#VLJUBCOIFSIBLLTBLMESWF\":%*/:\":*/-\"3*OBBJUUJSTBZMZBTBOOIÐLÐNMFSJOF HËSFLJUBCOEÐ[FOJ NFUOJ TPSVWFõFLJMMFSJLTNFOEFPMTBIJ¿CJSõFLJMEFBMOQZBZNMBOB- NB[ GPUPLPQJZBEBCBõLBCJSUFLOJLMF¿PóBMUMBNB[ :BZO4PSVNMVTV $BO5&,÷/&- :BZO&EJUÌSÑ %J[HJ–(SBGJL5BTBSN .FINFU÷MLFS¦0#\"/ *4#//P :BZOD4FSUJGJLB/P \"ZEO:BZOMBS%J[HJ#JSJNJ #BTN:FSJ ÷MFUJöJN &SUFN#BTN:BZO-UEõUJr \":%*/:\":*/-\"3* JOGP!BZEJOZBZJOMBSJDPNUS 5FMr 'BLT 0533 051 86 17 aydinyayinlari aydinyayinlari * www.aydinyayinlari.com.tr %¸O¾P.DSDáñ·/÷7&34÷5&:&)\";*3-*, ÜNİVERSİTEYE HAZIRLIK 4. MODÜL MATEMATİK - 1 Alt bölümlerin 1PMJOPNMBS¦BSQBOMBSB\"ZSNB KARMA TEST-3 Karma Testler EDĜOñNODUñQñL©HULU POLİNOMLAR 1. P( x ) bir polinom olmak üzere, E) 5 4. 1 Y 2 Y WF 3 Y CJSCJSJOF FõJU PMNBZBO QPMJ- Modülün sonunda ÇARPANLARA AYIRMA x .P( x ) + P( -x ) = 2x2 - x + 1 OPNMBSES PMEVôVOBHÌSF 1 LBÀUS EFS[1 Y ] =EFS[2 Y ] =EFS[3 Y ] =OFöJUMJ- ôJOEF A ) 15 B ) 10 C ) 8 D ) 6 I. der [ P ( x ) + Q ( x ) ] +EFS<3 Y >âO ³ Polinom Kavramı t 2 II. der [ P ( x ) ] . der [ R ( x ) ] = n2 III. der [ P ( x ) . Q ( x ) + R ( x ) ] = 3n JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS tüm alt bölümleri L©HUHQNDUPDWHVWOHU ³ Polinomlarda İşlemler t 8 2. P ( x - 1 ) polinomunun, x – 3 ile bölümünden kalan \" :BMO[* # :BMO[** $ :BMO[*** 4, Q ( x + 1 ) polinomunun, x - 3 ile bölümünden kalan -PMEVóVOBHËSF 2Q [1 Y+ ] +Y2 + 1 D) I ve II E) I ve III ³ Polinomlarda Bölme İşlemi t 16 6ñQñIð©LðĜOH\\LĜ QPMJOPNVOVOY-JMFCÌMÑNÑOEFOLBMBOOF- ³ Polinomların Çarpanlara Ayrılması - I t 26 www.aydinyayinlari.com.tr EJS ·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" A ) -2 B ) -1 C ) 0 D ) 1 E ) 2 10-÷/0.,\"73\".* 5. 1 + x1 + x2 + x3 + ... + x7 : x6 + x4 + x2 + 1 \\HUDOñU x2 - 2x - 3 x2 - 9 ³ Polinomların Çarpanlara Ayrılması - II t 30 ÖRNEK 2 TANIM JGBEFTJOJOFOTBEFöFLMJBöBôEBLJMFSEFOIBOHJ- 15 TJEJS ³ Polinomların Çarpaan0,laa1r, aa2,Aay3,r.ı..lm, ana-s1,ıa-n IHIFIS¿FtLTBZ4MB1S O 1 Y = x 2n–1 +Y2mYn +12 %XE¸O¾PGHNL¸UQHN EPóBMTBZWFYEFóJõLFOPMNBLLPõVMVJMF polinomunun derecesi en çok kaç olabilir? A) x + 3 B) x + 1 C) 1 – x 1 Y = anYn + an-1Yn-1 + ... + a1 Y+ a0 3. D) 1 E) 1 ³ Karma Testler t 49 CJ¿JNJOEFLJGPOLTJZPOMBSBHFS¿FMLBUTBZMWFCJS x +1 x +3 EFóJõLFOMJ polinom ¿PLUFSJNMJ EFOJS VRUXODUñQ©¸]¾POHULQH <(1m1(6m/6258/$5 1PMJOPNMBS¦BSQBOMBSB\"ZSNB <HQL1HVLO6RUXODU DNñOOñWDKWDX\\JXODPDVñQGDQ ³ Yeni Nesil Sorular t 55#ÐUÐO QPMJOPNMBS GPOLTJZPOMBSO Ë[FM CJS UÐSÐ 1. 6. x4 + x3 + 5x2 + 3 = ( x +31.) (x3 + ax2 + bx + c) + d 0RG¾O¾QJHQHOLQGH\\RUXP PMBSBLFMFBMOBCJMJS ¥FWSFTJYCSPMBO*õFLJMEFWFSJMFOFõLFOBSпHFO- :BS¿BQCSPMBOCJSLÐSFNFSLF[JOEFOYCSV[BL- \\DSPDDQDOL]HWPHYE 1 Y = anYn + an-1Yn-1 + ... + a1Y+ a0QPMJ- EHFHULOHUL¸O©HQNXUJXOX OPNVOEB (x – 1) MFSõFLJMEFLJHJCJCPZBONõUS olduôVOBHÌSF B C D MELUBJGBCEJSFETÐJO[MJFONEMFFôLFFSTJJMJZPS VRUXODUD\\HUYHULOPLĜWLU a0, a1Y B2Y2, ... , anYnJGBEFMFSJOJOIFSCJSJOF (x + 2) QPMJOPNVOterimleri, LBÀUS a0, a1, a2, ... , anHFS¿FMTBZMBSOB JMHJMJUFSJN- A) 6 B) 13 C) 7 D) 15 E) 8 MFSJOLBUTBZMBS XODĜDELOLUVLQL] 22 ar `3WFS`/PMNBL1Ð[FSF ÖRNEK 3 &WJOCJSLFOBSV[VOMVóV O1 arYrUFSJNJOEFLJYJOÐTTÐPMBOSZF CVUFSJNJO P ( x ) = x3 + ax2 + ( b + 1 ) x + 3 x derecesiEFOJSWFder [ P ( x ) ] WFZBd [ P( x )] CJ- Aşağıdaki polinomların başkatsayısını, derecesini ve sabit terimini bulunuz. CJSJNPMBOLBSFõFLMJOEFLJËOEVWBS FIOJ Y+ 1) bi- O2 ¿JNJOEFHËTUFSJMJS a ) 1 Y =Y4 -Y2 +Y+ 2 b ) 2 Y =Y5 -Y2 +Y3 -Y7 + 1 - 2 rim, boyu (x + CJSJNPMBOEJLEËSUHFOõFLMJOEFLJ II &OCÐZÐLEFSFDFMJUFSJNJOLBUTBZTPMBOBnLBU- c ) 1 Y Z =Y3 y5 +Y2 y -Y4 y7 TBZTOB CBöLBUTBZ EFóJõLFO J¿FSNFZFO B0 NBOUPMBNBNBM[FNFTJJMFUBN\"BZNOFLOVLSBBQMBMBHOËBSDFB*L*UõSFLJMEF¿7FW. SFTYJ ZYWFCS[PQMBPO[JFUJõG-UBNTBZMBSPMNBLÑ[FSF UFSJNJOFEFsabit terimEFOJS LFOBSпHFOMFSCPZBONõUS Y` N+)-x2 + y2 - z2 = 13 +2xz #VOBHÌSF PMVöBOBSBLFTJUJOBMBOOOYJOCJS #VOBHÌSF B+CLBÀPMNBMES QPMJOPNVPMBSBLJGBEFTJ \" Y JÀJO Buna göre, P ( x ) = ¥FWSFTJYPCMESVPôMBVOOFBõHLÌFSOFB SZLBÀUS I. \" Y -YJMFLBMBOT[CËMÐOÐS $OW%¸O¾P7HVWOHUL A) 0 B) 3 C) 8 пHDF)O1E0FLJ CEP)ZB1O2NBNõ пHFO TAB)Z5T öFLBMJO) E6F C) 7 D) 8 E) 9 II. \" Y QPMJOPNVOVOLBUTBZMBSUPQMBNÕEJS ÖRNEK 1 UBONMBOBO QPMJOPNVO LBUTBZMBS UPQMBN LBÀ- TEST - 17- 11 , x7 , 1 , x , 2 3, 29 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT US III. \" Y QPMJOPNVOVOCBõLBUTBZTÕEJS x3 x 5. x2 + 4x - 3 = 0 ise ifadelerinden hangileri bi1r .polinoam4u+nbt2er-imai2dbir2? - a2 1. E 2. A 3. D 51 4. D 5. A 6. E 7. CJGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS $ % & \" # \" :BMO[* # *WF** $ *WF*** JGBEFTJOJO ÀBSQBOMBSOEBO CJSÖJ BRöNBEôKEB4LJMFSEFO x2 + 9 % **WF*** & * **WF*** $\\UñFDPRG¾OVRQXQGD x2 IBOHJTJEFôJMEJS? \"öBôEBLJJGBEFMFSEFOIBOHJMFSJQPMJOPNEVS Her alt bölümün A) a2 + b2 B) a + b a ) 1 YC )=a-Y2b- 2 + 3 JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS x D) a + 1 b )E)1a Y- =1Y3 -Y2 + 2 A) 22 B) 21 C) 20 D) 19 E) 18 2. ôFLJMEF Fõ EJLEËSUHFOEFO PMVõBO PZVO QBSL 4. ,VõCBLõ HËSÐOÐNÐ õFLJMEFLJ HJCJ PMBO CJS SFTJN x +3 NPEFMJWFSJMNJõUJS BUËMZFTJOJO LFOBS V[VOMVLMBSOO CB[MBS õFLJMEF HËTUFSJMNJõUJS WDPDPñ\\HQLQHVLOVRUXODUGDQ c ) 1 Y = x3 - x2 + x + 2 VRQXQGDRE¸O¾POHLOJLOL b ROXĜDQWHVWOHUEXOXQXU 32 xx ba d ) 1 Y = 3 x + 2 x xx x WHVWOHU\\HUDOñU ba a x y 2. 214 - 28 + 1 6. a3 + b3 = 40 ve ab (a + b) = 8 210 - 8 y JöMFNJOJOTPOVDVLBÀUS PMEVôVOBHÌSF B+CBöBôEBL JMFSEFOIBOHJTJ 1. x7, 2 3 , 29A) 27 B) 127 C) 92 D2). 64 3. aE) )3 8 / 2 b) –8 / 7 / 1– OFFöJUUJS 4. c, d #VQBSLOIFSEJLEËSUHFOJOLFOBSMBSOBCJSTSBPMB- x 4 88 20 /4 2 c) –1 / 11 / 0 DBLõFLJMEFUPQMBNNFUSF¿JUMFSMF¿FWSJMFDFLUJS x 77 \"UÌMZFOJO BMBO CJSJNLBSF ÀFWSFTJ CJSJN PMEVôVOBHÌSF YLBÀCJSJNEJS A) 6 B) 4 C) 3 D) 2 E) 1 #VOBHÌSF QBSLOUPQMBNBMBOOOBUÑSÑOEFO FöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & A) 520a - 13a2 # B-B2 3 $ B-B % 310a - 12a2 3 3. 1 a- a-6 & 530a - 12a2 + 3 a-4 a-2 7. 1969.1973 + 4 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS JöMFNJOJOTPOVDVLBÀUS 56 3. B 4. C 1. E 2. A A) a + 2 B) a - 3 C) 1 A) 1970 B) 1971 C) 1972 E) 1975 D) a - 4 E) 3 a D) 1974 4. x – 5 = 4 8. x + 1 =-1PMEVôVOBHÌSF x x PMEVôVOBHÌSF x + 25 JGBEFTJOJOEFôFSJBöBô x2007 + 1 x x2007 EBLJMFSEFOIBOHJTJEJS JGBEFTJOJOEFôFSJLBÀUS A) 12 B) 15 C) 18 D) 24 E) 26 A) 2 B) 1 C) 0 D) -1 E) -2 \" B C E 40 \" B B \"
ÜNwİwVwE.ayRdinSyaİyTinlaEri.YcoEm.trHAZIRLIK ·/÷7&34÷5&:&)\";*3-*, MATEMATİK - 1 4. MODÜL POLİNOMLAR ÇARPANLARA AYIRMA ³ Polinom Kavramı t 2 ³ Polinomlarla İşlemler t 8 ³ Polinomlarda Bölme İşlemi t 16 ³ Polinomların Çarpanlara Ayrılması - I t 26 ³ Polinomların Çarpanlara Ayrılması - II t 30 ³ Polinomların Çarpanlara Ayrılması - III t 41 ³ Karma Testler t 49 ³ Yeni Nesil Sorular t 55 1
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr 10-÷/0.,\"73\".* Polinom ÖRNEK 2 TANIM 15 a0, a1, a2, a3, ... , an-1, an HFS¿FL TBZMBS O 1 Y = x 2n–1 +Y2mYn +12 EPóBMTBZWFYEFóJõLFOPMNBLLPõVMVJMF 1 Y = anYn + an-1Yn-1 + ... + a1 Y+ a0 polinomunun derecesi en çok kaç olabilir? CJ¿JNJOEFLJGPOLTJZPOMBSBHFS¿FMLBUTBZMWFCJS 1 Y CJSQPMJOPNJTFIFSCJSUFSJNJOÑTTÑEPôBMTBZPM EFóJõLFOMJ polinom ¿PLUFSJNMJ EFOJS NBMES#VOBHÌSF 15 ! N ve n + 12 `/PMNBM #ÐUÐO QPMJOPNMBS GPOLTJZPOMBSO Ë[FM CJS UÐSÐ 2n - 1 PMBSBLFMFBMOBCJMJS 1 Y = anYn + an-1Yn-1 + ... + a1Y+ a0QPMJ- 15 ES ! N ise (2n - 1) ifadesi 15 in pozitif tam sa OPNVOEB a0, a1Y B2Y2, ... , anYnJGBEFMFSJOJOIFSCJSJOF 2n - 1 QPMJOPNVOterimleri, ZCÌMFOMFSJPMNBMES a0, a1, a2, ... , anHFS¿FMTBZMBSOB JMHJMJUFSJN- MFSJOLBUTBZMBSEFOJS O halde, 2n - 1, {1, 3, 5, 15}EFôFSMFSJOJBMBCJMJS 2n - 1 = 1 j n = 1 , 2n - 1 = 3 j n = 2 2n - 1 = 5 j n = 3 , 2n - 1 = 15 j n = 8 olur. n = 8 için 15 = 1 olurken n + 12 = 20 olur. 2n - 1 O halde polinomun derecesi en çok 20 olur. ar `3WFS`/PMNBLÐ[FSF ÖRNEK 3 arYrUFSJNJOEFLJYJOÐTTÐPMBOSZF CVUFSJNJO Aşağıdaki polinomların başkatsayısını, derecesini derecesiEFOJSWFder [ P ( x ) ] WFZBd [ P( x )] CJ- ve sabit terimini bulunuz. a ) 1 Y =Y4 -Y2 +Y+ 2 ¿JNJOEFHËTUFSJMJS b ) 2 Y =Y5 -Y2 +Y3 -Y7 + 1 - 2 c ) 1 Y Z =Y3 y5 +Y2 y -Y4 y7 &OCÐZÐLEFSFDFMJUFSJNJOLBUTBZTPMBOBnLBU- TBZTOB CBöLBUTBZ EFóJõLFO J¿FSNFZFO B0 UFSJNJOFEFsabit terimEFOJS ÖRNEK 1 B #BöLBUTBZT EFSFDFTJ TBCJUUFSJNJEJS C #BöLBUTBZT-8, derecesi 7, sabit terimi 1 - 2 dir. - 11 , x7 , 1 , x , 2 3, 29 D #BöLBUTBZT- EFSFDFTJ TBCJUUFSJNJES x3 x ÖRNEK 4 ifadelerinden hangileri bir polinomun terimidir? \"öBôEBLJJGBEFMFSEFOIBOHJMFSJQPMJOPNEVS r - 11 = - 11x-3 teriminde -3 n/PMEVôVOEBOCJS a ) 1 Y =Y2 - 2 + 3 3 x b )1 Y =Y3 -Y2 + 2 x + 3 x c ) 1 Y = x3 - x2 + x + 2 polinomun terimi olamaz. 32 r Y7 teriminde 7 `/ES1PMJOPNVOUFSJNJPMBCJMJS d ) 1 Y = 3 x + 2 r 1 = –1 teriminde -1 n/PMEVôVOEBOCJSQPMJ B 1PMJOPNEFôJMEJS x C 1PMJOPNEFôJMEJS x c) Polinomdur. d) Polinomdur. nomun terimi olamaz. 1 r 2 x=x 1 teriminde b NPMEVôVOEBOCJSQPMJOPNVOVO 2 terimi olamaz. r 2 3 ve 29 teriminde x li ifadenin derecesi 0 ol EVôVOEBO `/ QPMJOPNVOUFSJNJPMBCJMJS 1. x7, 2 3 , 29 2 2. 20 3. a) 3 / 4 / 2 b) –8 / 7 / 1– 2 c) –1 / 11 / 0 4. c, d
