMajy je,n;HaKe KaHOHCKe 08n11Ke.

HeKa R IIpH Y p;aTC

• deg

• B07 IlpeTXO,n;Ha TeopeMa Ce BeOMa '!eCTO KOpHCTH IIpH pap;y ca IIOJIHHOMHMa IIITO neMO IDI)'CTPOBaTH Ben y Hape,n;ttOM rrpMMepy. . IlpHMep3.

O,n;pe,n;HMo KOHCTaHTe a H p TaKo ,n;a rroJIHHOMH P(x) =x3-2.x2 +a H Q(x) =(x+ l)(.x2 + px +y) 8y,n;y eKBHBaJieHTHH. Krum je Q(x) =x3+CP+1) x2 + (y + p)x +y, ,n;a 811 811no x3-2x2 +a=x3+CP+1) x2 + (y +p)x+y, rrpeMa nperxo,n;ttoj TeopeMM HMaMo ,n;a je -2 =P + l, o=y+~, a=y. J.13 OBJ.fX je,n;ttaKOCTH je,n;ttOCTaBHO ,n;o811jaMO ,n;a je: p =-3, y=3, a =3. +

266

· R=

Qp;aB~

r,n;e cy ):(aKJie O)l;HOC

f\eno 3=a 2=a 4=-a

-5=~

Konw

~

Q)

Jt

AKO je tteKJ.1 noJIHHOM Moryhe pacTaBHTJ.1 Ha 'llffiHon;e, ott,n;a cy 'lliHJ.10.QM TaKol)e IlOJIJ.1HOMJ.1 ca J.1CTOM npoMeHJbJ.1BOM, IIpJ.1 'leMY je CTeneH CBaKOr 'llflIJ10D;a Belli o,n; ttyne, aJIJ.1 je MalhH o,n; cTeneHa noJIHHOMa. Ilo attanornjH ca penan;HjoM ,n;eJbHBOCTif n;em1X 8pojeBa K(l)KeMo ,n;a csaKJ.1 o,n; 'IJ.1HJ.1J1an;a ,n;eJIJ.1 noJ1J.1HOM, op;Hoctto, ,n;a je nomrnoM p;eJl>HB csaKHM o,n; 'IHHHnan;a Ha Koje je pacTaBJbeH. HeKa cy A J.1 B noJIHHOMJ.1 npH qeMY B HHje ttyJITJ.1 nonHHOM. IloJIHHOM A je ,n;eJl>HB nonHHOMOM B, o,n;ttoctto, B p;eJIH A aKo nocTojH noJIHHOM Q TaKaB ,n;a je A =B · Q. IloJIHHOM Q ce Ha3J.1Ba KOJIH'IHHK npH p;eJbelhy A ca B.

I

.l(e¢HHlfqff.

Ueo c5poj a je ;:i:e.-i.HB l\emrn c5pojeM b,

IlpHMep4.

IloJIHHOM A(x)

,Ue;beH>e no11J1Boxa

pa3;nfCIJmtM o;:i: Hy,Ie,

=r -1 je ,n;eJbHB noJ1J.1HOMOM B(x) =x + 1 jep je A

,::i:ellia:

B

axo nocrojn l\eO c5poi q TaKaB ;:i:a je a =b · q.

