TON THAN (Chu bien) SUI VAN TUY~N - NGUY~N DlJC TRUONG CAc chuven ae CHON LOC.I" TOAN Ti:\PMQT NHA XUAT BAN GIAO DL,JC VI~T NAM

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1 TON THAN (Chu bien) SUI VAN TUY~N - NGUY~N DlJC TRUONG CAc chuven ae CHON LOC.I" TOAN Ti:\PMQT NHA XUAT BAN GIAO DL,JC VI~T NAM

2 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} CDTO AN8TAP1 16

3 D~ giup cac em hqc sinh hqc t~p tot mon Toan 6 Trung hqc ca so (THCS) hi~n nay va 6 Trung hqc ph6 thong (THPT) sau nay, chung t6i bien SO<:ln b sach gom 8 cuon: "Cue chuyen de' cltf?lllf?c To/m 6, 7, 8, 9 t(tp mc)t l'cl t(tp lwi ". Moi cuon trong b sach c6 cac chuang tuang ung vai cac chuang trong sach giao khoa Toan. Cac chuang deu duac viet theo nhilng chuyen de (ca bim va nang cao) rna cac tac gia cho ding d6 Ia nhfrng chuyen de can thiet cho vi~c hqc va hi~u sau kien thuc cua chuang. Moi chuyen de gom ba phan: A. Kie/1 tlurc ujn nit&: Phan nay dua ra nhfmg kien thuc ca ban, nhilng kien thuc b6 sung can thiet d~ c6 th~ giai duqc cac bai t~p, cac d':lng toan cua chuyen de. B. Mc)t so' l'i dt.t : Phan nay trinh bay nhilng vi d1:1 chqn IQc minh hqa cho nhilng dang toan di~n hlnh cua chuyen de vai each trlnh bay IOi giai chu~n muc kern theo nhilng nlt~j/1 xh, /t(/1 y, hinlt lu(tn,... ve phuang phap giai, Ve cac sai lam hqc sinh CO th~ mac, Ve vi~c tlm toi them cac each giai khac,... Nhieu vi du 6 ph an nay duqc trich trong cac de thi hoc sinh gi6i Toan 6 THCS, trong cac de thi vao lap 10 THPT chuyen. C. Bui t(tp: Phan nay dua ra h~ thong cac bai t~p duqc phan lo':li theo cac d':lng toan d~ hqc sinh de Slr dl:'ng. H~ thong cac bai t~p nay kha da d':lng, bao gom cac bai t~p ca ban va cac bai t~p nang cao cho hqc sinh kha, gi6i. Nhieu bai duqc trich tlr cac de thi hqc sinh gi6i Toan 6 trong va ngoai nuac. Moi cuon sach deu cung cap m t soluqng Ian cac bai t~p vm huang dan giai kha chi tiet, minh hqa cho phuang phap gi:h cac d<:1ng toan, cac chuyen de dade c~p. Cuoi sach la phan Hl((Jng dan gidi - Dup so' cho cac bai t~p 6 cac chuyen de. Qua nhfmg huang dan giai Cl:l th~, hqc sinh se nam ro han each giai cho moi d':lng toan. Cac kien thuc trong moi cuon sach duqc sap xep tu cte den kh6, duqc trlnh bay dan gian, de hi~u, dap U'ng cho nhieu doi tuqng hqc sinh. 3

4 Oic tac gia cua b(> sach Ia nhfrng thay co giao da c6 nhi~u kinh nghi~m trong vi~c giang d~y, boi du6ng h<;>c sinh gioi Toan 6 THCS, d6 Ia cac thay co giao: PGS.TS NGND Ton Than (Chu bien b(> sach), NGVT Bui Van Tuyen, NGVT Nguyen Ng<;>c D~m, Ths. Nguyen Due Truang, Ths. Nguyen Due Tan, Ths. Nguyen Anh Hoang, Ths. D~ng Van Quan, Ths. Ph~m Thj L~ Hang. M~c du da c6 nhi~u co gang, song b(> sach kh6 tranh khoi nhgng thieu s6t. Cac tac gia rat mong nh~n duqc thu g6p y cua cac em h<;>c sinh, cac thay co giao va cac bac phl_l huynh. Moi y kien xin glri v~ theo dja chi :Bun To/m -Tin, Nlu1 xu/it hdn Giuo d1_1c Viet Num - 187B Gidng Vo -Hd N(Ji. Hi v<;>ng rang, b(> sach se Ia tai li~u tham khao thiet thl_l'c, hfru fch d6i v6'i cac em h<;>c sinh THCS, cac thay co giao d~y Toan va b~n d<;>c yeu thfch Toan. H 1 Ni)i, thling 3 nclm 2013 cac TAc GIA 4