www.aydinyayinlari.com.tr 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 5 ÖRNEK 7 1 Y =Yn–2 +Y2-n +Y2 + 1 1 Y = N- Y2 + ( n - Y+NO JGBEFTJ CJS QPMJOPN PMEVôVOB HÌSF 1 O - O EFôF JGBEFTJTBCJUQPMJOPNPMEVóVOBHËSF SJLBÀUS m + n +1 UPQMBNOOEFôFSJLBÀUS P(x) bir polinom ise 5xn-2 ve 4x2-n terimlerinin üssü 1 Y QPMJOPNVTBCJUQPMJOPNPMEVôVOBHÌSF Y2 ve x in n -äWF-OäPMNBMES0IBMEFO=PMNBMES P ( x ) = 5.x0 + 4.x0 + x2 + 1 = x2 + 10 bulunur. LBUTBZTPMNBMES N- 2 = 0 ve n - 3 = 0 P ( 2 ) = 14 tür. P ( 2 ) - 2 = 14 - 2 = 12 bulunur. olur. 4BCJU1PMJOPN4GS1PMJOPNV TANIM #VSBEBON= 2, n = 3 bulunur ve P ( x ) = m.n den P( x ) = 6 olur. 1 Y = an Yn + an-1 Yn-1 + ... + a2 Y2 + a1 Y1 + a0 QPMJOPNVOEB O halde, m + n + P ( 2018 ) = 2 + 3 + 6 = 11 bulunur. an = an -1 = ... = a2 = a1 = 0 ÷LJ1PMJOPNVO&öJUMJôJ PMEVóVOEBFMEFFEJMFO1 Y = a0QPMJOPNVsa TANIM bit polinomdur. Sabit polinomun derecesi 0 ES 1 Y = anYn + an-1Yn-1 + ... + a2 Y2 + a1 Y1 + a0 1 Y QPMJOPNVOEB 2 Y = bNYN + bN-1YN-1 + ... + b2 Y2 + b1 Y1 + b0 an = an-1 = ... = a2 = a1 = a0 = 0 QPMJOPNMBSJ¿JO PMEVóVOEBFMEFFEJMFO1 Y =YQPMJOPNVTGS polinomdur. 1 Y =2 Y l n =NWF 4GSQPMJOPNVOVOEFSFDFTJCJMJOFNF[ an = bN , an-1 = bN-1 , ... , a1 = b1 , a0 = b0 PMNBMES ÖRNEK 6 ÖRNEK 8 1 Y =Y2 -Y2 – a + ab - 3 1 Y =Y4 -Y3 -BY2 + ( b - Y+ 3 2 Y =DY4 + ( d - Y3 +Y2 -Y+ e ifadesi 4. dereceden ve sabit terimi 75 olan bir po MJOPNPMEVôVOBHÌSF B+CLBÀUS QPMJOPNMBSCJSCJSJOFFöJUPMEVôVOBHÌSF B C D EWF FTBZMBS OCVMVOV[ P( x ) polinomu 4. dereceden bir polinom olabilmesi için -3x2-a teriminin üssü 2 - a =PMNBMES0IBMEF 1 Y WF 2 Y QPMJOPNVOVO BZO EFSFDFMJ UFSJNMFSJO a = -2 olur. P( x ) polinomunun sabit terimi 75 olma LBUTBZMBSFöJUPMBDBôOEBO TJÀJO ab - 3 ifadesi 75 = 5 3 PMNBMES - 2b - 3 = 5 3 ise b = - 3 3 olur. c = 5 , a = -2 , e = 3 O halde, a + b = - 2 - 3 3 bulunur. -4 = d-1 b-1 =-1 fp fp d =-3 b=0 bulunur. 5. 12 6. - 2 - 3 3 3 7. 11 8. a) a = –2, b = 0, c = 5, d = –3, e = 3
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 9 ÖRNEK 12 1 Y =Y3 - ( 2a - Y2 +Y- c + 3 1 Y =Y3 -Y2 +Y- 4 2 Y = ( b - Y4 + ( d - Y3 +Y2 +Y- 1 PMEVôVOB HÌSF P (1 - 3 x) JGBEFTJOJO FöJUJOJ CVMV nuz. QPMJOPNMBSCJSCJSJOFFöJUPMEVôVOBHÌSF B C DWFE TBZMBSOCVMVOV[ P ( x ) = (x3 - 3x2 + 3x - 1) - 3 = (x - 1)3 - 3 P^ 1 - 3 x h = ^ 1 - 3 x - 1 h3 - 3 1 Y WF2 Y QPMJOPNMBSOOBZOEFSFDFMJUFSJNMFSJOJO LBUTBZMBSFöJUPMBDBôOEBO = -x - 3 bulunur. b - 1 = 0 , 3 = d - 1 , -2a + 1 = 5 , -c + 3 = -1 b=1 d=4 a = -2 c=4 ( P ( x ) polinomunda 4. dereceden terim yoktur.) %m/*m ÖRNEK 13 1 Y WFSJMEJóJOEF 1 2 Y JCVMNBLJ¿JO 1 Y 1 Y Z =Y2 y + y2 -Y+ 1 QPMJOPNVOEBYZFSJOF2 Y ZB[MS PMEVôVOBHÌSF P( 1 - x, y + JGBEFTJOJOFöJUJOJCV lunuz. ÖRNEK 10 P ( x, y ) = x2y + y2- 2x + 1 1 Y =Y2 -Y+ 3 P ( 1 - x, y + 1 ) = (1 - x)2 (y + 1) + (y + 1)2 - 2(1 - x) + 1 PMEVôVOBHÌSF P ( 2)LBÀUS = x2y + x2 + y2 - 2xy + 3y + 1 bulunur. P^ 2 h = 5^ 2 h2 - 4 2 + 3 %m/*m = 13 - 4 2 bulunur. 2 Y JO UFSTJOJ 1 2 Y QPMJOPNVOEB ZFSJOF ZB[BSTBL 1 Y JCVMVOV[ [ Q o Q-1 ] Y =YPMEVóVOVIBUSMBZO[ ÖRNEK 11 ÖRNEK 14 1 Y =Y2 -Y+ 2 1 Y- =Y2 -Y+ 2 PMEVôVOBHÌSF 1 Y+ JGBEFTJOJOFöJUJOJCVMVOV[ PMEVôVOBHÌSF 1 Y JOFöJUJOJCVMVOV[ P ( x + 1 ) = ( x + 1 )2- 2 ( x + 1 ) + 2 3x - 1 in tersi olan x+1 ü polinomda yerine yazar = x2 + 1 bulunur. 3 sak; P(x) = 9d x+1 2 x+1 n+ 2 n - 6d 33 = x2 + 1 bulunur. 9. a = –2, b = 1, c = 4, d = 4 10. 13 - 4 2 11. x2 + 1 4 12. –x–3 13. x2y + x2 + y2 – 2xy + 3y + 1 14. x2 + 1
www.aydinyayinlari.com.tr 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 15 ÖRNEK 18 P( 1 -Y =Y2 -Y+ 4 1 Y = Y- Y- Y- Y- PMEVôVOBHÌSF 1 LBÀUS QPMJOPNVWFSJMJZPS #VOBHÌSF 1 -YJGBEFTJOJZBQBOYEFôFSJOJZFSJOFZB[BSTBL 1 - 2x = 3 1 +1 +1 + ... +1 +1 x = -1 UPQMBNLBÀUS P(3) = (-1)2 - (-1) + 4 = 6 bulunur. P(1) = (<1 - 1) (1 - 2) . . . (1 - 150) = 0 0 P(2) = (2 - 1) (<2 - 2) . . . (2 - 150) = 0 0 h P(150) = (150 - 1) . (150 - 2) ... (1145402- 145403) = 0 0 P(151) = (151 - 1) . (151 - 2) ... (151 - 150) = 150.149...1 = 150! bulunur. ÖRNEK 16 ÖRNEK 19 1 Y+ =Y2 -Y+ 1 1 Y QPMJOPNVJMFJMHJMJPMBSBLBõBóEBLJCJMHJMFSWFSJMJZPS PMEVôVOBHÌSF 1 Y+ JGBEFTJOJOFöJUJOJCVMVOV[ r der [1 Y ] =UÐS Polinomda x yerine x - 1 yazarsak; P ( x + 1) = (x - 1)2 - 3 ( x - 1 ) + 1 r ,BUTBZMBSCJSCJSJOEFOGBSLMES = x2 - 5x + 5 bulunur. r ,BUTBZMBS{0, 1, 2, 3, 4, 5}LÐNFTJOJOCJSFMFNBO- ES ÖRNEK 17 #VOB HÌSF ZVLBSEBLJ LPöVMMBS TBôMBZBO LBÀ GBSLM P ( 3 x ) =Y2 -Y+ 5 1 Y QPMJOPNVWBSES PMEVôVOBHÌSF 1 Y2 OJOFöJUJOJCVMVOV[ P^ x h = 4 + 3 + 2 + dx + e Polinomda x yerine x6 yazarsak, P(x2) = (x6)2 - 3(x6) + 5 ax bx cx = x12 - 3x6 + 5 bulunur. . . . .. 5 . 5 . 4 . 3 . 2 = 600 (0 hariç ) GBSLMQPMJOPNWBSES 15. 6 16. x2 – 5x + 5 17. x12 – 3x6+ 5 5 18. 150! 19. 600
TEST - 1 1PMJOPN,BWSBN 1. \"öBôEBLJJGBEFMFSEFOIBOHJTJQPMJOPNEVS 5. * 1 Y =Y4 -Y3 + 3 x \" P (x) = 4x3 - 5x2 + 1 - 3 II. P Y =Y6 -Y2 - 4 5x x x # P (x) = 2x - x + 3 III. P Y = 4 2 x3 + 3 3 x2 + $ P (x) = 2 IV. 1 Y = 2 + 1 x + x2 V. P Y = 9 - x2 % P (x) = 2 x - 3 2 x2 + 1 x +1 23 & P (x) = 3x2 - 1 :VLBSEBLJJGBEFMFSden kaç tanesi polinomdur? x \" # $ % & 12 6. 1 Y = ( 2a - Yb - 3 + ( a - Y6 + 2b - 1 2. P^ x h = x 3n–2 n–2 JGBEFTJ JLJODJ EFSFDFEFO CJS QPMJOPN PMEVôVOB HÌSF 1 LBÀUS +x +2 \" - # - $ % & JGBEFTJ CJS QPMJOPN CFMJSUUJôJOF HÌSF EFSFDFTJ 7. #JMHJ öLJ QPMJOPNVO FõJU PMNBT J¿JO QPMJOPNMBSO LBÀUS BZO EFSFDFMJ UFSJNMFSJOJO LBUTBZMBS CJSCJSJOF FõJU \" # $ % & PMNBMES 1 Y = N- Y4 + ( n - Y3 +Y+ k - 5 3. 1 Y Z =Y9 +Y4 y2 +Y8 y3 -Y6 y5 2 Y =Y3 + U- Y+ 7 QPMJOPNMBSWFSJMJZPS QPMJOPNVOVOEFSFDFTJLBÀUS P ( x ) =2 Y PMEVôVOBHÌSF N+ n + k + t top MBNLBÀUS \" # $ % & \" # $ % & 4. 1 Y =Y4 - ( a -C Y3 +Y2 +Y- c + 3 24 2 Y =CY4 +Y2 + ( d -D Y+ 5 8. P(x) = ^ a - 2 h x a + ^ a - 24 h a – 2 PMNBLÐ[FSF QPMJOPNMBSCJSCJSJOFFõJUPMEVóVOBHËSF x a + b + c +ELBÀUS 1 Y CJSQPMJOPNEVS #VOBHÌSF 1 Y QPMJOPNVOVOEFSFDFTJen çok kaç olabilir? \" # $ % & \" # $ % & 1. D 2. C 3. & 4. & 6 5. C 6. \" 7. & 8. #
1PMJOPN,BWSBN TEST - 2 1. \"öBôEBLJMFSEFOLBÀUBOFTJQPMJOPNEVS 5. #BõLBUTBZT TBCJUUFSJNJPMBOEFSFDFEFO I. P (x) = 4x2 - 3 x + 2 1 Y =NY5 + ( n - Y4 + ( k - Y+U- 2 II. P (x) = 5x3 - 2x + 3 QPMJOPNVWFSJMJZPS x 1 Y QPMJOPNVOVOLBUTBZMBSUPQMBNPMEVôV III. P (x) = 3 7 + 4x OBHÌSF N+ n + k +UUPQMBNLBÀUS IV. 1 Y =Y-4 + Y-2 +Y+ 4 V. P (x) = 17 \" # $ % & VI. P (x) = x3 - 27 x-3 \" # $ % & 12 6. BWFCCJSFSUBNTBZPMNBLÑ[FSF p–4 1 Y = (a +C Y3 + ( a - 2 x2 + ( b + 3 Y 2. P (x) = 3x p –1 + 2x QPMJOPNVUBNTBZLBUTBZMCJSQPMJOPNPMEVôV polinomunun derecesi en çokLBÀUS OBHÌSF B+CUPQMBNLBÀUS \" - # - $ - % & \" # $ % & 3. 1 Y =Ya+5 +Ya -Y24-3a +Y+ 1 7. 1 Y = N-O Y3 + N- Y2+ n + k -1 JGBEFTJ CJS QPMJOPN PMEVôVOB HÌSF B OO LBÀ QPMJOPNVTGSQPMJOPNPMEVôVOBHÌSF GBSLMEPôBMTBZEFôFSJWBSES m + n +LUPQMBNLBÀUS \" # $ % & \" # $ % & 4. 1 Y = ( n - Y2 + N+ Y+ k - 2 8. \"öBôEBLJMFSEFOIBOHJTJQPMJOPNEVS TGSQPMJOPNVWFSJMEJôJOFHÌSF n + m + k topla \" P (x) = 2x7 - 3x + 4x + 3 NLBÀUS # P (x) = 5 + 5x + 1 \" # $ % & x $ P (x) = 3x3 - 3 x2 + x4/3 + 4 % P (x) = 3 x7 + 8x2 - 4x - 13 & P (x) = 15x4 - x + 7 + 3 x 1. # 2. # 3. C 4. \" 7 5. & 6. C 7. D 8. D
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr 10-÷/0.-\"3-\"÷õ-&.-&3 TANIM TANIM 1PMJOPNMBSEB5PQMBNBWF¦LBSNB÷öMFNJ 1PMJOPNMBSEB#ÌMNF÷öMFNJ 1PMJOPNMBSEB UPQMBNB WFZB ¿LBSNB JõMFNMF- 1 Y CËMÐOFO 2 Y CËMFO 3 Y CËMÐN WF SJZBQMSLFOBZOEFSFDFMJUFSJNMFSJOLBUTBZMBS , Y LBMBOES UPQMBOSWFZB¿LBSMS BZOEFSFDFMJUFSJNFLBU- TBZPMBSBLZB[MS1PMJOPNMBSEBCVMVOBOGBSLM 1 Y 2 Y EFSFDFMJUFSJNMFSUPQMBNBWFZBGBSLBBZOFOZB- 3 Y [MS , Y 1PMJOPNMBSEB¦BSQNB÷öMFNJ der [1 Y ]ãEFS[2 Y ]UJS 1 Y WF2 Y QPMJOPNMBSOO¿BSQNCVMVOVS- LFO 1 Y JO CÐUÐO UFSJNMFSJ 2 Y JO CÐUÐO UF- 2 Y áPMNBLÐ[FSF SJNMFSJJMF¿BSQMS&MEFFEJMFOUFSJNMFSJOUPQMBN 1 Y 2 Y QPMJOPNVOVWFSJS der [, Y ] < der [2 Y ]UJS 1 Y =2 Y 3 Y +, Y FõJUMJóJOEFLJ , Y =JTF1 Y 2 Y FUBNCËMÐOÐS ÖRNEK 3 ÖRNEK 1 \"öBôEBWFSJMFO1 Y QPMJOPNMBSO2 Y QPMJOPNMB SOB CÌMÑQ CÌMÑNÑO EFSFDFTJOJ CÌMÑNÑ WF LBMBO 1 Y =Y2 -Y 2 Y =Y3 +Y- 4 PMNBLÑ[FSF BöBôEBLJJöMFNMFSJOFöJUJOJCVMVOV[ bulunuz. a)1 Y -2 Y a) 1 Y =Y5 -Y4 +Y3 - 2 Y =Y- 2 b) 1 Y =Y4 -Y3 +Y2 + 2 Y =Y2 +Y- 2 c) 1 Y =Y3 -Y2 +Ym 2 Y =Y- 1 b)1 Y 2 Y ¦Ì[ÑN x–2 c) 1 Y+ -Y2 Y 3x4 + 2x3 + 5x2 + 10x + 20 d)1 Y2 +Y2 Y 3x5 – 4x4 + x3 – 2 a) -3x3 + 2x2 - 8x + 12 a) b) x5 - x4+ 2x3 - 6x2 + 4x c) -x4 - x2 + 5x 3x5 – 6x4 d) 3x4 + 3x2 - 8x 2x4 + x3 – 2 2x4 – 4x3 5x3 – 2 5x3 – 10x2 10x2 – 2 10x2 – 20x 20x – 2 20x – 40 38 :VLBSEBLJCËMNFJõMFNJOJ Y5-Y4 Y3 - 2 ÖRNEK 2 = Y- (134x44+442x434+452x24+44104x4+42403) + 638 bölüm kalan Y- Y2 - +1 Y -Y2 + 3 =Y3 -Y 2 +Y PMEVôVOBHÌSF 1 Y LBÀUS CJ¿JNJOEFJGBEFFEFCJMJSJ[#ËMÐOFOJO. derece- EFOWFCËMFOJOEFSFDFEFOQPMJOPNPMEVóVCJS CËMNFJõMFNJOEFCËMÐNQPMJOPNVOVOEFSFDFTJ ( 2x - 1 ) ( x2 - 3 ) + 2P ( x ) - x2 + 3 = x3 - x 2 + x 5 - 1 =CVMVOVS b) #ËMÐNY2-Y+ LBMBO-Y+ 16, 2P(x) = -x3 + x2 + 7x - 6 CËMÐNÐOEFSFDFTJ - x 3 + 2 + 7x - 6 c) #ËMÐN 2 5 3 CËMÐNÐOEFSFDFTJ2, x x - x+ P(x) = 2 bulunur. 24 8 kaMBO- 5 8 1. a) –3x3 + 2x2 – 8x + 12 b) x5 – x4+ 2x3 – 6x2 + 4x c) –x4 – x2 + 5x 8 d) 3x4 + 3x2 – 8x 2. a - x3 + x2 + 7x - 6 k / 2