C

~~~

x2-l =(x+l) · (x-1). IloJIHHOM x2+1 HHje ,n;eJbHB ca x + 1. AKo bJ.1 noJIHHOM x2 + 1 8Ho ,n;eJbHB ca x + l, ott,n;a bH nocTOjao noJIHHOM Q(x) TaKaB ,n;a je x2 + 1 = (x + 1) · Q(x). Mel)pHM, Ta,n;a bJ.1 3a x = -1 MO pan a bHTJ.1 (-1) 2 +1 = ((-1) + 1) . Q(-1), IIITO HHje Ta'IHO . •

P(x)

He,n;eJbHBOCT tteKor noJIHHOMa HeKHM ,n;pyrHM noJIHHOMOM ,n;oso,n;H Hae ,n;o TaK03BaHor ,,p;elbelba ca oCTaTKOM': HaHMe, Ta'IHa je cne,n;eha TeopeMa. 3a cBaKa ,n;sa noJIHHOMa A J.1 B, npH qeMY B HHje ttyJITJ.1 noJIHHOM, nocToje je,n;HHCTBeHJ.1 noJIJ.fHOMJ.1 Q J.1 R TaKBH ,n;a je A = B · Q + R, r,n;e je R =0 HJIJ.1 je deg (R) 10

HeKa je A(x) =x4 +3x3 +2x2 +4x-5 J.1 B(x) =x2+x+1. O,n;pe,n;HMo KOJIJ.flIIDfK Q H ocTaTaK R npH ,n;elhelhy A ca B. Y ,n;aTOj rnTyan;HjH ttenocpe,n;Ho 3aKlhyqyjeMo ,n;a je: •deg (Q) =2, • BO,n;enJ.1 KoeHn;HjeHT IlOJIJ1HOMa Q jep;HaK l, • R =0 HJIJ.1 deg (R) < 2. O,n;as,n;e rne,n;H ,n;a je noJIHHOM Q(x) obJIHKa x2 +ax+ a ,n;a je noJIHHOM R o8JIJ.f:Ka yx -r r,n;e cy a, y, o KOHCTaHTe Koje Tpe8a o,n;pe,n;MTJ.1. ,D;aKJie, x4 +3x3 +2x2 +4x-5 =(x2 +x+ l~ · (x2 +ax+p) +yx+o, O)l;HOCHO,

p,

p,

a R ocraTaK

npu ;:i:einelby A ca B. Axo je deg (A) 1. Jep;HHH p;eJI11on;11 IIOJIHHOMa A(x) Koj11 cy creneHa n jecy nomrnoMM aA(x), 3a npoH3BOJbaH peaJiaH 5poj a pa3JIJ1qJ1T op; ttyJie.

je

IIommoM D je 3aje1J;llJl'IKH p,e11Km1~ noJIJ1HOMa A 11 B aKo cy 11 AH B p;elbHBH ca D. IlomrnoM S je 3aje)J;HJ111I01 cap,p*arra~ noJIHHOMa A 11 B aKo je S p;elbHB 11 ca A 11 ca B. ).(x

,.....__,.

D(x)

1(0,1B"IBJl]o;

~

~

r-1 =lx-1)-(x+l)

Ilp11Mep l. HeKa je A(x) =x3-x M B(x) =x"-1. H11je TeIIIKO np0Bep11rn p;a je jep;att 3ajep;tt11qK11 p;emman; nommoMa A(x) 11 B(x) noJIHHOM D(x) =r-1;

A(x)

D(x)

A(x)

D(x)

r-A---.

r-"---.

r-A---.

r-"---.

x3 - x =cxz. - 1) . x 11

r

-1

=cxz. - 1) . cxz. + 1)

l1Majy mf IIOJUfHOMlf A(x) 11 B(x) H p;pynrr 3ajep;tt11qKJ1X p;eJIHJian;a? (}q:JrrJiep;Ro je ,a;a cy CBH p;emwl.-\J.f IIOJIHHOMa D(x) =r -1 TaKol)e 3aje,a;tt11qK11 p;eJI11on;11 nommoMa A(x) H B(x). KaKo cy x -1 H x + 1 p;elI11on;11 IIOJIHHOMa D(x), OHH cy 11 3ajep;im'l1m p;emwl.-\J.f IIOJUfHOMa A(x) 11 B(x):

IlpeTxq 3ajep;m ::xeJIHJHU B.ajiem ·e;ri;HOCT paCTaB

!!p HpO~

.::e 3ap,pl

IIp x3-x=(x- l) · (:x!+x) 11 x"-1 =(x-1) - (x3+:x!+x+ 1), .x3-x=(x+l)- (:x!-x) 11.x"-l =(x+l) · (x3-r+x-l).

0p;pe,ml pacTaBJ

A(x HMje TeIIIKo 3aKJb)"UITJ.f p;a cy a, ax-a, ax +a, a:x!-a, a E JR\ {O} CBJ.f p;eJIJ10D;M p;ana: IIOJIJ1HOMa A(x) 11 B(x). IlpHMeTHMO p;a HaKo 3ajep;tt11qKHX ;i;emrnaQa ,n;aTMX IlOJIJ.fHOMa HMa 5ecKOHaqHo MHOrO, IIOCTOje caMO qeTHpl1 pa3JIJ1qJ1Ta nrna p;emmaQa: KOHCTaHTe, AJ3a TMna p;eJIMJiaQa rrpBor pep;a H jep;att THII p;eJIHJian;a p;pyror pep;a.

:1 A(x)

J1aKo 3a HajBerrn 3aje,n;ttHqKH p;emrnan; noJIHHOMa A(x) H B(x) MO)KeMo y3eTH forno Koj11 p;emrnan; MaKCHMaJIHor cTenetta - y OBOM cnyqajy cTerretta p;Ba, yo811qajetto je p;a ce y3Me ottaj q}'[jH je Bop;etrn Koeqmn;11jeHT jep;ttaK jep;HHHQH. ,ll;aKJie,

H3,ll;(x3-x,

r

-1) =r -1.

IIpHMeTMMO p;a je ttajBetrn 3ajep;tt11qKH p;eJIHJian; IIOJIJ1HOMa A(x) 11 B(x) p;elbHB CBHM OCTaJIHM 3ajep;HJ1qKJ1M p;eJIJ10QJ1Ma OBa p;Ba IlOJIJ1HOMa. •

A(x

HeKa cy A(x) 11 B(x) rronHHOMH KOjH HHcy Hynrn.

HajBehH 3aj ep;HH'IKJ1 p;eJIJUian; rronHHOMa A(x) H B(x) je CBaIUI rronHHOM KojH 2

je p;elbHB CBHM ocTanHM p;enHou;HMa rronHHOMa A(x) H B(x). Ca H3JJ:(A(x), B(x)) 03HaqaBaneMo 8Hno KOjH HajBeh11 3ajep;HHqKJ.I AenHnau; rronHHoMa A(x) u B(x).

HajMalbH 3aj ep;HH'IKH caµ;p)l(aJia~ rronHHOMa A(x) H B(x) je CBaK11 rromrnoM

:ie4>nmntnJa

KojH je p;e7bl1B rronHHOMHMa A(x) H B(x) H Aen11 CBaK11 p;pyrn cap;p)f(anau; OBa ABa ~)',Ie.

rron11H0Ma. Ca H3C(A(x), B(x)) 03HaqaBaneMo 8Hno KOjH HajMalbH 3ajep;HHqKJ.I cap;p)f(anau; rronHHOMa A(x) 11 B(x).

J

aD. IlpoH3Bop; p;Ba rronHHOMa je p;eJbHB lbHXOBHM HajBenHM 3ajeAHffqKJ1M Aemrn~e.M n rrpH TOMe je KOnffqHHK HajMalbH 3ajeAHK'llill CaAP)f(anau; Ta ABa rrommoMa.

1f