5 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} Y/Jhuang I PHEP NHAN VA PHEP CHIA cac DA TH(/C Chuyendil PHEPNHANCACDATHUC A. KIEN THUC CAN NHO 1. Muon nhan m(>t don thuc v6i m(>t da thuc ta nhan don thuc v6i ttrng h;~mg ti1 cua da thuc r6i c(>ng cac tfch v6i nhau. 2. Muon nhan m(>t da thuc v6i m(>t da thuc, ta nhan m6i h~ng ti1 cua da thuc nay v6i ttrng h;:tng tlr cua da thtrc kia r6i c6ng cac tfch v6i nhau. -..!, B. MQT SO VI Dl) Vi dt,~ 1. Cho bon so, so sau han so tru6c Ia 2. Chtrng minh rang hi~u cua tfch hai so 0 gifra va tfch cua so dau v6i so cuoi lu6n kh6ng d6i. Gidi. G9i bon soda cho la X, X + 2, X + 4 va X + 6. Hi~u cua tfch hai so 0 giua va tfch cua so dau v6i so cuoi la : (x + 2)(x + 4)- x(x + 6) = x 2 + 4x + 2x + 8- x 2-6x = 8 (kh6ng d6i). Vi dt,~ 2. Cho m so, m6i so bang 3n + 1 va n so, m6i so bang 9-3m. Biet t6ng tat d cac so d6 bang 5 Ian t6ng m + n. Chtrng minh rang m = n. Gidi. T6ng cua m so (3n + 1) v6i n so (9-3m) Ia m(3n + 1) + n(9-3m). Theo de bai ta c6 m(3n + 1) + n(9-3m) = 5(m + n) ~ 3mn + m + 9n - 3mn = 5m + 5n ~ 9n - 5n = 5m - m ~4n =4m ~n =m. 5

6 Vi dt,~ 3. Tfnh t6ng cac h~ so cua luy thtra b~c ba, luy thtra bac hai va luy thtra b~c nhat trong ket qua CUa phep nhan (x 2 +X + 1 )(x 3 - X+ 1 ). Gidi Ta c6 (x 2 + x + 1 )(x 3 - x + I) =x -x +x +x -x +x+x -x =X +X+. H~ so cua luy thtra b~c ba Ia 0, h~ so cua luy thtra b~c 2 Ia 0, he so cua luy thtra b~c nhat Ia 0 nen t6ng cac h~ so nay bang 0. Vi dt,~4. Cho M = (x + a)(x 2 + bx + 16) van= x a) Viet bi~u thuc M du6i d~g mot da thuc thu g<;m theo luy thtra giam dan cua X. b) V6i gia tr! nao cua a va b thl hai da thuc M van 1u6n c6 gia tr! btmg nhau v6i m<;>i gia trj CUa X? Gidi a) Ta c6 M = (x + a)(x 2 + bx + 16) = x 3 + bx x + ax 2 + abx + 16a = x 3 +(a+ b)x 2 + (ab + 16)x + 16a. b) M = N v6i m<;>i gia trj cua X <=> x 3 +(a+ b)x 2 + (ab + 16)x + 16a = x 3-64, Vx a+ b = 0 a- -4 <=> ab + 16 = 0 <=> { - { b = 4. 16a = -64 Nhfj.n xet: Hai da thuc viet du6i d(;lng thu g<;>n c6 gia tr! bang nhau v6i m<;>i gia tr! cua bien khi va chi khi cac h so cua cac luy thtra cung b~c bang nhau. Vi dl,l 5. Cho bi~u thuc A= (4m- l)(n- 4)- (m- 4)(4n- 1). Chung minh rang A : 15 v6i m<;>i gia tr! nguyen cua m va n. Gidi A=(4m-l)(n-4)-(m-4)(4n-l) = 4mn - 16m - n ( 4mn - m - 16n + 4) = 4mn - 16m - n + 4-4mn + m + 16n - 4 =-15m+ 15n = -15(m- n) : 15. Vi dl,l 6. Cho bon so nguyen lien tiep khong chia het cho 5, khi chia cho 5 duqc nhfrng so du khac nhau. Cht!ng minh rang hi~u cua tfch hai so cuoi v6i tich cua hai so dau Ia m(>t so c6 t~n cung dung m(>t chfr so 0. 6