www.aydinyayinlari.com.tr 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, TANIM ÖRNEK 6 der [1 Y 2 Y ] = der [1 Y ] + der [2 Y ] 1 Y WF2 Y QPMJOPNMBSJ¿JO der> P^ x h der [ P Y ] - der [2 Y ] der6^ P (x) h@ = 3WFder9P (x) . Q2 (x)C = 10 der6^ Q (x) h@ H= Q^ x h PMEVôVOBHÌSF der6P (x)@ LBÀUS der [1 Y + 2 Y ]âNBY[EFS1 Y EFS2 Y ] der [1 Y ] n = n. der [1 Y ] der [P( x )] = m, der [Q ( x )] = n olsun. der6P^ xn h@ = n.der6P^ x h@ der7 P^ x hA m = 3 & = 3 olur. der7 Q^ x hA n der [B1 CY+D ] = der [1 Y ] der[P( x ).Q2( x )] = 10 j m + 2n = 10 olur. der [B1 CY+D n] = n.der [1 Y ] ÷LJJGBEFZJPSUBLÀÌ[FSTFL O= 2, m = 6 bulunur. der[P(x)] = 6 ÖRNEK 4 der [ P ( x ) ] = 3, der [ Q ( x ) ] =PMNBLÑ[FSFBöBôEB ÖRNEK 7 LJQPMJOPNMBSOEFSFDFTJOJCVMVO V[ a)1 Y 2 Y b)1 Y +2 Y öOUFSOFUÐ[FSJOEFOFmUJDBSFUZBQBOCJSJõMFUNF JISB¿FUUJ- P(x) d) P3 Y óJCJSÐSÐOÐOYBEFEJOJY5-ZFNBMFUNFLUFEJS#VÐSÐ- c) f ) P4 Y + Q2 Y OÐOYBEFEJOJOTBUõOEBOGJSNBOOHFMJSJJTF Q(x) h) 3P2 Y+ 23 Y2 Y2 +JGBEFTJJMFCVMVOBCJMNFLUFEJS e) P2 Y . Q3 Y P4 (x2) #VOBHÌSF g) 4Q3 Y2 k) P3 (x) a) ·SÑOÑO TBUöOEBO FMEF FEJMFO L»S JGBEF FEFO Q3 (x3) polinomu bulunuz. Q2 (x) b) #V ÑSÑOEFO UBOF TBUBSTB JöMFUNFOJO LBÀ 5-L»SFMEFFEFDFôJOJCVMVOV[ B C D E F G H I L ÖRNEK 5 B ÷öMFUNFOJO L»SO WFSFO QPMJOPN ÑSÑOMFSJO TBUöO 1 Y WF2 Y QPMJOPNMBSJ¿JO EFS[1 Y 2 Y ] = 3, den elde edilen gelir polinomu ile maliyet polinomu der > P (x) H = 1 PMEVôVOBHÌSF EFS[ P( x ) ] kaçUS OVOGBSLJMFCVMVOVS Q (x) #VOBHÌSF L»SQPMJOPNV P(x) = 0,02x2 + 175 - 7x der[P( x )] = m, der[Q ( x )] = n olsun. = 0,02x2 - 7x + 175 tir. der[P( x ).Q( x )] = m + n = 3, C ÷öMFUNFOJO ÑSÑOEFO UBOF TBUUôOEB FMEF FEF DFôJL»S L»SQPMJOPNVOVOY=JÀJOBMBDBôEFôF der> P(x) H = m - n = 1 olur. Taraf tarafa toplarsak; SFFöJUUJS #VOBHÌSF 1 Y = 0,02x2 - 7x + 175 polinomunda Q(x) P(2000) = 0,02 . (2000)2 - 7.2000 + 175 m = 2, n = 1 bulunur. = 80000 - 14000 + 175 der[P( x )] = 2 olur. =5-L»SFMEFFUNJöUJS 4. B C D E F G H I L 5. 2 9 6. 6 7. a) 0,02x2 – 7x + 175 b) 66175 TL
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 8 ÖRNEK 10 \"#$CJSпHFO Y2 -BY+ BY3 +Y2 -Y+ A BÀMNOEBO FMEF FEJMFDFL QPMJOPNEB x3 lü terimin LBUTBZTPMEVôVOBHÌSF BLBÀUS 2x + 1 [AH] m [ BC] (x2 - ax + 3) (ax3 + 2x2 - 5x + 1) BH | |AH = Y+ | |BC = Y+ BÀMNOEB -5x3 -2ax3 + 3ax3 = (-5 -2a + 3a) x3 = (a - 5)x3 bulunur. x3MÑUFSJNJOLBUTBZTPMEVôVO C 1 Y =\" \"#$ dan a - 5 = 12 ve a = 17 bulunur. x+7 PMEVôVOBHÌSF 1 Y QPMJOPNVOVOLBUTBZMBSUPQMB NOCVMVOV[ P(x) =\" \"#$ = ^ x + 7 h.^ 2x + 1 h dir. 2 P^ 1 h = 8.3 = 12 bulunur. 2 1PMJOPNVO,BUTBZMBS5PQMBNOWF %m/*m 4BCJU5FSJNJOJ#VMNB 1 Y QPMJOPNVOVO TANIM ¥JGUEFSFDFMJUFSJNMFSJOJOLBUTBZMBSUPQMBN P^ 1 h+ P^ -1 h #JSQPMJOPNVOTBCJUUFSJNJCVMVOVSLFOEFóJõLF- 2 OJOJOZFSJOF TGS ZB[MS1 Y QPMJOPNVOVO 5FLEFSFDFMJUFSJNMFSJOLBUTBZMBSUPQMBN TBCJUUFSJNJ1 EFóFSJOFFõJUUJS P^ 1 h- P^ -1 h EJS #JS QPMJOPNVO LBUTBZMBS UPQMBN CVMVOVSLFO 2 EFóJõLFOJOJO ZFSJOF ZB[MS 1 Y QPMJOPNV- OVOLBUTBZMBSUPQMBN1 EFóFSJOFFõJUUJS ÖRNEK 9 ÖRNEK 11 1 Y =Y2 -Y+ 5 1 Y = Y3 -Y2 -Y+ 2 QPMJOPNVOVOTBCJUUFSJNWFLBUTBZMBSUPQMBNOCV lunuz. QPMJOPNVOVO ÀJGU EFSFDFMJ UFSJNMFSJOJO LBUTBZMBS UPQMBNLBÀUS P ( 0 ) = 02 - 4.0 + 5 = 5 P ( 1 ) = 12- 4.1 + 5 = 2 bulunur. P^ 1 h+ P^ -1 h = ÀJGU EFSFDFMJ UFSJNMFSJO LBUTBZMBS 2 UPQMBN P(1) = 4 ve P(-1) = 4 tür. P^ 1 h+ P^ -1 h 4 + 4 O halde, = = 4 bulunur. 22 8. 17 9. 4BCJUUFSJNJ LBUTBZMBSUPQMBN 10 10. 12 11. 4
www.aydinyayinlari.com.tr 10-÷/0.-\"3¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 12 ÖRNEK 15 1 Y = Y2 +Y+ 3 Y3m 2 ôFLJMEFFOJ Y+ WFCPZV Y- CJSJNPMBOCJSCBI- ¿FOJOIBOHJNFZWFMFSMFFLJMFDFóJOFEBJSZBQMBOCJSLSP- QPMJOPNVOVO UFL EFS FDFMJ UFSJNMFSJOJO LBUTBZMBS LJWFSJMNJõUJS UPQMBNLBÀUS 8x – 2 P(1) = 0 ve P(-1) = 4 P^ 1 h- P^ -1 h 0 - 4 Üzüm Elma O halde, = = - 2 bulunur. Ceviz Limon 22 Kivi 6x + 18 ÖRNEK 13 ¶[ÐNWFDFWJ[FLJNBMBOMBSCJSCJSJOFFõJUPMVQFMNB MJ- NPOWFLJWJBMBOMBSOOпÐCJSCJSJOFFõZBQMNõUS Y3 -Y2 + 9 #VOB HÌSF Ñ[ÑN MJNPO WF LJWJ FLJMFDFL BMBOMBSO JGBEFTJOJOBÀMNOEBLJÀJGUEFSFDFMJUFSJNMFSJOLBUTB UPQMBNOWFSFO1 Y QPMJOPNVOVCVMVOV[ ZMBSUPQMBNOCVMVOV[ Üzüm j (3x + 9) (4x - 1) = 12x2 +32x - 9 P(1) = (4 - 5 + 1)9 = 0 Limon j = (2x + 6) (4x - 1) = 8x2 + 22x - 6 ,JWJj (2x + 6) (4x - 1) = 8x2+ 22x - 6 P(-1) = (-4 - 5 + 1)9 = (-8)9 = (-23)9 = -227 P(x) = (12x2 + 33x) - 9) + 2(8x2+ 22x - 6) P^ 1 h+ P^ -1 h 0 - 227 = - 26 bulunur. = 28x2 + 77x - 21 bulunur. = 22 2 ÖRNEK 14 ÖRNEK 16 P( x ) polinomu için, #JS1 Y QPMJOPNVY+JMFCËMÐOEÐóÐOEFCËMÐNY2 - 2 WFLBMBOEJS 1 Y- +1 Y+ =Y+ 16 #VOBHÌSF 1 Y QPMJOPNVOVCVMBMN PMEVôVOBHÌSF 1 LBÀUS <P ( x) = >(x + 2) .>(x2 - 2) + 51 P(x - 1) ve P(x + QPMJOPNMBSOOUPQMBNCJSJODJEF receden 12x + 16 polinomu ise P(x) = ax +CöFLMJO Bölünen Bölen Bölüm Kalan de birinci dereceden bir polinomdur. Polinomlar fonk TJZPOMBSO Ì[FM CJS UÑSÑ PMBSBL EÑöÑOÑMFCJMJS #V ZÑ[ P ( x ) = (x3 - 2x + 2x2 - 4) + 1 EFO GPOLTJZPOMBSEBLJ CJMFöLF JöMFNJOF CFO[FS PMBSBL P ( x ) = x3+ 2x2 - 2x - 3 bulunur. P(x) = ax + b ise P(x - 1) = a(x - 1) + b = ax - a + b ve P(x + 3) = a(x + 3) + b = ax + 3a + b elde edilir. P(x - 1) + P(x + 3) = 2ax + 2a + 2b olur. 2ax + 2a + 2b = 12x + 16 j 2a ve 12 = 2a + 2b = 16 j a = 6 ve b = 2 dir. O halde, P ( x ) = 6x + 2 polinomu bulunur. P ( 2 ) = 14 olur. 12. –2 13. –226 14. 14 11 15. 28x2 + 77x – 21 16. x3+ 2x2 – 2x – 3
TEST - 3 1PMJOPNMBSMB÷öMFNMFS 1. 1 Y =Y5 -Y4 +Y2 -Y+ 3 5. 1 Y =Y3 +Y2 +Y+ 2 PMEVôVOBHÌSF 1 - LBÀUS PMEVôVOBHÌSF P_ 3 2 - 1 iLBÀUS \" # $ % & \" 3 2 + 1 # 3 2 $ % & 2. 1 Y =Y2 -Y+ 3 6. Y>PMNBLÐ[FSF PMEVôVOBHÌSF 1 Y- BöBôE BLJMFSEFOIBO P_ 4 x i =Y3 -Y2 + 2 gisidir? PMEVôVOBHÌSF 1 Y3 BöBôE BLJMFSEFOIBOHJ sidir? \" Y2 -Y+ # Y2 -Y+ 5 \" Y36 +Y24 - 2 # Y36 -Y24 + 2 $ Y2 -Y+ % Y2 -Y+ 1 $ Y9 -Y3 - % Y9 -Y3 + 2 & Y2 -Y+ 3 & Y18 -Y6 + 2 3. ( a -Y CY2 +D =Y3 -Y2 +Y- 3 7. Y3 +BY2 + Y4 +BY3 -Y2 - PMEVôVOBHÌSF B+ b +DLBÀUS BÀMNZBQMEô OEBFMEFFEJMFOQPMJOPNEBY6M \" - # - $ - % - & -1 teriminin katsaZT-PMEVôVOBHÌSF BLBÀ US \" - # - $ - % - & -6 8. %JEFN CJS ZBõOEBLJ L[ J¿JO CJS UBOHSBN ZBQNBL JTUJZPSY>PMNBLÐ[FSF 4. der [1 Y ] = 5, der [2 Y ] =WF Q_ x i CJSQPMJ- A | |AB =Y+ 2 DE | |AC =Y- 12 P_ x i | |BC = Y + 14 OPNPMEVóVOBHËSF V[VOMVLMBS WFSJMJ- I. EFS<1 Y 2 Y > II. EFS<1 Y 2 Y > yor. III. der > Q_ x i H=3 P_ x i BF C ifadelerJOEFOIBOHJMFSJEPôSVEVS 5BOHSBN EÌSU QBSÀBZB BZSMEôOEB PMVöBDBL PMBO EÌSU ÑÀHFOJO ÀFWSFMFSJO UPQMBNO WFSFO \" :BMO[* # *WF** $ **WF*** QPMJOPNBöBôEBLJMFSEFOIBOHJTJEJS % *WF*** & * **WF*** \" Y+ # Y+ $ Y+ 20 % Y+ & Y+ 20 1. \" 2. # 3. # 4. D 12 5. & 6. # 7. C 8. D
1PMJOPNMBSMB÷öMFNMFS TEST - 4 1. Pf 1 - 7x p =Y3 +Y2 -Y+ 1 P (x - 1) + x.Q (x) 2 5. = 4 PMEVôVOBHÌSF P (- LBÀUS \" - # - $ % & x+2 olmak üzere, P ( x ) polinomunun sabit terimi 3 olduôVOBHÌSF 2 Y JOLBUTBZMBSUPQMBNLBÀ US \" # $ % & 2. der [1 Y ] =WFEFS[2 Y ] =PMNBLÐ[FSF 6. Y5 +Y2 -Y- 2n P3 (x2) QPMJOPNVOVO ÀJGU EFSFDFMJ UFSJNMFSJOJO LBUTBZ Q2 (x3) MBSUPQMBNPMEVôVOBHÌSF OLBÀUS JGBEFTJ CJS QPMJOPN PMEVôVOB HÌSF CV QPMJOP mun EFSFD FTJLBÀUS \" # $ % & \" # $ % & 3. P( 1 -Y =Y2 -Y+ 1 7. Y3 -Y2 +Y- Y4 +Y5 -Y3 +Y2 + PMEVôVOBHÌSF 1 Y+ BöBôEBLJMFSEFOIBO ÀBSQNOEBY5MJUFSJNJOLBUTBZTLBÀUS gisidir? \" # $ % & A Y2 -Y+ # Y2 -Y- 1 $ Y2 +Y+ % Y2 +Y+ 1 & Y2 -Y+ 1 8. ôFLJMEFZBOZBOBCVMVOBOJLJLVUVEBZB[MQPMJOPN- MBSUPQMBOZPSWFUPQMBN LPNõVMBSPMBOBMULVUVZB ZB[MZPS Q(x) x–5 2x + 3 + + 2x + 1 4. 1 Y WF2 Y QPMJOPNMBSJ¿JO + der [ P3 Y . 2 Y ] = 14 der> P2 (x) H = 2 P(x) Q3 (x) #VOBHÌSF 1 Y +2 Y QPMJOPNVOVOLBUTBZ MBSUPQMBNLBÀUS PMEVôVOBHÌSF EFS[ P( x ) + Q ( x ) ] LBÀUS \" # $ % & \" # $ % & 1. C 2. \" 3. D 4. # 13 5. & 6. C 7. \" 8. &
TEST - 5 1PMJOPNMBSMB÷öMFNMFS 1. \"öBôEBLJJGBEFMFSEFOIBOHJTJZBOMöUS? 5. 4BCJUPMNBZBO CJSEFOGB[MBQPMJOPNVO¿BSQNCJ- \" 1 Y+ QPMJOPNVOVOLBUTBZMBSUPQMBN1 ¿JNJOEFZB[MBNBZBOQPMJOPNMBSBJOEJSHFOFNFZFO UÐS QPMJOPNMBSEFOJS # 1 Y+ QPMJOPNVOVOTBCJUUFSJNJ1 UÐS #BõLBUTBZTCJSPMBOJOEJSHFOFNFZFOQPMJOPNMBSB $ 1 Y- QPMJOPNVOVOLBUTBZMBSUPQMBN1 - BTBMQPMJOPNEFOJS UÐS 1 Y WF2 Y QPMJOPNMBSEFSFDFTJPMBOBTBMQP- % 1 Y+ QPMJOPNVOVOLBUTBZMBSUPQMBN1 MJOPNMBSES EJS 1 Y 2 Y =Y2 +Y+ 2 & 1 Y+ QPMJOPNVOVOTBCJUUFSJNJ1 EJS PMEVôVOB HÌSF 1 Y + 2 Y BöBôEBLJMFSEFO hangisidir? \" Y+ # Y+ $ Y+ 3 % Y+ & Y+ 5 2. P (x)= _ 3a - 9 i 3 x + _ b - 2 i x + _ 2a + 3b ix + a - 2b 6. Y2 -Y+ Y4 -Y3 +Y+ P ( x ) bir polinPNPMEVôVOBHÌSF 1 LBÀUS JGBEFTJOJOÀBSQNOEBY5MJUFSJNJOLBUTBZTLBÀ US \" # $ % & \" - # - $ - % - & -1 3. 1 Y WF2 Y QPMJOPNMBSJ¿JO der [P (x)] = 3WF der [Q (x)] de r [ P2 Y+ 2 Y3 + ] = 18 7. 1 Y3 -Y2 =Y3 -Y2 + 5 PMEVôVOBHÌSF EFS[ P( x ) ]LBÀUS PMEVôVOB HÌSF 1 BöBôEBLJMFSEFO IBOHJTJ dir? \" # $ % & \" # $ % & 18 8. 1 Y =Yn +Yn-1 +Yn-2 +...+Y3 +Y2+Y+1 4. n+1 + 3 . x2n–6 + x2 JGBEFTJOEFSFDFEFO LBUTBZMBSCFMJSMJCJSLVSB MBHÌSFZB[MNöCJSQPMJOPNPMEVôVOBHÌSF P (x) = 4.x der [ P ( x ) ]LBÀUS CJSQPMJOPNPMEVôVOBHÌSF 1 Y QPMJOPNVOVO derecesi en azLBÀUS \" # $ % & \" # $ % & 1. D 2. C 3. & 4. # 14 5. C 6. # 7. D 8. \"
1PMJOPNMBSMB÷öMFNMFS TEST - 6 1. A = {1, 2, 3} PMNBLÑ[FSF BöBôEBLJMFSEFOIBO 5. 1 Y =YN +Y2 +YWF gisi her n ` \"JÀJOCJSQPMJOPNEFSFDFTJPMBCJ 2 Y =Y4 +Y+QPMJOPNMBSES lir? der [1 Y +2 Y ] =N \" -O # n2 + 1 $ 6 2 n PMEVôVOBHÌSF [P ( x2 ) ] 3 polinomunun derece % - n2 & 2n + 1 si en azLBÀUS 3 \" # $ % & 2. 1 Y+ =Y2 +Y+ 7 PMEVôVOBHÌSF 1 -Y QPMJOPNVBöBôEBLJ 6. 1 Y = Y2 -Y- 3 Y3 -Y- 5 lerden hangisidir? QPMJOPNVOEB UFL EFSFDFMJ UFSJNMFSJO LBUTBZMBS \" Y2 + # Y2 +Y+ 3 UPQMBNLBÀUS $ Y2mY+ % Y2 +Y+ 3 \" - # - $ % & & Y2 + 3 3. 1 Y =Y4 -Y2 +Y- 5 7. 1 Y = N+ Y4 +Y3 -Y2 -Y+ 3 2 Y =Y2 +Y-QPMJOPNMBSWFSJMJZPS 2 Y =OY4 + N- Y3 + ( 2 +O Y2 +Y- 2 , Y =1 Y- +2 Y PMEVôVOB HÌSF , Y QPMJOPNVOVO LBUTBZMBS QPMJOPNMBS için d [P( x ) + Q ( x )] = PMEVôV UPQMBNLBÀUS OBHÌSF 2 Y QPMJOPNVOVOLBUTBZMBSUPQMBN LBÀUS \" - # - $ - % & \" # $ % - & -2 4. 1 Y Z =Y3 y2 -YZ3 +YZ+Y- 3y 8. 5BN TBZ LBUTBZM JLJ WFZB EBIB GB[MB QPMJOPNVO polinomu için P ( 1, - JGBEFTJOJOEFôFSJLBÀUS ¿BSQNõFLMJOEFJGBEFFEJMFNFZFOWFCBõLBUTBZT PMBO QPMJOPNMBSB JOEJSHFOFNFZFO QPMJOPN EF- OJS #VOBHÌSF BöBôEBLJMFSEFOIBOHJTJJOEJSHFOF meyen polinomdur? \" # $ % & \" Y2 -Y- # Y3 -Y $ Y2 + 1 % Y4 + & Y5 1. C 2. & 3. \" 4. D 15 5. D 6. D 7. & 8. D