IlpeTXO)l;HO TBpl)elbe HaM OMoryhaBa p;a Ha jep;HOCTaBaH HaqHH op;pep;l1MO HajMalbl1 3ajep;HHqKH cap;p)f(anau; ABa rron11HOMa aKo rrpBo op;pep;HMO lbHXOB HajBenH 3ajep;HHqK11 p;en11nau;. HajBenH 3ajep;HHqK11 p;en11nau; H HajMalbH 3ajep;H11qKH cap;p)f(anau; p;Ba rron11H0Ma jep;HOCTaBHO op;pel)yjeMO aKO rronHHOMe pacTaBHMO Ha qffHl1ou;e KOj l1 ce p;aJbe He Mory pacTaBJbaTH H rrocTyrrHMO aHanorHo Kao rrpHnl1KOM OAPelJHBalba H3~ -a H H3C-a

~

mu;11

rrp11pop;HHX 8pojeBa KOjH cy paCTaBJbeHl1 Ha rrpocTe 'lliHHou;e. 0BOM rrpHnKKOM heMO ce 3ap;p)f(aTH CaMO Ha TaKBl1M cnyqajeBHMa.

Eyium~oB

a;rrop1ua.\I je IIOCTynaK 3a HaJiaJKeibe ttajBeher Jaje,nmf'IKor p;emmn;a ASa u;ena Opoja. Onncatt je Ha CTpam1 53. Ilonryirn aHanorHo ce MO)l{e Haillf HajBehH 3aje)J;lrn'IKH p;emrnan; HeKa wia IIOnHHOMa (BH)l;H 3ap;aTaK 26. Ha CTpaHH 306 H IheroBO pellle:tbe). AaKne, EyKnHp;OBanropHTaM je yttHBepJanHH noCTynax H Ja op;pel)HBa:tbe ttajBeher Jajep;HWIKor p;enHou;a (na THMe H ttajMaiber 3aje,nmfl1Kor ca,IlJ>)l{aoua) 4Ba IIOillfHOMa.

IlpHMep2. Op;pep;HMO HajBehH 3ajep;HHqKJ.I p;enHnau; H HajMal:b11 3ajeAHffqK}f caAP)f(anau; rromrnoMa pacTaBJbalheM Ha q11HHou;e. 1

Wll1Ta

;a

mo Koj11 ce

l1M

A(x) =x4-x2, B(x) =x3-2.x2 +x A(x) =x4-x2 =x2(x2 -1) =x2(x- l)(x + 1) B(x) =x3-2x2 +x=x(x2-2x+1) =x(x-1)2 H3JJ:(A(x),B(x)) =x(x-1), H3C(A(x),B(x)) =x2(x-1 )2(x + 1) 2) A(x) =2x3 +x2+2x+1, B(x) =x5-x A(x) =2x3 +x2+2x+1=x2(2x+1) + (2x+ 1) = (x2 +1)(2x + 1) B(x) =x5-x =x(x4-1) =x(x2- l)(x2+1) =x(x- l)(x+ l)(.x2+1) H3,D;(A(x),B(x)) =x2 + 1 H3C(A(x),B(x)) =x(x- l)(x + 1)(2x + 1)(.x2+1) 3) A(x) =x2+1, B(x) =x2 +x+1 IlonHHOMH A(x) H B(x) ce He Mory pacTaBKTH Ha qffmmu;e. H3,D;(A(x),B(x)) = 1 H3C(A(x),B(x)) =(x2 +1)(.x2+x+1) )

+

271

Pa~oHaJIHH arrre6apcKH H3pa3H

.

Ilp

Bpe,iJ;I:Iocr fomo Kor n;enor anre8apcKor H3pa3a MO:>KeMo o,n;pe,n;HTH aKo 8mm Koje peaJIHe 8pojeBe ;:{OAeJIID.10 rrpoMeHJbHBHM Koje ce y lheMy rrojaBlhyjy. To HHje CJI)"Iaj

,Uo8poj

ca pan;nomunIJtM arrre8apcKHM H3pa3HMa 8y,n;ytrn ,n;a y lhIDCOBOM rpaJ)elhy )"IeCTByje

sa:>KH je

,n;emelhe, a Kao rrrro je no3HaTo, ,n;elbelbe ttyJIOM HHje ,n;eqmHHCaHo.

IIp1D1ep r.

,UpyrHN

HeKaje

w,n;eHTJf

x+yz R(x,y,z) =-~-(x-2)(y-z)

Jifa rrpe-

Oq}fi'rrewrn, Bpe,n;HOCT H3pa3a R(x,y,z) He rrocTOjH aKo je rrpoMeHJbHBOj x ,n;o,n;eJbeHa

•p/\--iq

·e IafIOJionrja Il03Harn

Kao !le MopraHoB 3aKOH (BHiUi CTpaHy 21).

Bpe,n;HOCT

2 IDrn cy rrpoMeHJbJIBHM y Hz ,n;o,n;eJbeHe HCTe Bpe,n;HOCTH. ,UaKJie,

"Bpe,n;ttOCT l1:3pa3a R(x,y,z) He IIOCTOjH" ~ X = 2 Vy= Z.

MO:>Ke N ,n;e¢Jrn~

0,n;aBAe ,n;albe 3aKlb)"I}'jeMo ,n;a He

,.,Bpe,n;ttOCT H3pa3a R(x,y,z) IIOCTOjH" ~ •(x = 2 Vy =z) ~

x:t:-2/\y:t:-z.

Ka:>Ke ce ,n;a je ¢opM)'Jla

x:t:-2/\y:t:-z ycnos nop; KojMM je H3pa3 R(x,y,z) ,n;e¢HHHCaH. IlpHJIHKOM HaBol)elha ycrroBa 3HaK ,, /\" ce Hajqemb.e 3aMelhyje 3ape3oM. HaBO,n;JiMO jom HeKoJIIDe]

Bt:-0,Dt:.O

[ cac511:palhe H op;y3HMaH>e]

Bt:-0,Dt:.O

[MHO)l(elhe}

B t:. 0, D t:. 0, Ct:. 0 (npHMeTHTe p;a BOAJIMO paqytta 0 CBHM MeCTHMa rp;e ce nojaBn,yje p;en,elbe)

[p;en,e1De]

1

273'

Ilp

Il.,.-ep4. 3a CBaKJ.1 Og paqnoHamrn:x aJITeOapcKHX H3pa3a HallH neMO eKBHBaJieHTaH pa.QHOHaJIHH

HeKaje

H3pa3 y OCHOBHOM OOmucy J1 OAJ>eWfTH YCJIOBe IIOA KOjHMa eKBHBaJieHTHOCT BroKH.

1) Tpattc

1 + (x-y z

l=--x-.e±y

r-f

(x-y)2 x2-f

y OCHOBHJ1 00Jll1K.

r-f + x2-2xy+f r-f x2-f r-f +x2-2xy+f

,,MeCTa•

x2-f

HMeHHG

2x2-2xy

,D;a OH~

x2-f

l -3x -:t:-•

2x(x-y) aKO OTK

x-y)(x+y) 2x

,[l;a 0HC1

= - - , x-:t:-y, x-:t:--y

x+y

l-3l=

a2 2) Tpattca H O,qy3KMalba K3pa3a

=

Mopa

a(a+2 ) -a2 (a -2) (a + 2) a2 +2a-a2 (a-2)(a+2)

=-----

IpCUICKMO HajMalbH

3ajewnfq](J{ caJqJxanan;.

2a -----,a-:t:-2,a-:t:--2 (a-2)(a +2) 3) Tpattc11K oBe 3aB11CHocT11 H11je rrpaBa V = 2 - 0,25 t, Ben ,n;eo oBe rrpaBe, Tj. lJ.Y>K·

+

Ilp11Mep3.

qme,

I CBaKl1

[)ja

x oja j e

IlpeTIIOCTaBl1MO ca,n;a ,n;a ce y C~, y KOjl1 MO)l(e ,n;a CTaHe 3i Bop;e, ymma Bop;a paBHoMepHOM 11K oBe 3aBl1CHOCTl1. fpacl>11K je .n;eo rrpaBe

v

v = 0,5 + 0,25t.

rJ

r-

AKo je rrpoTeKJIO t M11HyTa o.n; rro'!eTHor TpeHyTKa, OHp;a je 3a TO BpeMe l1CTeKJIO 0,25 . t Jll1Tapa BO,n;e. IlOIIITO je y rrocy,!l;J1 y IIO'IeTHOM TpeHyTKY aHTf, AOBOJbBO je AOKa3aTH Aa je Ke p;obHTH HH 3a KaKaB H3bop KOHCTaHTH k J1 n.