7 Gidi. G<;>i bon so nguyen lien tiep khong chia het cho 5, khi chia cho 5 duqc nhfrng so du kh:k nhau Ian Iuat Ia 5k + 1, 5k + 2, 5k + 3, 5k + 4 (k E Z). Ta c6 (5k + 3)(5k + 4) - (5k + 1)(5k + 2) = (25k k + 15k + 12)- (25k k + 5k + 2) = 25k k k 2-15k - 2 = 20k + 10 = 1 0(2k + I). Vi 2k + I Ia m(>t so le nen I 0(2k + I) c6 t~n cung boi dung m(>t chfr so 0. c. BAI TAP 1.1. Viet cac bieu thuc sau du6'i dl;lng da thuc sap xep theo Iuy thua giam cua bien x a) (3x + a)(2x - Sa) - 6a(2x - a) ; b) (9x- 5y)(2x + 7y)- (4x + 3y)(8x- y) Cht1ng minh ding thuc (x + a)(x +b)= x 2 +(a+ b)x + ab. Ap dyng tfnh nhanh : a) (x + 5)(x + 2) ; c) (x + 8)(x - 3); b) (x - 7)(x - 4) ; d) (x- 9)(x + 1) Cho da thuc A= x 2 + Ilx + m trong d6 m Ia m(>t so nguyen duong. Tim gia trt nho nhat cua m, gia tr! 16'n nhat cua m de da thuc A Ia tfch cua hai da thuc v6'i h~ so nguyen Xac d!nh cac h~ so a, b, c biet rang v6'i m<;>i gia trt cua X thl : a) (5x- 3)(2x- c)= ax 2 + bx + 21 ; b) (ax+ 4)(x 2 + bx- 1) = 9x x x +c. 1 5 Ch b. h, A :1 3 n+ I ( n-1 n) n( n+ I n) 3 d, ~1 *.. o 1eu t uc = x x - y + y x - y trong o n H- E 1 ~ ay thu g<;>n bieu thuc A de cht1ng to ding khi thay cac gia trt cua X va y boi cac so doi cua chung thl gia trt cua bieu thuc A v:in kh6ng d6i M(>t khu dat hlnh chfr nhc:;tt c6 chu vi Ia 100 m. Neu chi~u dai va chi~u r(>ng cung giam di a (met) trong d6 a < 50 thl di~n tich khu dat nay giam di bao nhieu met vu6ng? Ch o x + y = 2, c h' ung mm h dang z t h' uc : 2(x + I )(y + 1) = (x + y)(x + y + 2). 7

8 1.8. Cho biet (x + y)(x + z) + (y + z)(y + x) = 2(z + x)(z + y). Cht!ng minh rang 2 x2 +/ z = -----= Tfnh gia tri cua bi~u thuc sau bang each hqp If : a) A= x 5-70x 4-70x 1-70x 2-70x + 29 t(;li x = 71; b) B = x 5-36x x 1-69x 2-34x + 15 tai x = Cho bi~u thuc A= 3(x 2 + x + 2)(x 1 - x 2 - x +I) Hay thi,i'c hi~n phep nhan roi viet ket qua theo luy thua giam dan cua X. Cho biet h~ so cua luy thua bac 4, cua luy thua b~c 3, cua luy thua b~c 2 trong ket qua Cht!ng minh rang gia tri cua cac bi~u thuc sau khong phy thu(>c vao gia tri CUa bien X: a) A = ( 4x - 5)(2x + 3) - 4(x + 2)(2x - I) + (I Ox + 7) ; b) B = (7x- 6y)(4x + 3y)- 2(14x + y)(x- 9y)- 19(13xy- 1) Tim x, biet: a) 4x(5x + 2) - (lox- 3)(2x + 7) = 133; b) 3(6x - 5)(4x + 1) - (8x + 3)(9x - 2) = Cho bi~u thuc B = (n - I )(n + 6) - (n + I )(n - 6). Cht!ng minh rtmg v6'i m<;>i gia tri nguyen cua n thl B : Cho ba so nguyen lien tiep. L~p cac tfch cua hai trong ba so d6. Biet tong cua ba tfch nay Ia 242. Tim ba so nguyen d6. Gqi y: G<;>i so nguyen a gifra Ia a Cho bi~u thuc P = (x- a)(x- b)(x- c), trong d6 : a+ b + c = 12; ab +be+ ca = 47 va abc= 60. a) Hay viet P du6'i dl;lng m(>t da thuc thu g<;>n, sap xep theo luy thua giam cua bien x. b) Tfnh gia tr~ cua P khi I xi = Cht!ng minh d&ng thuc (a- 1)(a- 2)(a + 3)- (a+ 1)(a + 2)(a- 3) = 12. Ap dt,mg ket qua tren M cht!ng minh rang : = Cho cac so X, y, z ti I~ v6'i cac so a, b, c. Cht!ng minh rang : 8 (x 2 + 2/ + 3z 2 )(a 2 + 2b 2 + 3c 2 ) =(ax+ 2by + 3czl