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr 10-÷/0.-\"3%\"#²-.&÷õ-&.÷ #ÌMNF÷öMFNJ:BQNBEBO,BMBO#VMNB ÖRNEK 2 ÖRNEK 1 1 Y =Y3 +Y2 -Y- 3 1 Y =Y2 -Y- 6 QPMJOPNVOVOTGSMBSOCVMVOV[ QPMJOPNVOVOTGSMBSOLÌLMFSJOJHSBGJLÀJ[FSFLCVMB ÷LJODJ EFSFDFEFO 1 Y QPMJOPNV TGSMBSO 1 Y = 0 MN EFOLMFNJOJUBNLBSFZFUBNBNMBZBSBLCVMBMN P(x) = 0 j x2- 4x - 6 = 0 ¦Ì[ÑN x2 - 4x + 4 - 10 = 0 y x2 - 4x + 4 = 10 (x - 2)2 = 10 3 ^ x - 2 h2 = 10 2 1 | |x - 2 = 10 j x - 2 = 10 veya x - 2 = - 10 –3 –2 –1 0 12 3 x j x = 2 + 10 veya x = 2 - 10 dur. –1 P(x) Polinomunun ax + C JMF #ÌMÑONFTJOEFO –2 &MEF&EJMFO,BMBOO#VMVONBT –3 TEOREM 7FSJMFO QPMJOPNVO HSBGJóJOJ ¿J[JN QSPHSBN LVMMBOB- #JS1 Y QPMJOPNVOVOBY+CJMFCËMÐNÐOEFO SBL¿J[FMJN¥J[JMFOHSBGJóJOYFLTFOJOJLFTUJóJOPLUB- FMEFFEJMFOLBMBOCVMNBLJ¿JO1 Y QPMJOPNVO- MBSQPMJOPNVOTGSMBS 1 Y =PMEVóVOPLUBMBS ES EB YZFSJOFCËMFOQPMJOPNVOVOLËLÐPMBO #VOBHËSF Y= - Y= -WFY= 1 OPLUBMBS1 Y - b f ax + b = 0 & x = - b , a ≠ 0 pZB[MS QPMJOPNVOVOTGSMBSES aa NOT #BõLBCJSJGBEFZMF1 Y = BY+C # Y + k PMVS P Y =Y3 +Y2 -Y-QPMJOPNVOVOTGSMBS PMBO 1 B = 0 lY-B 1 Y QPMJOPNVOVOCJS¿BS- Y= -J¿JO Y- (- = Y+ QBOES Y= -J¿JO Y- (- = Y+ Y=J¿JO Y- õFLMJOEFCJSFS¿BSQBOPMEVóVOBEJLLBUFEJOJ[ ÖRNEK 1 ÖRNEK 3 1 Y =Y3 +Y2 +Y- 6 QPMJOPNVOVOLÌLMFSJOJCVMVOV[ 1 Y =Y3 -Y2 -Y+ 1 polinomunun x -JMFCÌMÑN ÑOEFOLBMBOLBÀUS x3 + 4x2 + x - 6 =PMEVôVOPLUBMBS x3+ 3x2+ x2+ x - 6 = 0 x -1=0 jx=1 x2 (x + 3) + (x + 3) (x - 2) = 0 P(1) = 13 - 3.12 - 5.1 + 1 = -6 bulunur. (x + 3) (x2 + x - 2) = 0 (x + 3) (x + 2) (x - 1) =öFLMJOEFÀBSQBOMBSWBSES#V OBHÌSF Y= -3, x = -2, x =OPLUBMBS1 Y QPMJOPNV OVOTGSMBSES 1. –3, –2, 1 16 2. 2 ± 10 3. –6
www.aydinyayinlari.com.tr 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 4 ÖRNEK 8 1 Y =Y2 -Y+ 3 1 Y- =Y3 -Y2 +Y- 3 polinomunun 2x -JMFCÌMÑNÑOE FOLBMBOLBÀUS WFSJMEJôJOFHÌSF 1 Y QPMJOPNVOVOVOY+JMFCÌMÑ NÑOEFOLBMBOLBÀUS 1 2x - 1 = 0 j x = x + 2 = 0 j x = -2 j P(- ZJCVMNBMZ[ x = 0 yazarsak; P(x - QPMJOPNVOEBCJ[FLBMBOWFSJS 2 P(-2) = -3 bulunur. 1 12 1 Pd n = 4·d n - 2·d n + 3 = 3 bulunur. 22 2 ÖRNEK 5 1 Y =Y12 +Y8 +Y2 -Y+ 2 QPMJOPNVOVOYJMFCÌMÑNÑOEFOLBMBOLBÀUS x = 0 j P ( 0 ) = 2 bulunur. ÖRNEK 6 ÖRNEK 9 1 Y =Y3 -Y2 +BYmB+ 1 P ( 1 -Y =Y2 -Y+ 7 polinomunun x +JMFCÌMÑNÑOEFOLBMBOPMEVôV WFSJMEJôJOFHÌSF1 Y- 2 ) polinomunun x +JMFCÌ OBHÌSF BLBÀUS MÑNÑOEFOLBMBOLBÀUS x + 2 = 0 j x = -2 , P(-2) = 3 x + 1 = 0 j x = -1 , P(x - 2) polinomunda x yerine -1 ZB[MSTB 1 - CJ[FLBMBOWFSJS O halde P(1-x) polinomunda x yerine 4 yazarsak P(-3) ÑFMEFFUNJöPMVSV[ x = 4; P(-3) = 42 - 4 + 7 = 19 bulunur. P(-2) = (-2)3- (-2)2 + a.(-2) - 3a + 1 = 3 a =- 14 bulunur. 5 ÖRNEK 7 ÖRNEK 10 1 Y =Y3mBY2 +CY+ 2 1 Y- +2 Y+ =Y3 -Y2 +WFSJMJZPS polinomunun x - JMF CÌMÑN ÑOEFO LBMBO Y + 1 P ( x ) in x -JMFCÌMÑNÑOEFOLBMBOPMEVôVOBHÌ JMF CÌMÑNÑOEFO LBMBO PMEVôVOB HÌSF B C JLJMJ re, Q ( x ) polinomunun x - JMF CÌMÑNÑOEFO LBMBO sini bulunuz. LBÀUS x - 1 = 0 j x = 1 j P(1) = 2 x - 1 = 0 j x = 1 j P(1) = 3 tür. x - 4= 0 j x = 4 j2 LBMBOOWFSFDFLUJS x + 1 = 0 x = -1 j P(-1) = 4 O halde P(x - 2) + Q(x + 1) = x3- x2+ 2 ifadesinde x =P^91 h + Q^ 4 h = 27 - 9 + 2 olur. P(1) = 1 - a + b + 2, P(-1) = - 1 - a - b + 2 3 + Q(4) = 20 j Q(4) = 17 bulunur. 2=3-a+b 4=1-a-b 1=a-b -3 = a + b j÷LJEFOLMFNEFPSUBLÀÌ[ÑNZBQBSTBL a = -1, b= -2 ve (-1, -2) bulunur. 4. 3 5. 2 6. - 14 7. (–1, –2) 17 8. –3 9. 19 10. 17 5
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 11 ÖRNEK 14 1 Y- =Y- 2 +2 Y+ 1 Y- =Y2 -BY- 3a -QPMJOPNVWFSJMJZPS 2 Y+ =Y+ 1 -1 Y+ 1 Y JOLBUTBZMBSUPQMBNPMEVôVOBHÌSF TBCJUUF SJNJLBÀUS PMEVôVOB HÌSF 1 Y + QPMJOPNVOVO Y JMF CÌMÑ NÑOEFOLBMBOLBÀUS P ( x ) polinomunda x =ZB[BSTBLLBUTBZMBSOUPQMB NOCVMVZPSEVL1 = 2, P(x) polinomunun sabit teri x = 0 j P(x + 1) polinomunda x = 0 yazarsak; mi ise P ( 0 ) olarak bulunuyordu. O halde 1 LBMBOOWFSFDFLUJS x =1 = 9 - 3a - 3a - 1 x =1 -1) = 3.1 - 2 + Q(1 + 1) 2 = 8 - 6a 2.P ( 1 ) = 1 + Q(2) j 2.P^ 1 h - Q^ 2 h = 1 a = 1 yerine yazarsak; x = -2 -1 + 3) = 5.(-1) + 1 - P(-1 + 2) P(x - 2) = x2 - x - 4 olur. P(0) = -2 bulunur. 3.Q ( 2 ) = -4 - P(1) j P^ 1 h + 3.Q^ 2 h = - 1 #VJLJJGBEFEF2 ZJZPLFEFSTFL P^ 1 h = - 1 bulunur. 7 ÖRNEK 12 ÖRNEK 15 1 Y WF2 Y QPMJOPNMBSOOY-JMFCËMÐONFTJOEFOFM- 1 Y+ = Y2 - 2 Y+ +Y-WFSJMJZPS EFFEJMFOLBMBOMBSTSBTJMFWFUJS 2 Y QPMJOPNVOVO LBUTBZMBS UPQMBN PMEVôVOB #VOBHÌSF 1 Y 2 Y QPMJOPNVOVOY-JMFCÌMÑ HÌSF 1 Y JOTBC JUUFSJNJLBÀUS NÑOEFOLBMBOLBÀUS Q(1) = 1 CVMBDBô[ x - 3 = 0 j x = 3 j P ( 3 ) = 2 ve Q ( 3 ) = 5 tir. x = -1 = ( ( -2 )2 - 1 ) . Q ( 1 ) + 4 ( -2 ) - 1 x =1 Y 2 Y JGBEFTJOEFZFSJOFZB[BSTBL P ( 3 ) . Q ( 3 ) = 2.5 = 10 bulunur. = 3.4 - 8 - 1 = 3 bulunur. ÖRNEK 13 ÖRNEK 16 1 Y WF2 Y QPMJOPNMBSOOY-JMFCËMÐONFTJOEFOFM- 1 Y =Y2 -Y+QPMJOPNVWFSJMJZPS EFFEJMFOLBMBOMBSTSBTJMFWF-EJS P ( x - QPMJOPNVOVOLBUTBZMBSUPQMBNLBÀUS 3P (x) + 1 x = 1 yazarsak P(x - 3) polinomunda P(-2) yi bula k Q (x) DBô[ x= -1 -2) = 2(-2)2 - 7(-2) + 4 = 26 bulunur. polinomunun x - 2 iMFCÌMÑONFTJOEFOLBMBOPM EVôVOBgÌSF LLBÀUS? x - 2 = 0 j x = 2, P(2) = 4 ve Q(2) = -1 dir. x = 3.P^ 2 h + 1 = 5 PMEVôVOBHÌSF k.Q^ 2 h 3.4 + 1 = 5 & 13 = - 5k & k = - 13 bulunur. k.^ - 1 h 5 1 12. 10 13 18 14. –2 15. 3 16. 26 11. - 13. - 7 5
www.aydinyayinlari.com.tr 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 17 ÖRNEK 20 1 Y- =Y3 -Y2 +QPMJOPNVWFSJMJZPS ôFLJMEFLJUSBLUËSÐOBSLBUFLFSJOJO¿FWSFTJ Y+ NFUSF P ( x + QPMJOPNVOVOTBCJUUFSJNJLBÀUS ËOUFLFSJOJO¿FWSFTJ Y- NFUSFEJS x = 0 yazarsa P(x + QPMJOPNVOEB1 JCVMBDBô[ x =ZB[BMN1 = 23 - 22 + 4 = 8 bulunur. ÖRNEK 18 (x + 2) m (x – 1) m 1 Y =Y2 +Y-QPMJOPNVOVO Y- OJOB[BMBOLVW- Y3+BY2 -Y+ b -NFUSFZPMHJEJODFËOUFLFSJOWFBS- WFUMFSJOFHËSFEÐ[FOMFONJõCJ¿JNJ LBUFLFSJOUVSTBZMBSUBNTBZES 1 Y- = Y- 2 + Y- -õFLMJOEFEJS #VOBHÌSF Y2 + 5x - 1 polinomunun (x + 1) in aza #VOBHÌSF BCEFôFSJOJCVMVOV[ MBOLVWWFUMFSJOFHÌSFZB[EôN[EBCVMEVôVNV[QP MJOPNVOLBUTBZMBSUPQMBNLBÀUS x3 + ax2 - 2x + b - 2 polinomu (x + 2) ve (x - 1) ile UBNCÌMÑONFLUFEJS P(x) = x2 + 5x - 1 x = 1; (1)3 + a(1)2 - 2(1) + b - 2 = 0 j a + b = 3 bulunur. = 1.(x + 1)2 + 5.(x + 1) - 1 x = -2 ; -8 + 4a + 4 + b - 2 = 0 j 4a + b = 6 bulunur. ÷LJEFOLMFNJÀÌ[EÑôÑNÑ[EF LBUTBZMBSUPQMBN+ 5 - 1 = 5 bulunur. a = 1, b = 2 bulunur ve a . b = 2 dir. ÖRNEK 19 ÖRNEK 21 x+4 Y- 1 Y =Y3 -BY2 +Y+ 1 x3 + 4x2 + 3x + 12 PMEVôVOBHÌSF 1 LBÀUS x–1ôFLJMEFHFOJõMJóJ Y- NFUSF V[VOMVóV x = 1 için Y3+ Y2+ Y + NFUSF PMBO CJTJLMFU ZPMV WFSJMNJõUJS #JTJLMFUZPMVOVOUBNBNFOJ Y- NFUSF CPZV Y+ 0 = 1 - a + 2 + 1 j a = 4 buluruz. NFUSF PMBO QBSLF UBõMBSZMB CPõMVL LBMNBZBDBL õFLJMEF (x - 1). P(x) = x3- 4x2 + 2x + 1 LBQMBOBDBLUS#VJöJÀJOLBÀUBOFQBSLFUBöHFSFLJS x3 – 4x2 + 2x + 1 x–1 x3 + 4x2 + 3x + 12 polinomunu x +QPMJOPNVOBCÌM x3 – x2 x2 – 3x – 1 EÑôÑNÑ[EF Y2 (x + 4) + 3(x + 4) = (x + 4) (x2+ CÌ lüm (x2+ 3), kalan da 0 olur. O halde (x2+ 3) tane par –3x2 + 2x + 1 LFUBöHFSFLJS –3x2 æY –x + 1 –x + 1 0 P(x) = x2- 3x - 1 olur. P(1) = 12 - 3.1 - 1 = -3 bulunur. 17. 8 18. 5 19. (x2+ 3) 19 20. 2 21. –3
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 22 ÖRNEK 25 1 Y +1 Y- = 5 -Y-Y2 1 Y- a - =Y3 +Y-QPMJOPNVWFSJMJZPS PMEVôVOBHÌSF 1 LBÀUS #VOBHÌSF 1 Y JOY+BJMFCÌMÑNÑOEFOLBMBOLBÀ P(x) = ax2+ bx + c olsun. US P(x - 1) = a(x - 1)2 + b(x - 1) + c olur. x + a = 0 j x = -a j P(-B LBMBOWFSJS P(x - a - 1) polinomu P(-a) olabilmesi için x = 1 için P(x) + P(x - 1) = ax2 + bx + c + a(x - 1)2+ b(x - 1) + c P(-a) = 13 + 1 - 4 = -2 bulunur. 5 - 2x - x2 = 2ax2 + (2b - 2a)x + (2c + a - b) 13 ve c = 2 bu 1PMJOPN FöJUMJôJOEFO a = - , b = - 22 lunur. 2 P^ x h = - - 3x +2 dir. x 22 P(2) = -3 bulunur. ÖRNEK 23 ÖRNEK 26 1 Y m1 Y- =YWF1 = 4 #JS1 Y QPMJOPNVOVO Y- Y+ JMFCËMÐNÐOE FO PMEVôVOBHÌSF 1 LBÀUS LBMBOY+NEJS x = P^ 2 h - P^ 1 h = 6 P ( x ) in x -JMFCÌMÑNÑOEFOLBMBOPMEVôVOBHÌ x = P^ 3 h - P^ 2 h = 9 re, x +JMFCÌMÑNÑOEFOLBMBOLBÀUS x=P^ 4 h - P^ 3 h = 12 P(4) - P(1) = 27 P(x) polinomunun (x - 3) (x + JMFCÌMÑNÑOEFOCÌMÑN P(4) = 4 + 27 j P(4) = 31 bulunur. #PMTVO1 Y = (x - 3) (x + 2 #+ (3x + m) dir. x - 3 = 0 j x = 3 j P(3) =ES P^ 3 h = 9 + m _ j P(x) = (x - 3) . (x + #+ (3x - 3) bb P(-2) = 3(-2) - 3 6=9+m ` = - 9 bulunur. bb m = - 3 tür. a ÖRNEK 24 ÖRNEK 27 1 Y +Y1 -Y =Y3 +Y2 +Y 1 Y QPMJOPNVOVO Y2 m Y + JMF CËMÐNÐOEF CËMÐN PMEVôVOBHÌSF P( - LBÀUS 2 Y LBMBOY+EJS x = - 3P^ - 1 h - P^ 1 h = - 2 Q ( x ) in ( x + JMFCÌMÑNÑOEFOLBMBOPMEVôVOBHÌ re, P( x ) in x3 +JMFCÌMÑNÑOE FOLBMBOOFEJS x = 3P^ 1 h + P^ - 1 h = 6 #VJLJEFOLMFNEF1 MFSJZPLFEFSTFL1 -1) = 0 bu P(x) = (x2 - x + 1) . Q(x) + (x + 2) olarak yazabiliriz. lunur. Q(x) in (x + JMFCÌMÑNÑOEFOCÌMÑN# LBMBOPMTVO Q(x) = (x + #+ 3 olur. P(x) te Q(x) ifadesini yeri ne yazarsak; P(x) = (x2 - x + 1) ((x + #+ 3) + (x + 2) = (x + 1) (x2 - x + #+ 3(x2 - x + 1) + (x + 2) = (x3 + #+ 3x2 - 2x + 5 olur ve kalan (3x2 - 2x + 5) bulunur. 22. –3 23. 31 24. 0 20 25. –2 26. –9 27. (3x2 – 2x + 5)