Ilp IIo;o,ceTJ ca jep;Ho caCBHM

4 x

2

Ca :>KeJI>OM p;a jep;Ha'lHHaMa omm:reMo cBe rrpaBe paBHH, 3a 011mTH oOJDIK jep;11a'IKHe rrpa.Be Y3HMaMO

ax+by+c=O,

I

x=-2

x=O

rp;e cy a, b H c HeKe KOHCTaHTe, rrpH qeMY KOHCTaHTe a J1 b HHCY HCTOBpeMeHO je.n;ttaKe ttym1 (a ;eO HJIH b ;eO, Tj. a2 + b2 ;eO).

X=3 CBa.Ka jewrn'Illlia o5mrna

J,aKJie, l HeII03Hl

ax+ by+ c =0 (a 2 + b2 ;eO) MO:>Ke ce TpaacopMHcaTH

HJIH

y je.n;tta~

a c aKo Je . b ;e 0, y=--,;x-b,

H a Map KOOpp;Jf .:i;aTeje

x=-~. aKo je b =0 (rra Mapa 511rn a ;eO). a

CBaKa npaBa p,em1 paBaH y Kojoj ce HaJia3H Ha p;Be rronypaBHH. Ey.n;ytrn p;a CMO j ep;Hatrn:HaMa OilliCaJIJ.i CBe rrpaBe y paBHH, IIOCTaBJba ce IIHTalbe KaKO bHCMO orrn:cJIBaJIJ.i rronypaBHH. Orrmry 11.n;ejy 11nycTpOBaneMo y ttapep;HoM rrp11Mepy.

Penud

IlpB11ep4. HeKa j e p;aTa npal}a x +y-1 = 0. Yo'IMMO Ta'lKe A, B 11 C (B11p;11 CJIHKY neBo) jep;He op; rronypaBHH Kaj a je op;pel)eHa p;aToM rrpaBoM. Koopp;11HaTe (x,y) OBHX rnqaKa 3ap;oBOJbaBajy Hejep;HaKOCT x +y-1 ~ 0:

A(-1,3): -1 +3-1=l~0, B(l,l): 1 +1-1=l~0 ... 3anpaBo, KoopwruaTe CBHX Ta'laKa rronypaBHH Kojoj rrp11nap;ajy yoqeHe TaqKe 3ap;OBOJbaBalle Hej e.n;ttaKOCT x +y-1 ~ 0, rrp11 qeMY jep;HaKOCT Ba:>KH 3a KOopp;HHaTe TaqaKa Koj e npKIIap,ajy p;aTOj rrpaBoj. Koop,rµrnaTe Ta'laKa ,n;pyre nonypaBHH 3ap;oB0JbaBajy Hejep;HaKocT x + y- 1 ~ 0. A.Ko IIOCMaTpaMo TaK03BaHe OTBopeHe rronypaBHH, Tj. rronypaBHH 5e3 rpaHJfqHJ1X TaqaJKe o,n;roBapajyhe cTpore Hejep;HaKOCTH.

+

IIorrypaBaH ax+ by+ c :5 0 LIPTaMO TaKO IIITO Hajnpe HaQpTa.MO npaBy ax+by+c=O, 3aTHM H3a5epeMo npoH3BOJbtty Ta~Y P(x0 ,y0 ) Koja He npHnap;a OBOj npaBoj H npoBepHMo p;a JIJ1 j e 5poj ax0 + by0 + c II03HTHBaH HJIJ1 HeraTHBaH. AKo je HeraTHBaH, o~a mpaqmpa.Mo norrypaBaH Kojoj OBa TaqKa npHnap;a, a aKo je no3HTHBaH, oHp;a mpaqmpa.Mo norrypaBatt Kojoj Ta~a P He npHnap;a. AKo rpe5a p;a HaQpTaMO oTBopetty rrorrypaBaH ax+ by+ c < 0, nocTyna.Mo Kao y npeno.n;ttoM c:nyqajy, ca.Mo mTO rrpaBy qpra.Mo HcnpeKHp;aHHM JIHHHja.Ma HCTH'lytrn Ha n j HaqJfH p;a cy Ta~e rrpaBe HCKJbyqeHe.

282

MO:>K;D,al TaKO p;j KOjH 0

je;o,ua

II

Jep;Ha~

Jep;Ha" HYJIO

Jep;HaaHCopM~aMa KojHMa ce je.n;Haqwtte npeBop;e y eKBHBaJieHTHe je.n;HaqwHe

je.n;ttocTaBHwj er o8JIHKa.

Axo cy L(x)

H

HeKa

D(x) ne1rn anre8apc1G1 H3pa3H, ocHOBHe eKBwBaneHTHe

rpaHcopM~Hje 8a3Hpatte cy Ha rne.n;ehHM oqJi:rJiep;HHM TBpl)elhwMa.

I

paimpop.~mmja.Ma

2 u 3 ;:xofatja.Mo eKBHBa.•em:~ujy:

I x)=D(xl ~r

x)-D(x1=0

L Je.n;ttaqlfHe L(x) =D(x) w D(x) =L(x) cy eKBwBaneHTHe. 2. AKo je L 1(x) neo anre8apcKH H3pa3 eKB1rnaneHTaH ca L(x), a D 1(x) neo arrre8apcKH K3pa3 eKBwBaJieHTaH ca D(x), oH.n;a cy je.n;HaqwHe L(x) =D(x) w L 1(x) =D,(x) eKBHBaJieHTHe. 1. AKo je A(x) 8HJio Kojw Qeo anre8apc!G1 K3pa3, oH.n;a cy je.n;HaqwHe L(x) =D(x) w L(x) +A(x) =D(x) +A(x) eKBHBaneHTHe. 4 AKo je c OHJio KOjH peanaH 8poj pa3JIJfqMT o.n; ttyne, OHp;a cy je.n;HaqwHe L(i) =D(x) Jf c · L(x) =c · D(x) eKBHBaJieHTHe.

Ilp

PycKw 1 Kona. l IIOCTajl .n;y8mu

Haiya HCilO,ll; ' Ha Kon

Ilp1D1ep3. IlpH pemaBaILy ·e.:u1a'Ilffia nonyr ooe y npHMepy 3. apmt:el:b}ie~m ex1nmaaeHTHe rpa.HapopMaIUfje TaKO :ra je::J,Ra'Ilmy CBeJ;eMo Ha je;matnmy o0m£Ka x = a 'Ilrje pemel:be Henocpemm ,,tnrraMo':

J1JiycrpoBabe.Mo npHMetty HaBep;eHHX eKBHBaneHTHHX TpaHcopMaQwja pernaBalheM jewra'!IDie

x- 2 2x -1 3x - 1 --+--=3x--3 6 2 ~6- -

(

03Ha'Il

3aKOH~

x - 2 2x - 1 3x - 1 --+--=3x---. 3 6 2

oqJffJI MeTapl l13Mel)j

3x -- 1 ) ........................ [TpaHcopManKja4: x --2 +2x ---1 ) =6· ( 3x-3

6

2

L(x) =D(x)

~ 6 · L(x) =6 · D(x)]

~ 2(x-2)+(2x-1)=6 · 3x-3(3x-l) ......................... [TpaHcopMaQwja2] ~ 2x-4+2x-l =18x-9x+3 ................................ [TpaHcopManwja2] ~ ~

4x - 5 = 9x + 3 ............•............................... [TpaHcopMaQwja 2] 4x - 5 -4x = 9x + 3 -4x .................................... [TpaHcopManwja 3: L(x) =D(x) ~ L(x)-4x=D(x)-4x]

~ -5 =5x+3 ............................................... [TpaHcopManwja 2] ~ -5 -3 =5x+3 -3 ......................................... [TpaHcopMaQwja 3:

L(x) =D(x)

~

L(x)-3 =D(x)-3]

~

-8 =5x.................................................. [TpaHcopManwja 2]

~

5x =- 8 ................................................. [TpaHcopManwja l]

~ ~ :>

· 5x =

~

· (-8) .......................................... [TpaHcopManHja 4:

:>

L(x) =D(x) ~ 1/5·L(x)=1/5 · D(x)]

~ x=-! .................................................. [TpaHcopMaQwja 2]

.5 . . s;: • 8 • Jl aKJie, Jep;HHo pernelbe p;aTe Jep;HaqlfHe Je upoJ --. 5

. 3

3arap po8y cyMe TOKO

Y nrnpeM CM11cny CBaKy jep;Ha'IBH)' o8nMKa L(x) =D(x) Koja ce HaBep;eH11M eKBl1BaJieHTHl1M Tpattcf ,I\ napaMe .n;o.n;ene j e.n;Ha'UI

PeIIrnMo je.n;Ha'IJ.iH)' (x-l)(x-5) =2(x- l)(x-3).

(x-l)(x-5) =2(x-l)(x-3) ~ (x-l)(x-5)-2(x-l)(x-3) =0 ~ (x-l)(x-5-2(x-3)) =0 ~ (x-l)(x-5-2x+6) =0

ce je.::ma~a H3 npm.tepa 6. He cMe ,,110::ie.,nITHD ca x - 1.

Ilp Pellll1M

~cx-1)(-x+l)=O

3anrro?

~

x-1 =OV -x+ 1 =0

~

X=l Vx=l

~ X=

)J;aK.Jie, .n;aTa je,IµIatUIBa J.1Ma je.n;Ho pernelhe -To je 8poj 1.

A(x) =0 B(x)

+

1

~ A(x) =OAB(x) ;t:O

ce KopJ.1CTJ.1 npJ.1JJMKoM pemaBalha je.n;Hat.IJ.1He Koja je ,,cacTaBJbeHa" o.n; pa~J.10HaJIHJ.1X an:re8apcKHX J.13pa3a.

ab=O~a=OVb=O

abxO

~

a:Of\b:;cO

. 1 1 2 2 PeJIDfMo Je,IµIa'IIDI)' - - = - - - - - . x x2-x x-1 1-_!_=_2_ _ _2_~~ 2 +-2-=0 x x2-x x-1 x x(x-1) x-1

Vq)

r~

pAr1V(q/\r)

1

~ (x-1 ) -2+2x 2

.lYp~p

IIonrro je Ta'Il!a 1a.uLTIIKa.IUfja

x=-l

~

x =O/\x ,z I,

1wa.Mo ;:ia je x=-1 Ax=O/\x:;c I~ X=-1

286

x2- l x(x-1 )

x(x-1)

-o

~x2-1=0Ax(x-l);t:O ~

Ilpe TOI p;aTe je} Hacmt . o.n;roBaj npaBe J

(y=2x,

perrpe3

HJ.1jeTI 3aje.n;HI aKoje l HeMajy he .n;an

ft.;t:l, a

o~ x2-2x+l-2+2x =O

x(x-1 ) ~

• aKO jl • aKO jl CJIJ.1'IH~

Ilp11Mep6.

IIpuMeTifMO ;:ia

Beh CMd Kojoj je , Ha TOj C' KOHKpe1 o.n;pel)ei! J1Mam1

(x-l)(x+ 1) =0/\x;t:O/\x;t: 1

,IJ;o J.1C1 je

pa3JIJ.1~

• aKO

~

(x-1=OVx+1=0)/\x;t:O/\x;t:1 ~ (x=l Vx=-1)/\x;t:O/\x;t:l ~ (x= 1 /\x;t:O/\x;t: l)V (x=-1/\x;t:O/\x;t:1) ~ 1- V(x=-1/\x;t:O/\x;t:1) ~ x=-1 )J;aK.Jie, jeAJ.IBO perne&e .n;aTe jep;Ha'IJ.1He je 8poj -1. +

• aKO

B ,IJ;Be

lJ1x

Jeµ,Ha'l:HHe ca napaMeTpHMa mrncyjy cKyn je.n;ttaqJUia HCTor o8Jrnxa. Ha cTprurn 285 Ben CMO pa3MaTpanH je.n;tty je.n;ttaq:irny ca rrapaMeTpHMa. Peq je o je,n;HaqHHH ax= b y Kojoj je x HeII03HaTa, a a J1 b HeKe peanHe KOHCTaHTe. y TeopeMH KOjy CMO J1CTaKnJ1 Ha rnj CTpatt:11 o.n;pel)etta cy perne1M oBe je.n;ttaqJ1He rrpH qeMY a H b HHcy 8:mrn HeKH KOHKpeTHJ1 8pojeB:11, Ben cy TpeT:11paHJ1 Kao rrpoH3BOJbHJ1 8pojeBJ1 Koj:11 3a,IJ;OBOJbaBajy o.n;pel)ette ycnOBe. J1ManH CMO TPH cnyqaja Kaja neMO IIOHOBJ1TJ1: • aKo je • aKo je • aKo je

Je»1a"'llrae ca rrapa.Merpnxa

a 7:- 0, je.n;ttaq:11Ha ax= b HMa je.n;tto pernelbe :11 TO je 8poj .!!._;

a a= 0 H b 7:- 0, je.n;ttaqJ1Ha ax= b HeMa pernelba; a= 0 :11 b = 0, cBaK:11 peanaH 8poj je pernelbe je.n;ttaqJ1He ax= b.

h-=x+2

Cn:11qtta attanH3a ce crrpoBo,n;H rrp:11nHKOM pa3MaTpalba nHHeapHKX j ewiaqJ1Ha tteKor .n;pyror 08n:11Ka y Koj:11Ma ce rrope.n; tterro3HaTe rrojaBJbyjy H cnoBa - TaK03BaIIH rrapaMeTp:11, Koj:11 Mory y3:11MaT:11 rrpott3BOJbHe peantte Bpe.n;ttoCTH. A.Ko ce rrapa..\feTPHMa .n;o.n;ene KOHKpeTHe Bpe,n;HOCTJ1, je,n;Haq:11Ha ca rrapaMeTpHMa IIOCTaje ,,Obttqtta" je,n;Haq:11Ha.

2.x=x-+-2 -3x=x-l

1

--x=x-J 2 .-x=x+2 x=x+2

1

flp1D1ep8.

Pern:11Mo je,n;Haq:11Hy A.x = x + 2 y Kojoj je

A. peanaH rrapaMernp. A.=-2

Ilpe Tora pa3MOTp:11Mo reoMeTpttjcKy HHTeprrpernn;:11jy ,n;aTe je,n;Haq:11He. Ha rn:111.i;:11 .n;ecHo je Han;prntta rrpaBa y =x + 2 Kaja o,n;roBapa :113pa3y ca .n;ectte CTpatte je.n;ttaq:11He, Kao :11 rrpaBe y =Ax 3a pa3He Bpe,n;HOCTH rrapaMeTpa A. (y=2x, y=3x, y=-3x, y=-2x :11T.n;.). 0Be rrpaBe perrpe3eHTyjy neBy cTpaHy je,n;HaqHHe.

A.=-3

A.=2

"=-2:r "=-3x /

/

/

A. =- 1

A.=0

=G H:11je TernKo rrp:11MeT:11T:11 .n;a he rrpaBa y = x + 2 HMaTH 3aje,n;H:11qKJ1X TaqaKa ca CBHM rrpaBaMa 08n:11Ka y = A.x aKo je A. 7:-1. HacynpoT TO Me, rrpaBe y = x + 2 :11 y = x HeMajy 3aje,n;H:11qKJ1X TaqaKa. 0,n;aB,n;e 3aKJbyqyjeMo .n;a he .n;arn je.n;ttaq:11Ha :11MaT:11 je,n;:11HCTBeHo pernelbe aKo je A. 7:- 1, a 3a A. = 1 Hene :11MaT:11 pernelba.