www.aydinyayinlari.com.tr 10-÷/0.-\"3WF¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 28 ÖRNEK 30 #JSOBLMJZFGJSNBTLÐQõFLMJOEFUBõNBLBCJOJZBQUSNB- 1 Y = Y2mYm 2 Y +Ym ZQMBOMBNBLUBES PMEVôVOBHÌSF 1 Y QPMJO PNV Y+ JMFCÌMÑO EÑôÑOEFFMEFFEJMFDFLCÌMÑNQPMJOPNVOVCVMVOV[ x x+1 P(x) = (x2 - x - 2) . Q(x) + x - 3 x–1 = (x - 2) (x + 1).Q(x) + x - 3 nun (x + JMFCÌMÑNÑO EFOCÌMÑN Y- 2).Q(x) + 1 bulunur. * ,BCJOJOCJSBZSU1 Y =Y3 +BY2 +CY+DCJSJNEJS ÖRNEK 31 ** ,BCJO BZSUMBSY+ Y-WFYCJSJNPMBOEJLEËSU- HFOMFS QSJ[NBT õFLMJOEFLJ LVUVMBSMB CPõMVL LBMNB- P (x) = a . (3 x) 2 - (b - 1) x - 4x - 5 EBOEPMEVSVMBCJMJZPS ifadesi bir poliOPNPMEVôVOBHÌSF 1 B + P ( b ) top MBNLBÀUS #VOBHÌSF IBDNJY3 - 8 birimküp olan küplerle dol EVSVMVSTBLBÀCJSJNLÑQCPöMVLLBMS 1 Y JOQPMJOPNPMNBTJÀJOB= 0 ve b =PMNBMES P(x) = -4x - 5 bulunur. P(0) + P(1) = -5 - 9 = -14 bu P (x ) polinomu x + 1, x -WFYFUBNCÌMÑOÑS1 -1) = 0, lunur. P(1) = 0, P(0) = 0 olur. ÖRNEK 32 P(0) = c = 0, P(-1) = -1 + a - b = 0 j a - b = 1 ve YQP[JUJGCJSUBNTBZPMNBLÐ[FSF P(1) = 1 + a + b = 0 j a + b = - ÑÀEFOLMFNJÀÌ[EÑôÑ Q = Y+ Y+ Y+ müzde a = 0, b = -1 ve c = 0 bulunur. PMBSBLCJS Q QPMJOPNVUBONMBOZPS P(x) = x3 -YCVMVOVS#JSBZSU1 Y PMBOLBCJOJOIBDNJ 2. Q = q PMEVóVOBHËSF (x3 - x)3 PMVS #V EFôFSJ IBDNJ Y3 - 8) br3 olan bir kü QÑOIBDNJOFCÌMEÑôÑNÑ[EFY3- 8 = 0 j x3 = 8 j x = 2 yi q - Q GBSLen az LBÀBFöJUUJS (x3 - x)3 te yerine yazarsak, (23 - 2)3 = 63 = 216 bulunur. x = 1 için p = 2.3.4 = 24 x = 2 için p = 3.4.5 = 60 ÖRNEK 29 x = 3 için p = 4.5.6 = 120 P [1 Y +mY]m1 Y+ =PMNBLÐ[FSF 2. p = q olEVôVOBHÌSF 1 Y JOYmJMFUBNCËMÐOEÐóÐCJMJOJZPS p = 60, q = 120 q - p = 120 - 60 = 60 (en az) olur. 1 Y JOYmWFYmJMFBZSBZSCÌMÑNÑOEFOFM EFFEJMFOLBMBOMBSUPQMBNPMEVôVOBHÌSF 1 LBÀUS P ( 1 ) = 0 ve P ( 2 ) + P ( 4 ) = 14 olur. x =P [=P^ 1 h + 2] - P^ 4 h = 2 j P ( 2 ) - P (4) = 2 0 CVMVSV[÷LJEFOLMFNEFO1 = 6 bulunur. 28. 216 29. 6 21 30. (x – 2).Q(x) + 1 31. –14 32. 60
TEST - 7 1PMJOPNMBSEB#ÌMNF÷öMFNJ 1. Y3 -Y2 +Y-QPMJOPNV Y- JMFCËMÐOEÐ- 5. ¶¿ÐODÐEFSFDFEFOCJS1 Y QPMJOPNVOVOTBCJUUF- óÐOEFCËMÐN2 Y WFLBMBO, Y QPMJOPNEVS SJNJEJS 1 Y QPMJOPNV Y- Y- WF Y- JMFUBN #VOBHÌSF 2 Y +, Y QPMJOPNVBöBôEBLJMFS CËMÐONFLUFEJS #VOBHÌSF 1 LBÀUS den hangisidir? \" - # - $ - % - & -12 \" Y2 -Y+ # Y2 -Y+ 4 $ Y2 -Y+ % Y2 -Y+ 4 & Y2 -Y- 5 2. 1 Y+ =Y2 -Y+ 8 6. 1 Y WF2 Y QPMJOPNMBSBSBTOEB polinomunun (x - JMFCÌMÑNÑOEFOLBMBOLBÀ P (2x + 1) . (x2 - 2) US = x + 1CBóOUTWBSES Q (x - 1) 1 Y QPMJOPNVOVOY-JMFCËMÐNÐOEFOLBMBO-6 ES \" # $ % & #VOBHÌSF Q ( x ) polinomunun sabit terimi kaç US \" # $ % & 3. 1 Y3 =Y12 -Y9 +Y6 -Y3 + 3 7. 1 Y =Y3 +Y2 +Y- 2 Y =Y2 +Y- 6 PMEVôVOBHÌSF 1 Y QPMJOPNVOVOY-JMFCÌ QPMJOPNMBSWFSJMJZPS MÑNÑOEFOLBMBOLBÀUS [ P ( Q ( x ) ) + 2x + 8] polinomunun ( x - JMFCÌ MÑNÑOEFOLBMBOLBÀUS \" # $ % & \" # $ % & 8. #JS GBCSJLBEB ÐSFUJMFO CPSVMBS Y2 + BY + NFUSF V[VOMVóVOEBES )FS CPSV Y - NFUSF V[VOMV- óVOEBLJ Y+C FõJUQBS¿BZBCËMÐOÐZPS x2 + ax + 2 4. 1 Y2 + =Y4 - 3 polinomu için P ( x - JGBEFTJOJOFöJUJBöBôEB kilerden hangisidir? Ymå Ymå \" Y2 -Y+ # Y2 -Y+ 13 1BSÀBMBSEBOCJSJOJOV[VOMVôVNPMEVôVOBHÌ SF CPSVMBSLBÀFöJUQBSÀBZBCÌMÑONÑöUÑS $ Y2 -Y+ % Y2 -Y+ 13 & Y2 -Y+ 15 \" # $ % & 1. # 2. C 3. C 4. D 22 5. & 6. \" 7. \" 8. D
1PMJOPNMBSEB#ÌMNF÷öMFNJ TEST - 8 1. 1 Y =Y4 -Y3 +Y2 -Y- 15 5. 1 Y QPMJOPNVOVO Y- JMFCËMÐNÐOEFOLBMBO QPMJOPNVOVOY-JMFCËMÐNÐOEFOLBMBON Y+ 1 UJS JMFCËMÐNÐOEFOLBMBOOPMEVóVOBHËSF m + n top MBNLBÀUS Y+ 1 Y+ =2 Y + 2 \" # $ % & PMEVôVOB HÌSF 2 Y QPMJOPNVOVO LBUTBZMBS UPQMBNLBÀUS \" # $ % & 2. 1 Y =Y3 +Y2 +Y+N+QPMJOPNVWFSJMJZPS 6. 1 Y =Y54 +Y27 -Y9 +Y3 - 5 1 Y QPMJOPNVOVO Y + JMF CËMÐNÐOEFO LBMBO polinomunun x +JMFCÌMÑNEFOLBMBOLBÀUS PMEVóVOB HËSF 1 Y QPMJOPNVOVO Y JMF CÌMÑ \" - # - $ - % - & -5 NÑOEFOLBMBOLBÀUS \" # $ % & 3. 1 Y =Y3 -Y2 +Y+QPMJOPNVWFSJMJZPS 7. 1 Y QPMJOPNVJ¿JO P (x + 3 ) polinomunun ( x + JMFCÌMÑNÑOEFO 1 Y = 3 P ( -Y +Y2 -Y LBMBOLBÀUS PMEVôVOBHÌSF 1 LBÀUS \" # $ % & \" - 15 # - 13 $ - % - 11 & -5 22 2 4. A K 2 B 8. öCSBIJN6TUBFOJ Y2 +N DN CPZV Y2 -Y- 2 2 G DNPMBOEJLEËSUHFOõFLMJOEFUBIUBUBCBLBMBSEBOCJS E2 F LFOBS Y- DNPMBOLBSFõFLMJOEFQBS¿BMBSLF- TFDFLUJS &MEF FUUJóJ QBS¿BMBS IB[SMBEó ¿FS¿FWF- MFSJ¿JOLVMMBOBDBLUS D CH (x2 + m) cm ôFLJMEF \"#$%CJSLFOBSOOV[VOMVóVYCSWF (x2 – 5x – 6) cm ,#'& CJS LFOBS CS PMBO LBSFEJS $)(' CJS EJL- Usta | |EËSUHFOPMVQ '( =CJSJNEJS %JLEÌSUHFOöFLMJOEFLJUBIUBUBCBLBEBOIJÀBSU 5BSBMBMBOLFTJMJQ¿LBSMELUBOTPOSBHFSJZFLBMBO NBNBTJÀJONLBÀPMNBMES BMBO1 Y QPMJOPNVJMFJGBEFFEJMNFLUFEJS \" - # - $ - % - & -1 #VOB HÌSF P ( x ) polinomunun x - JMF CÌMÑ NÑOEFOLBMBOLBÀUS \" # $ % & 1. C 2. D 3. # 4. \" 23 5. \" 6. D 7. # 8. #
TEST - 9 1PMJOPNMBSEB#ÌMNF÷öMFNJ 1. 1 Y QPMJOPNV Y12 + Y9 + Y3 - QPMJOPNV JMF 5. 1 Y =Y4 -BY3 +Y2 +CY+ 9 CËMÐOEÐóÐOEFCËMÐNWFLBMBOQPMJOPNMBSOOEFSF- QPMJOPNVCJSUBNLBSFPMEVôVOBHÌSF CBöBô DFMFSJFõJUPMNBLUBES dakilerden hangisi olabilir? #VOBHÌSF 1 Y QPMJOPNVOVOEFSFDFTJFOGB[ \" - # - $ % & la kaç olabilir? \" # $ % & 6. 1 Y =YN+ 2 +YN- 4 2. ¶¿ÐODÐ EFSFDFEFO 1 Y QPMJOPNV Y2 + QPMJOP- polinomunun x + 1 ve x -JMFCÌMÑNÑOEFOLB MBOMBS FöJU PMEVôVOB HÌSF N JÀJO BöBôEBLJMFS NVOBUBNCËMÐOFCJMNFLUFEJS EFOIBOHJTJLFTJOMJLMFEPôSVEVS P ( x ) polinomunun x - 3 ve x -JMFCÌMÑNÑO den LBMBOMBSTSBTZMBWFPMEVôVOBHÌSF \" ¥JGUTBZ # 5FLTBZ (x + JMFCÌMÑNÑOEFOLBMBOLBÀUS $ %PóBMTBZ % ¥JGUEPóBMTBZ \" # $ % & & 5FLEPóBMTBZ 3. 1 Y TBCJUPMNBZBOCJSQPMJOPN 7. N O LTGSEBOGBSLMSFFMTBZMBSES 1 1 Y - =1 Y - 5 1 NY2 +OY-L =Y4 +Y3 +NY2 +OY- k + 4 FöJUMJôJOJTBôMBZBO1 Y QPMJOPNVOVOY+ 3 ile QPMJOPNVWFSJMJZPS CÌMÑNÑOEFOLBMBOLBÀUS 1 Y QPMJOPNVOVOY-N+ n +LJMFCËMÐNÐOEFO LBMBO-EJS \" - # - $ % & #VOBHÌSF O+ k -NEFôFSJLBÀUS \" m # - $ % & 4. Y4 -Y3 -Y2 +NY+ n 8. #JS[FZUJOZBóGBCSJLBTOEBHÐOMÐLZBóÐSFUJNJ polinomu x3 + x2 -JMFUBNCÌMÑOEÑôÑOFHÌ Y180 - MJUSFEJS ¶SFUJMFO ZBóMBS UFOFLFMFSF FõJU NJLUBSEBQBZMBõUSMZPS n kaçtS )JÀZBôBSUNBNBTJÀJOCJSUFOFLFEFLJZBôNJL re, UBSBöBôEBLJMFSEFOIBOHJTJolamaz? m \" # $ % - & -4 \" Y9 + # Y9 - $ Y20 + 1 % Y30 + & Y45- 1 1. D 2. \" 3. \" 4. & 24 5. & 6. D 7. D 8. C
1PMJOPNMBSEB#ÌMNF÷öMFNJ TEST - 10 1. 1 Y - 2 P ( -Y =Y3 +Y 5. 1 Y пÐODÐEFSFDFEFOCJSQPMJOPNEVS PMEVôVOBHÌSF 1 - LBÀUS P ( - =1 =1 = 4 PMEVôVOBHÌSF P ( -2 ) +1 EFôFSJLBÀUS \" # - $ - % - & -5 \" # $ % & 2. 1 Y = Y3 - Y+ 5QPMJOPNVJ¿JO 6. #JSGJSNBHÐOMÐL Y3 +BY2 +CY- UBOFÐSÐOÐSF- I. 5FLEFSFDFMJUFSJNMFSJOJOLBUTBZMBSUPQMBN-2 UJZPS #V ÐSÐOMFSJ LPMJMFSF Y + FS UBOF EJ[JODF EJS ÐSÐO Y FSUBOFEJ[JODFÐSÐOFLTJLLBMNBLUBES II. Y6MUFSJNJOJOLBUTBZT-EJS #VOBHÌSF IJÀÑSÑOFLTJLLBMNBNBTJÀJOLPMJ III. ¥BSQBOMBSOEBOCJSJ Y2 +Y+ EJS MFSFEJ[JMFOÑSÑOTBZTBöBôEBLJMFSEFOIBOHJTJ olamaz? JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS \" Y+ # Y+ $ Y- 2 & Y2- 4 \" :BMO[* # :BMO[*** $ *WF** % Y- % *WF*** & * **WF*** 3. 1 Y QPMJOPNVOVOY3 +Y2 -Y-JMFCËMÐNÐO- 7. 1 Y =Y4 -Y2 -Y+ 16 EFOLBMBOY2 +Y-PMEVóVOBHËSF P2 ( x ) poli polinomunun x + 2 JMFCÌMÑNÑOEFOLBMBOLBÀ US nomunun x -JMFCÌMÑNÑOEFOLBMBOLBÀUS \" # $ % & \" # $ % 4 - 5 2 & 4 - 10 2 4. 1 Y =Y3 +BY2 -CY+ 1 8. 1 Y QPMJOPNV CBõLBUTBZTPMBOJLJODJEFSFDF- polinomunun x +JMFCÌMÑNÑOEFOLBMBO EFOCJSQPMJOPNEVS x - JMF CÌMÑNÑOEFO LBMBOO LBU PMEVôVOB HÌSF B-CLBÀUS 1 Y QPMJOPNVOVO TGSMBS WF - PMEVôVOB HÌSF 1 LBÀUS \" - # - $ - % - & -1 \" # $ % & 1. # 2. # 3. C 4. # 25 5. D 6. D 7. D 8. \"
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr 10-÷/0.-\"3*/¦\"31\"/-\"3\"\":3*-.\"4** 0SUBL ¦BSQBO 1BSBOUF[JOF \"MBSBL ¦BSQBOMBSB NOT \"ZSNB OCJSEPóBMTBZPMNBLÐ[FSF TANIM ( a -C 2n = ( b -B 2n 7FSJMFOCJSJGBEFEFPSUBLUFSJNMFSWBSTBJGBEFCV ( a -C 2n + 1 = -( b -B 2n + 1 PSUBLUFSJNMFSJOQBSBOUF[JOFBMOBSBL¿BSQBOMB- SOBBZSMBCJMJS ±SOFóJO BY+CY-DY=Y B+ b -D EJS ÖRNEK 1 ôFLJMEFCJSLFOBSV[VOMVóVNCSPMBOCJSLBSFJMFLFOBS ÖRNEK 3 V[VOMVLMBSNCSWFOCSPMBOEJLEËSUHFO õFLJMEFJTF õFLJM EFLJ EJLEËSUHFO WF LBSFOJO CJSMFõUJSJMNFTJZMF PMV- \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ õBOEJLEËSUHFO¿J[JMNJõUJS a) ( a -C 2 - ( b -B 3 ( a +C b) ( a -C 3 ( b -D 2 + ( c -C 3 ( b -B 2 mn m+n a) (b - a)2 - (b - a)3 (a + b) m mm (b - a)2 (1 - (b - a) (a + b)) (b - a)2.(1 - b2 + a2) ôFLJM ôFLJM b) (a - b)3 (b - c)2 + (c - b)3 (a - b)2 õFLJM EFLJ EJLEÌSUHFOJO BMBOO JLJ GBSLM CJÀJNEF (a - b)2 (c - b)2 (a - b + c - b) ifade edelim. (a - b)2 (c - b)2 (a + c - 2b) ¦Ì[ÑN ôFLJM*EFLJQFNCFLBSFOJOBMBON2 UVSVODVEJLEËSUHFOJOBMBO=NO 5ÐNBMBO=N2+NO ôFLJMEFLJEJLEËSUHFOJOBMBO=N N+O #VJLJJGBEFBZOBMBOCFMJSUUJóJOEFO N2+NO=N N+O EJS ÖRNEK 2 (SVQMBOESMBSBL¦BSQBOMBSB\"ZSNB \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ TANIM a) NO2-N2 n b) N B+C - n ( a +C +Q B+C 7FSJMFO CJS JGBEFOJO UÐN UFSJNMFSJOEF PSUBL CJS c) N-O 2 Q-O + Q-O 2 N-O ¿BSQBO CVMVOBNZPSTB JGBEF HSVQMBOESMBSBL d) ( a -C B- b +D + ( b -B B- b +D IFSHSVQLFOEJJ¿JOEF¿BSQBOMBSOBBZSMS a) 3mn (n - 2m) (SVQMBS PSUBL ¿BSQBO QBSBOUF[JOF BMOEóOEB b) (a + b) (m - n + 2p) JGBEF¿BSQBOMBSOBBZSMNõPMVS c) (m - n) (p -n) (m-n + p - n) = (m - n) (p - n) (m+p-2n) BY+ by + ay +CY= a Y+Z + b Y+Z d) (a - b) (a - b + c) - (a - b) (2a - b + c) = Y+Z B+C (a - b) (a - b + c - 2a + b - c) = (a - b) (-a) 2. a) 3mn (n – 2m) b) (a + b) (m – n + 2p) 26 3. a) (b – a)2.(1 – b2 + a2) b) (a – b)2 (c – b)2 (a + c – 2b) c) (m – n) (p – n) (m + p – 2n) d) (a – b) (–a)