' -- ......... ~

/

1 A=-3

.L

1=3

•

I

~ I

~

.

2

3

4

x

A.=

,IJ;o HCTor 3aKJbyqKa .n;ona3HMO aHam13:11pajyhH ,n;aTy jewiaqHHY Ha rne.n;enH HaqJ1H. KaKo je

A.x=x+2 ~ (A.-l)x=2, pa3n:11KyjeMo ,n;Ba cnyqaja:

2

• aKo je

A. 7:- l, oH,n;a je x = --

• aKo je

A. = 1, ,n;aTa je.n;ttaqJ1Ha ce CBO,n;tt Ha 0 · x = 2, na 3aKJbyqyjeMo .n;a HeMa pemelha.

A.-1

:11 .n;arn je,n;ttaqttHa HMa je,n;ttHCTBeHo pernelhe;

0Ba ,n;Ba 3aKJbyqKa rrpe.n;crnBJbajy rrornytto pernelbe ,n;aTe je,n;ttaqJ1He ca napa.MeTPm1a. •

287

Ji Ha je;ma'!IDie ca napaMeTpKMa qecTo HaIDia3HMO np:iurnKoM pernaBaIDa 3a,n;aTaKa npilMeHOM ¢opM}'Jla Koje IIOBe3yjy HeKOJil1KO BeJiwnrna. Ha IIpHMep, II03HaTa je ¢opM}'Jla

Ilo,n;d

v=v0 +at Koja JJ;aje Be3Y H3Me1)y noqeTHe 5p3HHe

je,n;HO

v0, y5p3aIDa a 11 5p3HHe v HaKoH BpeMeHa t, np11

paBHOMepHo y5p3aHOM KpeTaIDy Ka,n;a cy BeKTOPH noqeTHe 5p311He J1 y5p3aIDa HCTOr CMepa. 0By ¢opMJlIY KOpHCTHMO y CBHM rnTyau;HjaMa Ka,n;a cy HaM no3HaTe Bpe,n;HOCTJ1 HeKe TPH BeJIJ1qlfHe Koje ce nojaBJbyjy y ¢opMJ1IH. Ta,n;a 113 p;aTe ¢opMyne o,n;pel)yjeMo HeII03Hary BeIDf'fHHJ TaKO IIITO je ,,H3pa3HMO y ¢ym cy, :;>ex ex s; cy.

xE(-=,

11m11111mmumm111a -1

vmwmwwwmw1111w~

-2 -1

(--00, 2)

-2

f

~] (--00, ~]

x E (-=, 2) c

x < )'

-1

x-3 x-ls;-- /.4 4 4x-4 s; x-3 f -x 3x-4 s; -3 I +4 3x s; 1 I: 3 1 xs;3

5>2x+l /-1

h

I

0

[-1,+oo)

3HaqJ1

u.

I

x E [-1,+oo)

Vfll/Ull/WJll/Ull/WJll/W~

-1

~ ./"-, -r< -x

xs;2x+l f-2x x-2x s; 2x+ 1-2x -x s; 1 / -(-1)

1

2

~

3

4

1111/lllllflllll(D -2

-1

01

3

I

1

2

~

4

3

IlpHMeTHMO .n;a cy CKyrrOBH pelllelha Heje.n;ttaq11Ha 113 rrpeTXO,Il;HOr rrp11Mepa TaK03BaIDf Heorpatt11qeHH HHTepBaJIH Koje CMO .n;eqmtt11caJIJ1 Ha CTpaHH

65. IIpeMa THM

.n;eqmtt11~11jaMa je:

x ~ a x E [a, +oo)

x > a x E (a, +oo) xs;a x E (-=,a] x < a x E (--00, a).

+

11pDiep2. CKyrr pellle!ha HeKe tteje.n;ttaqJrne MO:>Ke 811T11 11 rrpa3att. Pe~o, ttejewraqJ{Ha

x + 2 < x + 1 HeMa pellle!ha. HacyrrpoT TOMe, CBaKH peanaH 8poj je pemelhe HejewraqJ{He

x + 2 > x + l, Tj.

CK}'II

CBHX pelllelha oBe tteje.n;ttaqJIBe jeCTe cKyrr CBHX peaJIIDfX 8pojeBa R KOjH rroHeKa,n;a 03HaqaBaMO J1 Kao HHTepBaJI

(--00, +oo ) . •

289

I!IlocrynaK pema.BaH>a HejewiaqJIIHa je 3aCHOBaH Ha rrpJ1IMeHJ1I eKBJIIBaJieHTHJIIX rpaaccpopM~ja Koje Hejewia~e rrpeBo,n;e y Hejewia'UlHe ca JIICTJIIM cKyrroM pemelba.

h O,n;pe~

IJoTpt AKo cy L(x) H D(x) u;emt aJirebapcKH H3pa3H, OCHOBHe eKBHBaJieHTHe 1J>aHccpopM3Wfje Heje]:Ufa'IHHa ba3Hpatte cy Ha cne,n;ehHM TBpl)elbHMa. l L(x) $ D(x) o D(x) :?: L(x) L(x) :?: D(x) o D(x) $ L(x)

L(x) < D(x) o D(x) > L(x) L(x) > D(x) o D(x) < L(x) 2. AKo je L 1 (x) u;eo aJITe5apcKH H3pa3 eKBHBaJieHTaH ca L(x), a D 1(x) J.\eO an:re5apCI(){ H3pa3 eKBHBaJieHTaH ca D(x), oH,n;a je L(x) $ D(x) o L 1(x) $ D 1(x), L(x):?: D(x) o L 1(x):?: D 1(x), L(x) < D(x) o L 1(x) < D 1(x), L(x) > D(x) o L 1(x) > D 1(x). 1. AKo je A(x) bHJio KojH u;eo aJITe5apcKH H3pa3, oH,n;a je L(x) $ D(x) o L(x) + A(x) $ D(x) + A(x), L(x) :?: D(x) o L(x) + A(x) :?: D(x) + A(x),

L(x) < D(x) o L(x) > D(x) o 4. AKo je c bHJIO L(x) $ D(x) o L(x) < D(x) o

L(x) +A(x) < D(x) +A(x), L(x) +A(x) > D(x) +A(x).

1 < 3-

KaK0 1 Hejew Hej,e,n;E Ha

KOjH II03HTHBaH peaJiaH 5poj, OH,n;a je c · L(x) $ c · D(x), L(x) :?: D(x) o c · L(x) :?: c · D(x),

CJ]

tteje,n;:i

L(x) > D(x) o c · L(x) > c · D(x).

3

5. AKo je c fomo KojH HerameaH peanaH 5poj, oH,n;a je L(x) $ D(x) e c · L(x) :?: c · D(x), L(x) :?: D(x) e c · L(x) $ c · D(x), L(x) < D(x) e c · L(x) > c · D(x), L(x) > D(x) e c · L(x) < c · D(x).

Ca.c ll1

c · L(x) < c · D(x),

. -ls

II CBaJa · 0 ::; x < 1. Ta.n;a je Ix - I I = -{x - I) H !xi =x, na .n;aTa je.n;Haq11Ha rrocTaje -x +I + x =I. IloIIITO ce y OBOM CJI}"lajy .n;aTa jeAJfaq11Ha CBO,ll;l1 Ha Ta'Itty je,a;HaKOCT, IbeHa perneiba cy CBH 8pojeBH x Koj11 3a,a;oBOJbaBajy ycnoB 0::; x < I; ,a;aKne cB11 8pojeB11 H3 [O, I ). 3 c~-qa· I ::; x. Ca.n;a je lx-1 I= x - 1 11 lxl =x, rra .n;aTa je,a;Ha'I11Ha rrocTaje x - I + x = 1. Pemeibe oBe jeAJfa'IlIBe je 8poj 1. IloIIITO 8poj I 3a,a;oBOJbaBa yCJioB x ~ I, OH je peIIIelbe H nona3He jep;Ha'!lme. •• Cil}"fdJ· x