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 4 ÖRNEK 5 \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ a) a2 - ba - a + b a) Y2+Y- 2 b) Y2 +Y- 6 b) 2ab - ac + a - 2b + c - 1 c) Y2 -Y 2 - Y2 -Y - 8 c) YZ [2 + -[ Y2 + y2 d) Y2 -Y- 12 d) 2y -Y-YZ+Y2 e) a3 - 3a2 - a + 3 a) x2 + x - 2 = (x + 2) (x - 1) b) x2 + 5x - 6 = (x + 6) (x - 1) a) a (a - b) - (a - b) = (a - b) (a - 1) c) (x2 - 3x - 4) (x2 - 3x + 2) b) a(2b - c + 1) - (2b - c + 1) = (2b - c + 1) (a - 1) c) xyz2 + xy - zx2 - zy2 (x - 4) (x + 1) (x - 2) (x - 1) d) 15x2 - 11x - 12 = (5x + 3) (3x - 4) zx (yz - x) - y(zy - x) = (yz - x) . (zx - y) d) 2y (1 - x) - 3x(1 - x) ÖRNEK 6 (1 - x) (2y - 3x) Y2+ y2 +Y- 10y + 29 = 0 e) a3 - 3a2-a + 3 PMEVôVOBHÌSF Y+ZLBÀUS a2 (a - 3) - (a - 3) (a - 3) (a2 - 1) 29 u 25 +PMBSBLBZSSTBL x2 + 4x + 4 + y2 - 10y + 25 = 0 (x + 2 ) (x + 2) + (y - 5) (y - 5) = 0 (x +2 )2 + (y - 5)2 = 0 x + 2 = 0 j x = -2 y-5=0jy=5 x + y = 3 bulunur. ax2 CY D õFLMJOEFLJ ÷GBEFMFSJ ¦BSQBOMBSB \"ZSNB TANIM ÖRNEK 7 BY2 +CY+DõFLMJOEFLJпUFSJNMJJGBEFMFSJ¿BS- 2a2 - b2 + ab + 8a - b + 6 QBOMBSOB BZSSLFO CJSJODJ WF пÐODÐ UFSJNJO JGBEFTJOJÀBSQBOMBSOBBZSO[ BY2 =QYUYWFD=NOõFLMJOEF¿BSQBOMBSTF- ¿JMJS 2a2 + ab - b2 - b + 8a + 6 ax2 + bx + c px m 2a * -b ** +2 tx n a +b +3 *** #V ¿BSQBOMBS ¿BQSB[ PMBSBL ¿BSQMQ UPQMBO- EóOEB PSUBODB UFSJN CVMVOVZPSTB ¿BSQBOMBS 2ab - ab =BC * EPóSVTF¿JMNJõUJS#VTF¿JMFO¿BSQBOMBSZBOZB- OBZB[MSTBBY2 +CY+ c = QY+N UY+O -3b + 2b = -C ** ¿BSQBOMBSOBBZSNBJõMFNJTPOMBONõPMVS 6a + 2a =B *** (2a - b + 2) (a + b + 3) bulunur. 4. a) (a – b) (a – 1) b) (2b – c + 1) (a –1) c) (yz – x) . (zx– y) 27 5. a) (x + 2) (x – 1) b) (x + 6) (x – 1) c) (x – 4) (x + 1) (x – 2) (x – 1) d) (1 – x) (2y – 3x) e) (a – 3) (a2 – 1) d) (5x + 3) (3x – 4) 6. 3 7. (2a – b + 2) (a + b + 3)
TEST - 11 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. ( b -B 3 (a -åZ 2 - (a -C 2 (y -B 3 5. ( a -C C+D 2 + ( b -B B-D 2 JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJMFSEFO JGBEFTJOJO ÀBSQBOMBSOB BZSMNö öFLMJ BöBôEB hangisi EFôJMEJS? kilerden hangisidir? \" C-B 2 # B-Z $ C- y \" ( a +C B-C C- a +D # ( a +D C-D C+ a -D % B-C & C+ y $ ( a -D C+D C+ c -B % ( a +C C-D B- b +D & ( b -D B-C B- b -D 2. m3 x2 - m2 x3 m2 x - mx2 JGBEFTJOJO TBEFMFöNJö CJÀJNJ BöBô EBLJMFSEFO hangisidir? \" Y # N $ Y2 % N2 & NY 6. B C DCJSFSHFSÀFMTBZPMNBLÑ[FSF a + b + c =WFab + ac = 30 PMEVôVOB HÌSF b + D UPQMBN BöBôEBLJMFSEFO hangisi olabilir? \" # $ % & 3. 1 + x 7. abc2 + ab - a2 c - b2 c x + 1 1+ 1 a - bc x x2 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # Y $ Y2 % Y2 + & Y2 +Y JGBEFTJOJOFOTBEFöFLMJBöBôEBLJMFSEFOIBOHJ sidir? \" BC-D # C-BD $ D+ ab % BD+C & CD- a 4. x3 + 2x2 - 3x - 6 x2 - 3 JGBEFTJOJOFOTBEFöFLMJBöBôdakilerden hangi 8. Y2 + 5y2 -YZ- 6y + 9 = 0 sidir? PMEVôVOBHÌSF Y+ZUPQMBNLBÀUS \" Y- # Y+ $ Y2 - 2 \" # $ % & % Y2 + & Y2 - 3 1. & 2. & 3. # 4. # 28 5. \" 6. & 7. # 8. C
1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT TEST - 12 1. 272 + 62 5. x2 - 5x +BJGBEFTJOJOÀBSQBOMBSOBBZSMNöI»MJ UPQMBNBöBôEBLJMFSEFOIBOHJTJJMFJGBde edile Y- Y+C bilir? PMEVôVOBHÌSF BCLBÀUS \" # $ \" - # - $ % & & % 2. N- 2 =OJTF N2 -N+ N2 -N 6. m - n = 2 , n + p = 5 ifadesinin O UÑSÑOEFO FöJUJ BöBôEBLJMFSEFO 36 PMEVôVOBHÌSF 18mn - 15n + 18mp ifadesinin hangisidir? EFôFSJLBÀUS \" O2 # O2 - $ O2 - 2n \" # $ % & % O2 - & O2 - 4n 3. 6a2 Y2 -BY+ 1 7. 1 + 1 1 - xa–b 1 - xb–a JGBEFTJOJO ÀBSQBOMBSOEBO CJSJTJ BöBôEBLJMFS JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS den hangisidir? \" - # $ & Yb -Ya \" BY- # BY+ $ BY- 1 % xa xb % BY+ & BY+ 1 4. N5 -N3 +N2 - 18 8. a2 - ab - 2b2 - a2 - ab JGBEFTJ BöBôEBLJMFSEFO IBOHJTJOF UBN CÌMÑ ab + b2 ab - b2 nür? \" N2 - # N2 - $ N2 - 6 JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS % N2 + & N2 + 6 \" B # C $ a % & -2 b 1. D 2. & 3. C 4. \" 29 5. \" 6. \" 7. C 8. &
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr 10-÷/0.-\"3*/¦\"31\"/-\"3\"\":3*-.\"4*** ²[EFöMJL,VMMBOBSBL¦BSQBOMBSB\"ZSNB ÖRNEK 3 TANIM Y+ y =WFYZ= 6 Tam kare PMEVôVOBHÌSF x2 + y2JGBEFTJOJOEFôFSJLBÀUS ( a +C 2 = a2 + 2ab + b2 x + y = 4, x . y = 6 ( a -C 2 = a2 - 2ab + b2 (x + y)2 = 42 ( a + b +D 2 = a2 + b2 + c2 + 2 ( ab + ac +CD x2+ 2xy + y2 = 16 j x2+ y2 + 12 = 16 ( a - b -D 2 = a2 + b2 + c2 + 2 ( -ab - ac +CD j x2 + y2 = 4 bulunur. ÖRNEK 1 ÖRNEK 4 \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ 2m + 1 = 5PMEVóVOBHËSF a) Y2 -Y+ 1 3m b) N2 n2 +NO+ 1 c) x4 + 16 + 8 4m2 + 1 JGBEFTJOJOEFôFri LBÀUS 9m2 x4 d) 16Y + 2Y+ 2 + 4 d 2m + 1 2 = 2 & 2 + 4 + 1 = 25 e) Y6 -Y3 y4 + 4y8 f) -Y3 y2 +Y2 y3 -YZ4 3m n 5 4m 3 2 9m 21 71 & 4m + = bulunur. a) x2 - 2x + 1 = (x - 1)2 2 3 9m b) 4m2n2 + 4mn + 1 = (2mn + 1)2 c) f 2 + 42 2p x x d) (22x + 2)2 e) ( 5x3 - 2y4 ) 2 f) -2y2 x(4x2 - 12yx + 9y2) = -2y2 x (2x - 3y)2 ÖRNEK 5 x - 1 = 3PMEVóVOBHËSF x x + 1 ifadesinin QP[JUJGEFôFSJLBÀUS x ÖRNEK 2 dx- 1 2 & x2 - 2 + 1 =9 öLJTBZOOGBSL LBSFMFSJUPQMBNEVS n = ^ 3 h2 #VTBZMBSOÀBSQNLBÀUS x2 x x - y = 7, x2+ y2 = 99 (x - y)2 = 72 j x2 - 2xy + y2 = 49 2 1 = 11 olur. d x + 1 n=^Ah olsun. j 99 - 2xy = 49 j x.y = 25 bulunur. x &x + 2 x dx+ 1 2 = 2 & 2 + 2 + 1 = 2 x n A x 2 A x j\"2 = 11 + 2 j A = 13 bulunur. 1. a) (x – 1)2 b) (2mn + 1)2 c) ^ x2 + 4 / x2 h 2 d) (22x+ 2)2 30 71 5. 13 e) (5x3-2y4)2 f) - 2y2 x (2x-3y)2 2. 25 3. 4 4. 3
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 6 ÖRNEK 9 N- n -Q=WFN2 + n2 +Q2 =PMEVóVOBHËSF ôFLJMEF \"#() WF (%&' CJSFS LBSF #$%( CJS EJLEËSU- mn + mp -OQJGBEFTJOJOEFôFSJLBÀUS HFOEJS AB C (m - n - p)2 = (8)2 HG D j m2 + n2 + p2+ 2 (-mn - mp + np) = 64 j 72 + 2 . (-mn - mp + np) = 64 j mn + mp - np = 4 bulunur. FE ÖRNEK 7 #V öFLMJO UÑN BMBO CJSJNLBSF WF ÀFWSFTJ CJ SJN PMEVôVOB HÌSF NBWJ CÌMHFOJO BMBO LBÀ CJSJN N- n +Q=WFN2 + n2 +Q2 = 20 karedir? PMEVôVOBHÌSF NO-NQ+OQJGBEFTJOJOEFôF SJLBÀUS yx Tüm alan = x2 + y2+ xy = 52 ¦FWSF= 4x + 4y = 32 (2m - n + 3p)2 = 62 j 4m2 + n2+ 9p2 + 2(-2mn + 6mp - 3np) = 36 y y x+y=8 j 20 + 2(-2mn + 6mp - 3np) = 36 j 2mn - 6mp + 3np = - 8 bulunur. yx (x + y)2 = (8)2 x2 + 2xy + y2 = 64 ÖRNEK 8 x x x2 + y2 = 64 - 2xy 52 = 64 - 2xy + xy xy = 12 bulunur. x x yz ÷LJ,BSF'BSL TANIM :VLBSEBLFOBSV[VOMVLMBSY ZWF[CJSJNPMBOпBEFU Y2 - y2 = Y-Z Y+Z LBSFWFSJMNJõUJS,SN[CËMHFOJOBMBOTBSWFNBWJCËM- HFMFSJOBMBOMBSUPQMBNOEBOCJSJNLBSFGB[MBES ÖRNEK 10 #VÑÀöFLJMZVLBSEBLJHJCJZBOZBOBCJSMFöUJSJMFCJMEJ \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ ôJOFHÌSF Z[ÀBSQNLBÀUS a) Q2 - 50k2 b) N4 - Y+ 2 x2 - (y2 + z2) = 36 ve x=y+z c) N2 - 9n2 j x2 - (x2 - 2yz) = 36 x2 = (y + z)2 d) a4 - 9b2+ 6b - 1 x2 = y2 + z2 + 2yz a) 2(p2 - 25k2) = 2(p - 5k) (p + 5k) j 2yz = 36 ve yz = 18 bulunur. b) (2m2 - 2x - 1) (2m2 + 2x + 1) c) (2m - 3n) (2m + 3n) d) a4 - (9b2 - 6b + 1) = a4 - (3b - 1)2 = (a2 - 3b + 1) (a2 + 3b - 1) 6. 4 7. –8 8. 18 31 9. 12 10. a) 2(p–5k) (p + 5k) b) (2m2 – 2x – 1) (2m2 + 2x + 1) c) (2m–3n) (2m + 3n) d) (a2–3b + 1) (a2+ 3b – 1)
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 11 ÷LJ,ÑQ5PQMBNWFZB'BSL \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ TANIM a) Y4 -Y2+ 25 b) a4 + 64 a3 + b3 = ( a +C B2 - ab + b2 a3 - b3 = ( a -C B2 + ab + b2 a) x4 - 10x2 + 25 - x2 (x2 - 5)2 - x2= (x2 - x - 5) (x2+ x - 5) ÖRNEK 14 b) a4 + 16a2 + 64 - 16a2 \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ (a2 + 8)2 - (4a)2 = (a2 - 4a + 8) . (a2 + 4a + 8) a) Y3+ 8 b) a3 - 27b3 ÖRNEK 12 c) N4 n -NO4 [- y =Y+ y =JTFx2 - 2y2 + z2 + 4 ifadesinin de a) x3 + 23 = (x + 2) (x2 - 2x + 4) ôFSJLBÀUS b) a3 - (3b)3 = (a - 3b) (a2 + 3ab + 9b2) c) 2mn (8m3 - 27n3) (x2 - y2) + (z2 - y2) + 4 2mn(2m - 3n) (4m2 + 6mn + 9n2) (x - y) (x + y) + (z - y) (z + y) + 4 6 (x - y) + 6(z + y) + 4 J z - y = 6 N K O K O K x + y = 6 O ÖRNEK 15 K z + x = 12 O N+ n =WFN2 + n2 = 10 L P PMEVôVOBHÌSF m3 + n3 ifadesinin EFôFSJLBÀUS 6 (x + z) + 4 = 6.12 + 4 = 76 bulunur. ÖRNEK 13 (m + n)2 = (5)2 j m2 + 2mn + n2 = 25 j 10 + 2mn = 25 ,FOBSV[VOMVLMBSYWFYCSPMBOEJLEËSUHFOõFLMJOEF 15 CJSBSTBÐ[FSJOFUBCBOLBSFPMBOCJSLFOBSOOV[VOMVóV j m.n = ZCSPMBOWJMMBMBSEBOCJSNJLUBSJOõBFEJMFDFLUJS 2 y m3 + n3 = (m + n) . (m2 - mn + n2) 6x = (5) . d 10 - 15 n = 25 bulunur. 22 y ÖRNEK 16 8x \"STBZBWJMMBMBSJOõBFEJMEJLUFOTPOSBHFSJZFLBMBOBSTBOO Y+ y =WFYZ= 3 BMBO_ 4 3 x - 2 7 y i _ 4 3 x + 2 7 y iPMEVóVOBHËSF PMEVôVOBHÌSF x6 + y6JGBEFTJOJOEFôFri LBÀUS CVBSTBZBLBÀBEFUWJMMBJOöBFEJMNJöUJS (x + y)2 = (4)2 j x2 + 2xy + y2 = 16 \"STBOOBMBO= 6x.8x = 48x2 br2 j x2+ y2 + 6 = 16 j x2 + y2 = 10 #JSWJMMBOOBMBO= y.y = y2 br2 \"UBOFWJMMBJOöBFEJMTJO (x2 + y2)2 = (10)2 j x4 + 2x2y2 + y4 = 100 (FSJZFLBMBOBSTBOOBMBO= j x4 + y4 + 18 = 100 j x4+ y4 = 82 a 4 3 x - 2 7 y ka 4 3 x + 2 7 y k = a 48x2 - 28y2 k br2 j 48x2 - 28y2 + y2\"= 48x2 x6+ y6 = (x2)3 + (y2)3 = (x2 + y2) (x4 - x2y2 + y4) j 28y2 = y2\"j\"=UBOFWJMMBJOöBFEJMNJöUJS = (10) . (82 - 9)= 730 bulunur. 11. a) (x2 – x – 5) (x2+ x – 5) b) (a2 – 4a + 8) . (a2 + 4a + 8) 32 14. a) (x + 2) (x2 - 2x + 4) b) (a - 3b) (a2 + 3ab + 9b2) 12. 76 13. 28 c) 2mn(2m - 3n) (4m2 + 6mn + 9n2) 15. 25 16. 730 2
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÷LJ5FSJNJO5PQMBNOOWF'BSLOO,ÑQÑ an ± bn#JÀJNJOEFLJ÷GBEFMFS TANIM TANIM ( a +C 3 = a3 + 3a2 b + 3ab2 + b3 n `/JTF ( a -C 3 = a3 - 3a2 b + 3ab2 - b3 an - bn = (a -C Bn - 1 + an - 2 . b + an -3.b2 + ... + bn-1 ÖRNEK 17 n `/WFOUFLJTF an + bn = (a +C Bn-1 - an - 2 . b + an - 3 . b2 - ... + bn - 1 993 + 3 . 992 + 3 . 99 + 1 JöMFNJOJOTPOVDVLBÀUS x = 99 olsun. ÖRNEK 20 993 + 3.992 + 3.99 + 1 = x3 + 3x2 + 3x + 1 \"öBôEBLJJGBEFMFSJÀBSQBOMBSOBBZSO[ = (x + 1)3= (99 + 1)3 = 1003 = 106 bulunur. a) Y5 + y5 b)Y6 + y6 ÖRNEK 18 c) Y7 - 1 6a2b - 2a3 =WFC3 - 6ab2 = 11 PMEVôVOBHÌSF B- b kaçUS a) x5+ y5 = (x + y) . (x4 - x3y + x2 y2 - xy3 + y4) b) x6 + y6 = (x2)3 + (y2)3 6a2b - 2a3 = 5 2b3 - 6ab2 = 11 = (x2 + y2) (x4 - x2y2 + y4) 2b3 - 6ab2 + 6a2b - 2a3 = 16 c) x7 - 1 = (x - 1).(x6 + x5 + x4 + x3 + x2 + x + 1) 2 (b3 - 3ab2 + 3a2b - a3) = 16 (b - a)3 = 8 j b - a = 2 j a - b = -2 bulunur. ÖRNEK 19 ÖRNEK 21 Y- 3 + Y- 2 + Y- - 44 210 + 29 + 28 + ... + 22 + 2 =Y ifadesinin deôFri x = 3 7 JÀJOLBÀUS PMEVôVOBHÌSF 211 in x türünEFOFöJUJOFEJS 1(x4-4 41)434+434.^4x42- 14h42 4+434.^4x4-441 3h + 1 - 45 ((x - 1) + 1)3 - 45 = x3 - 45 x = 2 + 22 + ... + 28 + 29 + 210 x = 3 7 için ^ 3 7 h3 - 45 = 7 - 45 = - 38 bulunur. x + 1 = 1 + 2 + 22 + ... + 28 + 29 + 210 17. 106 18. -2 19. -38 F1 (2 - 1)^ x + 1 h = (<2 - 1) .^ 1 + 2 + . . . + 10 h 2 1 211 - 1 = x + 1 211 = x + 2 bulunur. 33 20. a) (x + y) . (x4 - x3 y + x2 y2 - xy3 + y4) b) (x2 + y2) (x4 - x2y2 + y4) c) (x - 1).(x6 + x5 + x4 + x3 + x2 + x + 1) 21. x + 2
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr UYARI ÖRNEK 23 ôFLJMEFLJCÐZÐLLÐQÐOCJSLF- OBS B CJSJN LпÐL LÐQÐO CJS (ËSEÐóÐNÐ[ Ë[EFõMJLMFSEFO CB[MBSO EÐ[FO- b a LFOBSCCJSJNEJS MFZFSFLBõBóEBLJFõJUMJLMFSJFMEFFEFCJMJSJ[ ,ÑÀÑL LÑQ Lesilip çLBSUM A2 + B2 = ( A -# 2 + 2AB EôOEBHFSJZFLBMBOLTNO A2 + B2 = ( A +# 2 - 2AB IBDNJOJ WFSFO JGBEF BöBô ( A +# 2 = ( A -# 2 + 4AB dakilerden hangisidir? ( A -# 2 = ( A +# 2 - 4AB ( A +# 3 = A3 + 3A2 B + 3AB2 + B3 \" B-C B+C ( A +# 3 = A3 + B3 + 3AB ( A +# # B2 + b2 - 2ab ( A -# 3 = A3 - 3A2 B + 3AB2 - B3 $ B-C B2 + ab + b2 ( A -# 3 = A3 - B3 - 3AB ( A -# % B+C B2 - ab + b2 ÖRNEK 22 & B-C B+C B+C \"#$%LBSFTJOJOCJSLFOBSBCJSJNPMTVO ,BMBOIBDJN= a3 - b3 A Fb B = (a - b) (a2 + ab + b2) PMVQDFWBQ$öLLES b b Eb G D aC ÖRNEK 24 #V LBSFOJO LÌöFTJOEFO LFOBS C CJSJN PMBO &'(# a2 - 4a - 6 =JTFa2 + 36 JGBEFTJOJOEFôFSJLBÀUS LBSFTJBUMSTBHFSJZFLBMBOBMBOWFSFOÌ[EFöMJLBöB a2 ôEBLJMFSEFOIBOHJTJEJS \" B2 - b2 = (a -C B+C a2 - 4a - 6 = 0 j a2 - 6 =BIFSJLJUBSBGBZBCÌMFS # B-C 2 = a2 - 2ab + b2 $ B2 + b2 = ( a +C 2 - 2ab 2 - 6 4a 6 % B3 - b3 = ( a -C B2 + ab + b2 & ^ a – b h = a2 – b2 sek; a = & a - a = 4 tür. a a+b a kalan alan = a2 - b2 da- 6 2 = ^ 4 h2 & 2 - 12 + 36 = 16 = (a - b) (a + b) n a PMVQDFWBQ\"öLLES a2 a 2 36 & a + = 28 bulunur. 2 a 22. \" 34 23. C 24. 28
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 25 ÖRNEK 28 Y2 + 4y2 +Y- 16y + 25 = 0 #JS LFOBS Y CS PMBO LBSF õFLMJOEFLJ CJS LºóU BõBóEBLJ PMEVôVOBHÌSF x + y ifadesinin EFôFSJLBÀUS HJCJLBUMBOZPS (x2 + 6x + 9) + (4y2 - 16y + 16) = 0 ZBSTOO ZBSTOO x karesi karesi x (x + 3)2 + (2y - 4)2 = 0 I x + 3 = 0 j x = -3 y II 2y - 4 = 0 j y = 2 x + y = -1 bulunur. y III ÖRNEK 26 ***õFLJMEFFMEFFEJMFOEJLпHFOEFLFOBSV[VOMVLMBS ZCSPMBOTBSSFOLMJJLJ[LFOBSEJLпHFOLFTJMJQ¿LBSUM- a . b =WF a + b = c JTFa + b - c ifadesinin yor. FöJUJLBÀUS #VOB HÌSF LBMBO L»ôU UBNBNFO BÀMEôOEB BMBO a + b = c & ^ a + b h2 = ^ c h2 OWFSFODFCJSTFMJGBEFBöBôEBLJMFSEFOIBOHJTJEJS j a + 2 7ab + b = c & a + b + 18 = c \" Y-Z Y+Z # Y-Z Y+Z 81 $ Y2- 2y2 % Y-Z 2 j a + b - c = -18 bulunur. & Y-Z 2 x 2 *öFLMJOBMBO 2 2 br ÖRNEK 27 x 2 **öFLMJOBMBO YQPMNBLÐ[FSF 4 2 x + 12 = 19 x br PMEVôVOBHÌSF x + 4 x + 2 sonucu LBÀUS x 2 ***öFLMJOBMBO 8 2 br 2 = 2 & 2 = 2 , 2 = 2 y x 8y 2x 4y x 4BSCÌMHeninBMBO 28 x2 - 4y2 = 0 12 12 (x - 2y) (x + 2y) bulunur. x + = 16 + 3 & - 3 = 16 - x xx 4 4- x \"öLL & 3f - 1 p = 16 - x & 3f p=a4- x k^4+ xh xx 3 & = 4 + x & x + 4 x = 3 olur ve x & x + 4 x + 2 = 3 + 2 = 5 bulunur. 25. -1 26. -18 27. 5 35 28. \"
TEST - 13 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. N3 -N2 -N+ 8 5. Q2 -Q+ 3 = 0 JGBEFTJ BöBôEBLJMFSEFO IBOHJT JOF UBN CÌMÑOF PMEVôVOB HÌSF Q5 BöBôEBLJMFSEFO IBOHJTJOF mez? FöJUUJS \" N- # N+ $ N- 2 \" Q+ # Q- $ Q+ 15 % N+ 2 & N2 - 4 % Q- & -Q 2. x2 + 1 - 38 6. x2 - 4x - 5 x2 x2 - a JGBEFTJOJOÀBSQBOMBSOEBOCJSJTJBöBôEBLJMFSEFO rasyonFM JGBEFTJ TBEFMFöFCJMJS JTF B OO BMBCJMF DFôJEFôFSMFS UPQMBNLBÀUS hangisidir? \" - # - $ % & \" Ym # 1 $ x + 1 + 6 x+6 x % x + 1 - 6 & x - 1 - 6 xx 3. a - b =JTF 7. 263 + 3 . 262 + 3 . 26 = 3n - 1 & b-a PMEVôVOBHÌSF OLBÀUS a-7 b+7 \" # $ % JGBEFTJOJOEFôFSJLBÀUS \" -1 B $ % & 8. x y x (x + y)2 (x – y)2 y 4. \"öBôEBLJMFSEFOIBOHJTJ :VLBSEBUBNLBSFNPEFMMFNFTJHÌTUFSJMNJöUJS ( a2 - b2 + c2 2 - 4a2c2 4 10 A 5B JGBE FTJOJOCJSÀBSQBOEFôJMEJS? #VOBHÌSF \"+#UPQMBNen çokLBÀUS \" B- c -C # B- c +C $ B+ c - b % B+ b +D & B- c \" # $ % & 1. D 2. & 3. # 4. & 36 5. C 6. & 7. C 8. &
1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT TEST - 14 1. Y3 +Y2 y +YZ2 + y3 =JTF 5. 1 - 39 m2 x + m2 y + x + y 1 + 33 + 36 3m2 + 3 JöMFNJOJOTPOVDVLBÀUS JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" - # - $ - % - & - 26 \" - # $ % & 6. a + 1 = 3 JTF a fa- 1 2 2. a2 + ab + bc + c2 =WFC2 + 2ac + ab + bc = 14 a p PMEVôVOBHÌSF a + b +D OJOQP[JUJGEFôFSJ JGBEFTJOJOEFôFSJLBÀUS LBÀUS \" # $ % & \" # $ % & 7. xn mn (x + y)n yn (x + y)m x xm 3. Y= WFZ= JTF y ym x2 + xy + y2 x3 - y3 :VLBSEBNY YO ZN ZOJGBEFTJOJOHFPNFUSJL BMBOHËTUFSJNJPMBSBL Y+Z N+ Y+Z OJGBEFTJ- JGBEFTJOJOEFôFSJBöBôEBkilerden hangisidir? OFFõJUPMEVóVHËTUFSJMNJõUJS \" - # - $ % & #VOBHÌSF Y N+ n ) + y ( m + n ) ifadesinin alan PMBSBLHÌTUFSJNJBöBôEBLJMFSEFOIBOHJTJEJS A) m+n x B) x x x y m+n 4. 9a - 6a - 2 . 4a C) m n D) x + y x y JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJMFSEFO m+n hangisidir? \" a - 2a # a - 2a + 1 $ a + 2a + 1 m n E) % a + 2a – 1 & a + 1 - 2a x 1. # 2. D 3. \" 4. # 37 5. & 6. C 7. #
TEST - 15 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. Y= 2052 - 1952 5. 1 a2 - a3 - 1 PMEVôVOBHÌSF YBöBôEBLJMFSEFOIBOHJTJEJS a + 1 + \" # $ a 1 % & JöMFNJOJOTPOVDVBöBôEBLJMFSden hangisidir? \" -B # B $ & -a - 1 % B+ 2. c - a =WFC+ c = -2 6. Y+ y -[=WFY2 + y2 +[2 = 92 PMEVôVOBHÌSF a2 + b2 - 2c2 ifadeTJOJOEFôFSJ PMEVôVOBHÌSF xy - xz -Z[LBÀUS LBÀUS \" # $ % & \" # $ % & 7. YZ= 2, 1 + 1 = 8 3. x2 + ax - 12 x2 y2 PMEVôVOBHÌSF Y3 + y3BöBôEBLJMFSEFOIBOHJ x2 + 3x - 4 rasyonel ifadesi sadelFöFCJMJSJTFBOOBMBCJMF si olabilir? DFôJEFôFSMFSUPQMBNLBÀUS \" # $ % & \" m # m $ % & 8. ôFLJMEFLJ\"#$%LBSFTJ CJSFWJOCBI¿FTJOJOLVõCB- LõHËSÐOÐNÐEÐS DC EF 4. x4 + 2x3 + x2 + 2x HG x2 + 2x AB JGBEFTJOJOFOTBEFöFLMJBöBôEBLJMFSEFOIBOHJ #VCBI¿FOJOJ¿FSJTJOF&'()LBSFTJõFLMJOEFCJSIB- WV[ZBQMNõUS#BI¿FOJOWFIBWV[VO¿FWSFMFSJUPQ- sidir? MBNNFUSFEJS \" Y+ # Y+ $ Y2 + 1 #BIÀF JMF IBWV[ BSBTOEB LBMBO CÌMHFOJO BMBO JTFNFUSFLBSFPMEVôVOBHÌSF \"#$%CBIÀF % Y2 + & Y2 - 1 sinin çevresi kaç metredir? \" # $ % & 1. C 2. & 3. D 4. C 38 5. C 6. & 7. \" 8. C
1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT TEST - 16 1. Y2 - 2y2 +YZ+Y+ 5y + 3 5. x + 1 = 4 ise x3 + 1 JGBEFTJOJOÀBSQBOMBSOEBOCJSJBY+ by + c oldu x x3 ôVOBHÌSF B+ b +DUPQMBNLBÀUS JGBEFTJOJOEFôFSJBöBôd akilerden hangisidir? \" - # $ % & \" # $ % & 2. 6x2 + mx - 2 a+b = 3 _ bb 2x - 1 6. a2 + b2 + c2 = 25 ` ab + ac + bc = 12bba JGBEFTJ TBEFMFöFCJMJZPS JTF TBEFMFöNJö CJÀJNJ BöBôEBLJMFSEFOIBOHJTJEJS PMEVôVOBHÌSF DOJOBMBCJMFDFôJEFôFSMFSÀBS QNBöBôEBLJMFSEFOIBOHJTJEJS \" Y+ # Y- $ Y+ 1 % Y- & Y+ 2 \" # $ % - & -40 3. a2 - b2 - 4a + 4b =WFB+ b = 5 7. 8 x-4 +2 PMEVôVOBHÌSF B . b LBÀUS 16 x + 2 \" # $ % - & -100 JGBEFTJOJO TBEFMFöUJSJMNJö CJÀJNJ BöBôdakiler den hangisidir? \" 16 x - 2 # 8 x - 42 $ 16 x % 8 x & 4 x 4. BáCPMNBLÐ[FSF 8. ^ 361/8 - 36–1/8 h.^ 361/8 + 36–1/8 h BY- by = (a -C 2WFY- y = 4b 61/2 + 6–1/2 PMEVôVOBHÌSF YJOZEFOCBôNT[PMBOFöJUJ JöMFNJOJOTPO VDVLBÀUS BöBôEBLJMFSEFOIBOHJTJEJS \" B # C $ B- 4b \" 3 # 5 $ 5 % 3 & 1 7 7 6 66 % B+C & B- 2b 1. & 2. & 3. D 4. \" 39 5. # 6. & 7. C 8. #
TEST - 17 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. a4 + b2 - a2 b2 - a2 5. Y2 +Y- 3 =JTF JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJMFSEFO x2 + 9 x2 hangisi EFôJMEJS? JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS \" B2 + b2 # B+C $ B - b \" # $ % & % B+ & B - 1 2. 214 - 28 + 1 6. a3 + b3 =WFBC B+C = 8 210 - 8 JöMFNJOJOTPOVDVLBÀUS PMEVôVOBHÌSF B+CBöBôEBk ilerden hangisi OFFöJUUJS \" 27 # 127 $ 9 % 64 & 8 4 88 77 \" # $ % & 3. 1 a- a-6 + a-4 a-2 7. 1969.1973 + 4 JGBEFTJOJOFöJUJBöBôEBLJMFSd en hangisidir? JöMFNJOJOTPOVDVLBÀUS \" a + 2 # a - 3 $ \" # $ & % a - 4 & 3 a % 4. x – 5 = 4 8. x + 1 =-1 PMEVôVOBHÌSF x x PMEVôVOBHÌSF x + 25 JGBEFTJOJOEFôFSJBöBô x2007 + 1 x x2007 dakilerden hangisidir? ifadesinin EFôFSJLBÀUS \" # $ % & \" # $ % - & -2 1. \" 2. # 3. C 4. & 40 5. \" 6. # 7. # 8. \"
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, 10-÷/0.-\"3*/¦\"31\"/-\"3\"\":3*-.\"4**** %FôJöLFO %FôJöUJSNF :ÌOUFNJ JMF ¦BSQBOMBSB 3BTZPOFM÷GBEFMFSJ¦BSQBOMBSB\"ZSNB \"ZSNB TANIM TANIM 1 Y WF 2 Y CJSFS QPMJOPN WF 2 Y á PM- #JS JGBEFZJ CBõLB CJS EFóJõLFOMF EFóJõUJSFSFL EBIBCBTJUCJSJGBEFFMEFFUNFZFWFCVZËOUFN- NBLÐ[FSF P_ x i õFLMJOEFLJJGBEFMFSFrasyo MF JGBEFZJ ¿BSQBOMBSB BZSNBZB EFôJöLFO EF ôJöUJSNF ZÌOUFNJ JMF ÀBSQBOMBSB BZSNB de- Q_ x i OJS nel ifadelerEFOJS ÖRNEK 4 ÖRNEK 1 6a2b2 - 54b2 3a3b - 18a2b + 27ab 397 . 385 – 399 . 383 JöMFNJOJOTPOVDVLBÀUS SBTZPOFMJGBEFTJOJFOTBEFöFLJMEFZB[O[ 383 = y, 397 = x olsun. 397 . 385 - 399 . 383 = x.(y + 2) - (x + 2) y 2 2 - 2 6b 2 a 2 - 9 k j xy + 2x - xy - 2y = 2x - 2y = 2.(x - y) j 2 (397 - 383) = 2.14 = 28 bulunur. 6a b 54b a 3 2 = 3a b - 18a b + 27ab 3aba a2 - 6a + 9 k 2b^ a - 3 h^ a + 3 h 2b^ a + 3 h = = bulunur. a^ a - 3 h a^ a - 3 h2 ÖRNEK 2 ÖRNEK 5 f a + 2 . 1 - m2 p : m2 - 2m - 3 Y2 +Y 2 - Y2 +Y - 3 m–1 4 - a2 a2 - 3a + 2 JGBEFTJOJÀBSQBOMBSOBBZSO[ JöMFNJOJOTPOVDVOVFOTBEFöFLJMEFZB[O[ x2 + 2x = t diyelim. a + 2 ^ 1 - m h^ 1 + m h ^ a - 2 h.^ a - 1 h a - 1 t2 - 2t - 3 j (t - 3) (t + 1) ··= j (x2 + 2x - 3) (x2 + 2x + 1) j (x + 3) (x - 1) (x + 1)2 bulunur. m - 1 ^ 2 - a h^ 2 + a h ^ m - 3 h^ m + 1 h m - 3 bulu -1 -1 nur. ÖRNEK 3 ÖRNEK 6 20083 + 20073 - 2008 . 2007 + 1 > x + a - x - ax + bx H : a2 - 4x2 4015 ab ab b2 JöMFNJOJOTPOVDVLBÀUS iöMFNJOJOTPOVDVOVFOTBEFöFLJMEFZB[O[ 2008 = x , 2007 = y olsun. R V 2 2 Sx + a-x - ax + bx W : a - 4x b ab W x 3 + 3 ^ x + y h a 2 - xy + y 2 k S a 2 S ^ah y x W b x+y - x.y + 1 = - x.y + 1 SS ^ b h WW ^x+yh TX j x2- xy + y2- xy + 1 = (x2- 2xy + y2) + 1 xb + 2 - 2ax - b x 2 j (x - y)2 + 1, yerine yazarsak; j (2008 - 2007)2 + 1 = 12 + 1 = 2 bulunur. a b · ab a2 - 4x2 a ^ a - 2x h b2 b · = bulunur. a b ^ a - 2x h^ a + 2x h a + 2x 1. 28 2. (x + 3) (x - 1) (x + 1)2 3. 2 41 2b^ a + 3 h a-1 b 4. 5. 6. a^ a - 3 h m-3 a + 2x