)J;aKJie, CI 5.

1. cnyqaj: x < -1, Tj. x E (-oo, -1).

- :.·-, -l1·,

~-=-!_ - + ---:--- ----q.

LJ:aTa Hejep;Ha'IMHa ce CBO)J;Jil Ha HejeJJ.Ha'IHHY

-x + 1 -x - x - 1 > 5

' ..

5 . E( 5)

KOJa Je eKBJ!IBaJieHTHa ca x < --, TJ. x . LJ:aKJie, CBH 8pojeBJ!I J.13 3

(-00,-1)

n

::.

--

~--- -- +--:-.::=;~

-oo, - - . 3

cy peIIIelba p;aTe Hejep;Ha'IHHe.

1

~~-J.J++~++~

( 5)3 ( 5)3 -oo,-- = -oo,--

~

1

r~2

2. cnyqaj: -1::; x < 0, Tj. x E (-1,0). LJ:aTa Hejep;Ha'IHHa ce CBO)J;H Ha Hejep;Ha'IMHY

-x+l-x+x+l>5 Koja je eKBHBaJieHTHa ca x < -3, Tj. x E (-oo, -3 ). 0Baj cnyqaj HaM He p;aje peIIIelba IIOJia3He HejeJJ.Ha'IHHe jep je

[-1,0)

n

(-oo,-3) =0.

3. c:1yqaj: 0::; x < l, Tj. x E [0,1). LJ:arn Hejep;Ha'IHHa ce CBOp;H Ha HejeJJ.Ha'IllHY

-x + 1 + x + x + 1 > 5 Koja je eKBHBaJieHTHa ca x > 3, Tj. x E (3, +oo ). Kao 11 y IIpeTXop;HOM cnyqajy 11 oBora IJYTa He p;o811jaMo pemeH>a nona3He Hej eg:Haq}{He jep je

[O,l)

n

(3, +oo) =0.

4. cnyqaj: 1 ::; x, Tj. x E (1, +oo ). LJ:aTa Hejep;Ha'IHHa ce CBO)J;H Ha Hejep;Ha'IHHY a.

e3aTo

..

KOJa Je eKBHBaJieHTHa ca x CBM 8pojeBH 113

x-l+x+x+l>5

5 . (5 )

> -, TJ. x E -, +oo 3

.

3

(53 ) (53 )

CL +oo) n -, +oo cy peIIIelba p;aTe Hejep;Ha'IHHe.

= -, +oo

J+

LJ:aKne, CKYIJ peIIIelba p;aTe Hejep;Haq11He je (-oo, - : ) U (:, +oo

ep' He

;T,

>ojeBM

~ I.

~melbe

1

IIpHMep 12.

PeIIIHMO Hejep;Ha'IHHY ax+ 1 < x + a , rp;e je a IIapaMeTap. LJ:arn Hejep;Ha'IHHa je eKBHBaJieHTHa ca (a - l)x < (a - l)(a + 1). y 3aBHCHOCTJ1 op; Tora p;a JIH je a - 1 II03HTJ1BHO, HeraTMBHO J1JIJ1 j e j eJJ,HaKO irymt, pa3JIHKyjeMo cnep;ene cnyqajeBe. 1. cnyqaj: a < 1. Tap;a je a -1 < 0, na je CKYIJ pemelba p;aTe aej ep;Ha'!HHe (a + 1, +oo) . 2. cnyqaj: a > 1. Tap;a je CKYIJ perueH>a p;aTe HejeJJ,HaqHHe (-oo, a+ 1). 3. cnyqaj: a= 1. Y OBOM cnyqajy p;arn HejeJJ.Ha'IJ1Ha HeMa pemelha. 2

Hej e;{Ha1111He ca n apaMeTpJtXa

20· ../J

CHcTeMH JIHHeapHHX jep;HaqHHa CucTe

IlpDlq> l. O;:wewrno Macy CBaxor naxeTa axo cy y paBHOTe)KJ.f o8e Bare rrp11Ka3aHe Ha crr11u;11 neBo. AKo ca x 03Haqiwo Macy u;pBeHor rraxeTa, a cay Macy nrraBor rraKeTa, Ta,n;a rrp eMa npHIy rn:crer.rn B 0 aa rnaKo x 113 A, • HeraTHBHa Ha CKYIIJ A aKo je j(x) < 0 aa cBaKo x H3 A, • pacryha Ha CKYfiY A aKo je aa CBaKa p;Ba 8poja x 1 H x 2 H3 A TaqHa HMIIJIHKau,Hja x1 l, jep;ttaqHHa HMa p;Ba pernelba. +

303

3a;n;a~H

A

10. )J;oK . 1) (a.i

IlojaM 113pa3a. ,IJ;pBo H3pa3a - 3a CBaKO ;:qJBO o,i:pe;m: K3pa3 KOjH OHO o,qpel)yje.

ab

c

dab

c

dab

c

dab

c

d

11.

PacT~

1) x8 3) -acj 12. 1) .p,a

2) Pai

2. Haqpraj JlPBO H3pa3a; l) ab+c; 2) a+bc;

.3. ,lJ;OKa! :~ (a+b)c;

-t

a(b+c).

3. HaqpTaj JWBO K3pa3a (x + ay)(z + b), a 3aTHM H3paqy1rnj Bpe,qHOCT OBOr H3pa3a aKO je: l x=2, a=-3,y=O,z=-l, b=-3; 2 x=l,2, a=--0,l,y=l0,z=3,6, b=2,4.

QeJIH aJirebapcKH H3pa3H 8y;a;y 4. O;::q>ep;n cpel)eID1 obJIHK H cTeneH cne,qel'u1x MOHOMa: 1) 2x(-yx); 2) -3xz(-xy)yz; 3) --0,2.x2 y · 0,3xy; .,. 2,5ab · O,la 2 • 3b 3•

'-

- Mern nom

5. Hal)H cpel)em1 obmn< cne,qetrnx qem1X aJirebapcKMX H3pa3a: l) x"-x3_._1 +2.x2-x+3x3 +x; 2 x2y-xy+2-(2xy-x2y+xy2-2); 3} -(z2-z-(z- l +z2)); ,. -2a(ab-b + l)-b(a2 +a-1); 5) (x2+x+l) · (x2-x-l); 6 (2u+3v)(uv-u+v-l); I) (3ab + 2bc-ca)(a -b) -(a+ b)(ab-bc +ca).

1) ~ (!

2) AC 3) A

6 HeKajeA =2x-3y, B=2y-3x, C=3x+2y. O,qpe,qH cpel)eHH obJIHK H3pa3a: • A 2 -BC; : (A-B)(A-C); 3 AB-BC+CA.

";. Pacrasn aa qJifil{OQe cne,qehe H3pa3e: ax+ay+bx+by; 2 x5 +x4-x3-x2; : 36Y'-(Sv-1)2 ; 4 a2 - 7a +ab- 7b; 51 ac~bc+a1 +ab; 6 x3 +x2y-xy2-y3; 7) -18u4v+24u3v2-8u2v3; 8 3a7 b3 -27a 5 b5 ; 9} x2-2x-15; 10 y2+5y-6; 11) 4u 2 +4u-3; 13) .x4-8x2+16;

12 8z2 -6z+l; 14 (x3 +y2)(x2 + y3) -(x3-y2)(x2 -y3).

8. Koju ce oi:i; cnep;ehHx KBap;paTHKX TPmIOMa MO)Ke pacTaBHTH Ha 'fHHHoqe? 2 l) x2+4x+6; :, y2-7y+l2; 3) z2-z-2; "I u -2u+2.

E 304

S Pacrrum Ha 'fKHJ.lOQe cnep;ehe M3pa3e: • (a-b) 3 -8b3; 2 u3 +v6; 3 .x4+x2+1; 4: .x4-Sx2+4; 5) (y2 + y-t-1)(y2 +y+2) -12; 6) 2x2 +xy-y2-2x+y; /) 4a2 b2 -(a2 + b2-c2) 2 ; 8) x3 + x2 z + xyz +y2z -y3; 9) z+yz2 +xz2-x2z-x2y-xy2.

r

- O::;p x =

10. ,[{oK O;

2 x+2y-3 ~ O; 5 2x+y+3 ~ O;

3 -2x+y-3 < O; 6) x+2y+3 > 0.

YuyiUciUBo. Hajje,n;HOCTaBHHje je rrpoBepHTM ,n;a JIB Koop.n;HHaTe KoopwmaTHor rroqeTKa (O,O) 3a,n;oBOJbaBajy ,n;aTe Heje,n;HaKOCTM 11 Ha OCHOBY Tora o,n;peWfTJf Koj y rrorrypaBaH Tpe5a H3a5paTH.

307

= JianeapHe jep;Ha'IHHe ca jep;HoM Heno3HaToM

A

37. Penm: je,D;Haqime:

l) 3 +2x+5_x=x+2; 2 3

1) x-l+l-x=l+x; 2 3 3

l) x-l+2x+3=l+3x+5; 4 6

42. )];~ non c511 I c5a31

43.

4) O,S-0,4x 4; 5) l-2(1-2(1-x))=2-3(2-3x); 0,5 6) 0,35(x+0,34) -0,15x=0,2x-l,66; 7 l,73x+0,279(x-9) =2,09x.

3

lJ

JeAJIOM KIDIOrpa.M)' Bo.n;e .n;o.n;aTO je 2 KIDmrpaMa pacrnopa HeKe coJ1J1 op; 2% 11 y KIDiorpa.Ma pacrnopa coJrn o.n; 5%. Komi;eHTpa.i:.i;11ja con11 y p;oc511jeHoj MernaB11H11 je 4%. KoJrnXo Kl1Jlorpa.'da pacrnopa op; 5% je .n;o.n;aTO? 2) JeAJiO.M KJrnorpa.M)' Bop;e .n;o.n;aTO je x KIDiorpaMa pacrnopa HeKe con11 op; 2% 11 12 IGmorpa."1:a pacTBopa con11 o.n; 5%. Kom.i;eHTpau;11ja con11 y p;oc511jeHoj MernaB11H11 je 4%. KoJIID3x+4 5x·t-3~8x+21'

,., {7(x+l)-2x>9+4x 3(5-2x)-1~4-Sx'

3,

3x::::: 5-6x 4x - 1 ~ 1 - 3x . { 7 -2x > 2x+9 Ko~

52. Y 3a.BHCHOCTJ.f o;a; peamrnx rrapaMeTpa a l1 b perrrn Heje;a;HaqHHe l (a+ l)x+4 < (3-2a)x- l; ., (b-2)x- I~ 3-(b + l)x; 3) ax+ b1 ~ bx+ a 2 ; -± b - ax < a - bx. 5:?. Pe:arn: cM:cTe.M Hejewia'IKHa

CB a

59. Pen

{:~::~

y 3a.BHCHOCTH o,q peamior rrapaMeTpa

a.

C11cTeM11 1111HeapH11x jeµ;Haq11Ha 5-. Perrrn: cHcTeMe jewiaqffHa:

A

,- {x+y=2 ; 2x+3y=7

3) 3 {2x+5y=O;

2) {-x-2y=-l

2x-3y=4

3x+7y=l

4 {3x+y=O

6x+2y=O.

55 Perrrn CHCTeMe je;a;HaqffHa:

5 4 -;+y-=11. • 2 3 , { -+-=10 x y

2 2

1

x+2y + x-y =

1 1 { ----=l

x+2y

8

3 3

{

x-y

1

x+2y - 2x-y =O .

)

~+-2-=5 x+2y

2x-y

56 . .[LBa CTpenu;a cy Ha jeAHOM TaKMWielby Hcrramvrn no 30 MeTaKa. YcneIIIHOCT CBaKor crpenu;a H3pIDKeHa je pa3MepoM (KOJrnqJIHKOM) y 3a KpojeH>e, o.n;eJbeH>y 3a 1Jll1Belbe H o.n;eJbelhy 3a rraKOBalbe. Y rn8e1rn cy .n;arn BpeMeHa Koja cy rroTpe8tta 3a CBaKy ¢asy rrpoH3BO)l;Ibe Majn.n;e o.n;pe~eHor THna. 38or pa3JIH'HfTor 8poja 3anornem1X, CBaKO o.n;eJbelbe TOKOM He.n;eJbe MO)l(e MaKCHMaJIHO .n;a pa.n;H pa3JIJ1qJ1T 8poj cam:. YKynatt 8poj pap;IDIX caTM 3a cBaKo o.n;eJbeH>e HaBe.n;ett je y rrocne.n;H>oj KOJIOHH Ta8ene. J

MajHn;a THrra A

MajHn;a THrra E

MajHn;a THrra B

YKynatt8pojpa.n;ttmc caTH o.n;eJbeH>a

0,2h 0,3h O,lh

0,4h 0,5h 0,2h

0,3h 0,4h O,lh

l 160h l 560h 480h

O.n;eJbeH>e 3a KpojeH>e O.n;eJbeH>e 3a urnBeH>e O.n;eJbeH>e 3a rraKoBaH>e

I I I

KoJIHKO KOMa.n;a MajHn;a cBaKe BpCTe bH Tpe8ano tte.n;eJbHO rrpoH3BO.n;HTM .n;a bH CBaKo o.n;eJbeH>e Morno .n;a 3aBpurn nocao TOKOM pacrroJIO)l(JfBOr BpeMeHa?

59. Peurn cHcTeMe je.n;ttaqHHa: 1 2 3 --+--+--=l x+y y+z z+x

1)

1 4 6 --+--+--=5 x+y y+z z+x

<

2

6

x 1 +2x~ -3~3 +~4_=8 x1+x2 x3 x4- 1 x { 1 -x2-x3+x4 =7 x 1 +x2+x3+2x4 =4

2)

3

-----+--=0 x+y y+z z+x

'

11-aKOr

m I.Ka, ry/

2x+y=8 x-3 =11;

3)

{

4)

Y

{

x+y=3

B

_ x+ 2y-z-O . x+3y-5z= 1

~ 3aBHCHOCTH o.n; peanttor rrapaMeTpa a, peurn cne.n;ehe CMCTeMe j e.n;Ha'DIBa: -~\) x+y=O ;

G)J ca-2)x+ay=l; tvl_ (a 2 -4)x-y=O

c.Yl. x+ay=O

h\t _ __

r;_'

~

(a 1 )x + 2ay - 2 ; 2ax+(a-l)y=a-l

4)

{x+3y=l 2x - a = 10 .

Y

3x+ay= 5

61. Y 3aBHCHOCTH o.n; peanttor rrapaMeTpa a, peurn cne.n;ehe cHcTeMe je.n;ttaqHHa:

1)

ax+y+z=O x+ay+z=O; { x+y+az=a

2)

{x+y+2z=O x+ay+(3-a)z=O ; 2 x+y+(a +l)z=a-l

3)

{x+y+2z= 1 x+y+az=l . ax+2y+az=l

rror?

Jl:uHeapHe YH~:uje 62. J1crrHTaj ocobHHe H ttan;prnj rpa¢HK ¢yttKD;J1je f JR -+ JR, aKO j e: l) j(x)=-5x+l;

2) j(x)=3;

3 j(x)=2x-5.

E

63. 3a Koje Bpe.n;ttocrn k je ¢YHKD;Hja f JR -+ JR .n;aTa ca j(x) = (k- l )x + k 1) pacTylia; 2) orra.n;ajyha H tteraTMBHa Ha cryrry (0, + oo ) .

311

B

64 Mcmrraj ocoforne u Hai.q>Taj rpacpHK cpyma.vi:je f R -+ JR, a.Ko je: ~ j{x)=ll-x_+2; 2 j{x)=lx+ll-lx+2l-x; 3) j{x)=lx-ll-lx-2l+lx+ll; 'cl j{x)=ll2-xl-ll. 65. 1) H~Taj rpa¢HK cpy1n