·/÷7&34÷5&:&)\";*3-*, 4. MODÜL 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" www.aydinyayinlari.com.tr ÖRNEK 7 ÖRNEK 10 x2 - 6x + 5 Y= 7 , y =J¿JO x2 - a 1 rasyonel ifadesi sadFMFöFCJMEJôJOF HÌSF B ZFSJOF 3 x2 + 3 xy + 3 y2 HFMFCJMFDFLTBZMBSOUPQMBNLBÀUS ifadesiOJOFöJUJLBÀUS 2 ^ x - 5 h^ x - 1 h 3 x-3 y = & a = 25 ise x - 6x + 5 2 2 1 = x -a x -a 3 x2 + 3 xy + 3 2 y ka 3 2 y2 k x2 - 25 = (x - 5) (x + 5) olur. a3 x-3 + 3 xy + 3 y x 14444444442 4444444443 a = 1 ise x2- 1 = (x - 1) (x + 1) olur. a3 x-3 yk x–y BOOEFôFSMFSUPQMBN+ 1 =ES 3 x-3 y 3 7-3 3 3 7-3 3 x - y = 7 - 3 = 4 bulunur. ÖRNEK 8 ÖRNEK 11 48 2017.1985 – 1981.2021 a2 - 8a + 24 JGBEFTJOJOTPOVDVLBÀUS ifadesinin en büyük dFôFSJLBÀUS 1981 = x , 2017 = y olsun. bulu y.^ x + 4 h - x.^ y + 4 h = x y + 4y - x y - 4x (a2 - 8a + 16) + 8 = (a - 4)2+ 8 ifadesinin en küçük de ôFSJB= 4 için (a - 4)2 + 8 = 8 olur. = 4^ y - x h = 4.^ 2017 - 1981 h = 4.36 = 12 48 nur. = 6 FOCÑZÑLEFôFSJEJS 8 ÖRNEK 9 ÖRNEK 12 Y2 + y2 +Y– 2y + 22 A = ^ 16 5 + 1 h ^ 8 5 + 1 h^ 4 5 + 1 h^ 5 + 1 h ifadesinin en küçükEFôFSJLBÀUS PMEVôVOBHÌSF 16 5 JO\"UÑSÑOEFOFöJUJOJCVMVOV[ (x2+ 8x + 16) + (y2- 2y + 1) + 5 &öJUMJôJOIFSJLJUBSBGO^ 16 5 - 1 h ile çarparsak; = (x +4 )2 + (y - 1)2 +JGBEFTJOJOFOLÑÀÑLEFôFSJ ^ 16 5 - 1 hA = ^ 16 5 - 1 h^ 16 5 + 1 h ^ 8 5 + 1 h^ 5 + 1 h^ 5 + 1 h x = -4, y = 1 için 02 + 02+ 5 olur. 8 5-1 4 5-1 5-1 (5 - 1) ^ 6 5 - 1 h.A = ^ 5 - 1 h 6 5 = 4 + 1 = 4 + A bulunur. AA 7. 26 8. 6 9. 5 42 3 7-3 3 4+A 10. 11. 12 12. 4A
www.aydinyayinlari.com.tr 10-÷/0.-\"37&¦\"31\"/-\"3\"\":*3.\" 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 13 ÖRNEK 16 A = ( 5 + 2 + 4+ 8 + Y>ZJ¿JO PMEVôVOBHÌSF 16TBZTOO\"UÑSÑOEFOFöJUJOJCV lunuz. 4x f -2 + 4x + 4y p &öJUMJôJOIFSJLJUBSBGO - 1) ile çarparsak; 4–y 4y 4x JGBEFTJOJOFöJUJOJCVMVOV[ ^ 5 - 1 h.A = ^1 45 4-414h2.^ 54+4 414h3.a 2 + 1 k.a 5 4 + 1 ka 5 8 + 1 k 5 x y2 xy b 45 24–414l 4 x y .f 2 2 p = xy 2 2 - - 1 4 4 4 2 4 4 4 4 4 44 3 .4 yx 2 .2 . yx b 5444–144l2 22 142442+ 42443 1 4 4 4 4 4 4 8 4 4 4 4 4 4 4 4 44 3 1 4 4 4 4 4 4 4 4 454–4142 4 4 4 4 4 4 4 4 4 4 443 ^ x > y için h 516 - 1 2 xy 2 = x .2 y.f - p 16 16 yx 2 4.A = 5 - 1& 5 = 4A + 1 bulunur. 22 = (2x)2 - (2y)2 = 4x - 4y bulunur. ÖRNEK 14 215 + 28 + 2n JGBEFTJ CJS UBN LBSF PMEVôVOB HÌSF O EPôBM TBZT LBÀUS 28 = (24)2 olarak yazarsak; (24)2 + 4 + ? 2 >2.2 . ? ÖRNEK 17 215 = 25. ? Y+ y =WFYZ=JTF ? = 210PMNBMES O halde ?2 = 220PMNBMESn = 220PMEVôVOEBO 1+1 n = 20 bulunur. x4 y4 JGBEFTJOJOEFôFSJLBÀUS x+y 3 xy 11 = & xy + xy = 3 & y + x =3 x.y 1 ÖRNEK 15 &d 1 + 1 2 1 + 1 + 2 =9 Y3 +Y2 +Y- 6 JGBEFTJOJÀBSQBOMBSOBBZSO[ xy n = ^ 3 h2 & 2 2 6xy x3 + 3x2+ x2 + x - 6 x y 1 x2 (x + 3) + (x + 3) (x - 2) (x + 3) (x2+ x - 2) 11 & + = 7 bulunur. (x + 3) (x + 2) (x - 1) bulunur. 22 xy f 1 1 2 & 121 + + + = 49 2 p = ^ 7 h2 2 4 22 4 xy x =x y y 1 11 & + = 47 bulunur. 44 xy 13. \"+ 1 14. 20 15. (x + 3) (x + 2) (x - 1) 43 16. 4x - 4y 17. 47
TEST - 18 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT y-x x+y a- a 5. a + 3 a2 - 9 · a + 3 1. + x+y x-y 1 -1 3 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOhangisidir? a-3 \" # $ -1 JöMFNJOJOTPOVDVBöBôd akilerden hangisidir? 4xy 2xy \" - 1 # - a $ a % -B & B % & 3 33 x2 - y2 x2 - y2 2. a^ b2 + 1 h - b^ a2 + 1 h 9y2 - 1 - 24xy + 16x2 a2b2 - 1 6. 4x + 1 - 3y JGBEFTJOJO TBEFMFöNJö CJÀJNJ BöBôEBLJMFSEFO JGBEFTJOJOFöJUJBöBôEBLJlerden hangisidir? hangisidir? \" a + b # b $ a + b \" Z-Y- # Y- 3 ab a ab - 1 $ Y- 3y - % Y- 3y & - b % b - a a & Y- y ab + 1 3. 4 + 3 + x + x 7. \"öBôEBLJMFSEFOIBOHJTJB4 + 3a2 + 4 ifadesinin x-2 x+3 2-x ÀBSQBOMBSOEBOCJSJTJEJS JöMFNJOJOTPOVDVBöBôd akilerden hangisidir? \" +Y # Y- $ -Y \" B- # B+ $ B- 1 & B2 - a + 2 % 2 & Y % B+ x -2 1+ 2 8. 32 4. x : 1 x2 + 14x + 53 x - 4 2x - x2 ifadesinin en büyükEFôFSJLBÀUS x JöMFNJOJOTPOVDVBöBôEBLJMFSd en hangisidir? \" - # $ Y % Y- & -Y \" # $ % & 1. D 2. D 3. D 4. & 44 5. # 6. C 7. & 8. C
1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT TEST - 19 1. 1 - z2.t2 zt + 1 5. f x + 3 : y2 - y - 6 p : 4 - y2 : z- 1 zt y - 2 xy - 2y - 3x + 6 4x2 + 4x - 24 t JGBEFTJOJOFOTBEFöFLMJBöBôdakilerden hangi ifadesiOJOFOTBEFöFLMJBöBôEBLJMFSEFOIBOHJ sidir? sidir? \" [ U # - [ U2 $ - [ U % - [ & [ U2 \" - # - 1 $ 1 4 4 % -Y- 3 & x + 3 x-2 2. f x2 - x3 p : x2 6. x5 - x4z - xy4 + y4z x · x + 1 x - 1 x2 - 1 x4 - x3z - x2y2 + xy2z x2y + y3 JöMFNJOJOTPOVDVBöBôd akilerden hangisidir? ifadesinin en TBEFöFLMJBöBôEBLJMFSEFOIBOHJ sidir? \" - (1 +Y2 # +Y2 $ - 1 - Y # 1 $ x y y % +Y & \" % Y2 - y2 & Y 3. 2x2 + 5x - 12 · 6x2 - x - 15 7. Y2 +Y+ y2 - 4y + 25 = 0 3x3 + 7x2 - 20x 4x2 - 9 4 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" - # - Y $ Y % & 1 PMEVôVOBHÌSF Y+ 3y ifadesinin FöJUJBöBô x dakilerden hangisidir? \" - # $ % & ^ a + 8 h.f 1 - 64 p 8. a4 + 4b4 a2 4. 1 + 16 + 64 a a2 JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJlerden hangisidir? JöMFNJOJOTPOVDVBöBôEBLJMFSd en hangisidir? \" B2 - 2b2 + # B2 + 2b2 + 2ab \" - # $ B- 8 $ B2 + 2b2 + % B2 + b2 + 2ab & 8 % 1 & B2 - b2 + 2b 8-a a 1. # 2. \" 3. & 4. C 45 5. # 6. # 7. C 8. #
TEST - 20 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. a = WFC= PMEVóVOBHËSF 5. a4 + b2 - a2 b2 - a2 ^ a + b h2 - 4ab JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJMFSEFO 5a + 5b hangisi EFôJMEJS? ifadesiOJOFöJUJBöBô EBLJMFSEFOIBOHJTJEJS \" B2 + b2 # B+C $ B- b \" # $ % B+ & B- 1 % & 2. f x2 - 4 · 3x2 - 5x + 2 p : 3x2 + 4x - 4 x2 + x - 6 x2 + x - 2 x2 + 5x + 6 ifadesiOJO FO TBEF CJÀJNJ BöBôEBLJMFSEFO IBO 6. A = 292 - 112 ve B = 192 + 39 gisidir? PMEVôVOBHÌSF \"-#LBÀUS \" x - 1 # Y+ $ 3x - 1 \" # $ % & x+2 x-3 % & -1 3. a4 + 2a2 + 9 7. _ 8 7 + 1 i _ 4 7 + 1 i _ 7 + 1 i = x JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôdakilerden PMEVôVOBHÌSF _ 8 7 - 1 i JOYDJOTJOEFOEFôFSJ hangisidir? BöBôEBLJMFSEFOhangisidir? \" B2 + 2a - # B2 - 2a - 3 \" 6 # 4 $ 2 % Y & Y x x x $ B2 - 2a + % B2 + a - 3 & B2 - a + 3 x4 + x2y2 + y4 8. \"öBôEBLJMFSEFOIBOHJTJ 4. 1002 . 1004 . 1006 . 1008 x2 + y2 - xy ÀBSQNOEBOEBIBCÑZÑLUÑS \" 10032 . 10072 JGBEFTJOJO TBEFMFöNJö CJÀJNJ BöBôEBLJMFSEFO # 10012 . 10092 hangisidir? $ 1001 . 10052 . 1007 % 1001 . 10052 . 1009 \" Y2 + y2 +YZ # Y2 + y2 -YZ & 1001 . 1005 . 1007 . 1009 $ Y2 + y2 -YZ % Y2 -YZ- y2 & Y2 +YZ- y2 1. D 2. D 3. C 4. \" 46 5. \" 6. \" 7. \" 8. \"
1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT TEST - 21 1. 4a2 - b2 - 4a + 4b - 3 5. a4 + 4 2a - b + 1 JGBEFTJOJO ÀBSQBOMBSOEBO CJSJ BöBôEBLJMFSEFO hangisidir? JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" B2 - 2a + # B2 - a + \" B+ b - # B- b + $ B+ 1 $ B2 - 2a + % B2 - a + % B- & C- 3 & B2 + a + 2. a3 + 2a2 + 4a : a3 - 8 6. YWFZCJSFSEPóBMTBZES a2 + 4a + 4 4 - a2 Y2 + 4y = y2 +Y- 17 JöMFNJOJOTPOVDVBöBôd akilerden hangisidir? PMEVôVOBHÌSF YJOBMBCJMFDFôJGBSLMEFôFSMFSJO UPQMBNLBÀUS \" a # - a $ a \" # $ % & a+2 a+2 a-2 % - a & B a-2 x-y y y 7. x - 5 = 8 - -1 x-3 x x+y x 3. : x+y x +1 PMEVôVOB HÌSF (x - 3) 2 + 25 ifadesinin - x (x - 3) 2 y x-y y EFôFSJLBÀUS JöMFNJOJO FO TBEF CJÀJNJ BöBôEBLJMFSEFO IBO \" # $ % & gisidir? \" x # - x $ % - & Y y y 4. f x2 - 36 · x2 + 3x - 10 p : x2 + 4x - 12 8. Y2 -Y+ 1 = 0 x3 + 125 3x - 18 x2 - 5x + 25 PMEVôVOBHÌre, x2 - 1 ifadesinin pozitif de x2 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS ôFSJLBÀUS \" - # 1 $ % & \" 5 21 # 13 $ 17 3 % 4 7 & 15 1. \" 2. # 3. D 4. # 47 5. \" 6. & 7. \" 8. \"
TEST - 22 1PMJOPNMBSO¦BSQBOMBSB\"ZSMNBT 1. 4m2 - 1 5. x - 2 = 41 x (m + n) 2 - (m - n + 1) 2 JGBEFTJOJO TBEFMFöUJSJMNJö CJÀ JNJ BöBôEBLJMFS PMEVôVOBHÌSF x + 2 in po[JUJGEFôFSJLBÀUS den hangisidir? x \" # $ % & \" 2m + 1 # 2m - 1 $ 1 2n + 1 2n - 1 2n - 1 % 1 & 2m + 1 2. a2 - 1 · a4 + a2 + 1 6. x2 + 1 = 7 a3 - 1 a3 + 1 x2 PMEVôVOB HÌSF x3 + 1 ifadesinin pozitif de JöMFNJOJOTPOVDVBöBôEBLJlerden hangisidir? x3 \" B- # B+ $ B2 - a + 1 ôFSJLBÀUS \" # $ % & % B2 + a + & 3. 2a3 - 11a2b - 21ab2 : a2 - 3ab - 28b2 7. Y`;PMNBLÐ[FSF 4a2 - 9b2 2a2 + 5ab - 12b2 Y+ Y+ Y+ Y+ - 24 = 0 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS EFOLMFNJOJ TBôMBZBO Y EFôFSMFSJOJO UPQMBN \" B-C # -B $ BöBôEBLJMFSEFOIBOHJTJEJS \" - # - $ % & % B & C- 3a 4. f ^ x + y h2 - z2 . x2 + xy + xz p: x2 - xy + xz 8. a ve b ` Z ve 1 < n < 110 olmak üzere, ^ x + y + z h2 ^ x - z h2 - y2 ^ x - y h2 - z2 Y2 +Y- n = Y-B Y-C ifadeTJOJOFOTBEFöFLMJBöBôEBLJMFSEFOIBOHJ sidir? FöJUMJôJOJTBôMBZBOLBÀGBSLMOUBNTBZTWBS ES \" # Y $ Y+ y -[ % Y- y -[ & Z \" # $ % & 1. # 2. & 3. D 4. \" 48 5. D 6. C 7. \" 8. &
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