9 Chuyen de'2 '? NHffNG HANG DANG THUC DANG NHO A. KIEN THUC CAN NHO (A+ B) 2 =A 2 + 2AB + B 2 (A - B) 2 = A 2-2AB + B 2 A 2 - B 2 =(A+ B)(A- B) (A+B) 3 =A 3 +3A 2 B+3AB 2 +B 3 = A 3 + B 3 + 3AB(A + B) (A - B) 3 = A 3-3A 2 B + 3AB 2 - B 3 = A 3 - B 3-3AB(A- B) A 3 +B 3 = (A+B)(A 2 -AB+B 2 ) A 3 - B 3 = (A - B)(A 2 + AB + B 2 ) _,, _., B. KIEN THUC BO SUNG (I) (2) (3) (4) (5) (6) (7) 1. Binh phuong cua do thvc (a 1 + a ~) = a 1 + a an + 2a 1 a 2 + 2a 1 a a 1 an D~c bi~t, v6'i n = 3 ta c6 : + 2a 2 a 3 + 2a 2 a a 2 a a 11 _ 1 an. (a+ b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc. 2. Luy thua b c n cua mqt nhj thvc (nhj thvc Niu-ton) n 11 n-ib n(n- 1) n-2b2 n(n- 1)(n- 2) n-3b3 bn ( a + b) = a + na + a + a Cho n cac gia trj tu 0 den 5 ta duqc : V6'in=0 thl (a+b) 0 = V6'in=l thl I (a+ b) = a+b V6'in=2 thl (a+ b) 2 = 2 2 a + 2ab + b V6'i n = 3 thl (a+b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 V6'i n = 4 thl (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 V6'i n = 5 thl (a+b) 5 = a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + b 5 9

10 Ta nhan thay khi khai tri~n (a+ b) 11 ta duqc m(>t da th(rc c6 n + I h<:mg tii, hang tl:r dau Ia a 11, h~ng tii cu6i Ia bn, cac hang tii con l~i deu chua cac nhan tii a va b. VI vay (a+ b) 11 = B(a) + b 11 = B(b) +an. 3. Bang cac h~ so khi khai trh3n (a + b)n Voi n = 0 : Voi n = I Voi n = 2 I 2 Voi n = I I Voi n = I Voi n = S s s - Moi dong deu bat dau btmg I va ket thuc b~ng I - Moi s6 a mot dong k~ ttr dong thu hai deu b~ng s6 lien tren c(>ng voi s6 ben tnii cua s6 I ien tren. Bang tren day duqc gqi Ia tam giac Pa-xcan.. s. MOT so vi ov Vi d~:~ 7. Chtlng minh dng neu m(>t tam giac c6 d dai ba quill Ia a, b, c thoa man : (Sa - 3b + 4c)(Sa- 3b- 4c) = (3a- Sb) 2 thi tam giac d6 Ia tam giac vuong. Gidi. Ta c6 (Sa- 3b + 4c)(Sa- 3b- 4c) = (3a- Sb) 2 <=> I(Sa- 3b) + 4cJ[(Sa - 3b)- 4c] = (3a- Sb) 2 <=>(Sa- 3b) 2 - (4c) 2 = (3a - Sb) 2 <=> 25a 2-30ab + 9b 2-16c 2 = 9a 2-30ab + 25b 2 <=> 2Sa 2-9a 2 + 9b 2-2Sb 2 - l6c 2 = 0 <=> 16a 2-16b 2-16c 2 = 0 <=> 16i = 16b c 2 <=> a 2 = b 2 + c 2. Do d6 tam giac c6 d dai ba c~nh Ia a, b, c chfnh Ia m(>t tam giac vuong. Vi dl;l 8. Cho X+ y = -9; xy = 18. Khong tfnh cac gia tr! cua X va y, hay tfnh gia tri cua cac bi~u thuc sau : a) M = x + y ; b) N = x + y ; c) P = x - y. 10

11 Gidi. De bai cho gia trj cua tong X + y va tfch xy nen muon tfnh duqc gia tri Clla cac bieu tht'rc M, N, pta phai bi ~ u dien cac bi ~ u thuc nay du6i d<,mg cac bieu tht'rc c6 (x + y) va xy a) M = x + y = x + 2xy + y - 2xy = (x + y) - 2xy = (- 9) = b) N = x + y = x + 2x y + y - 2x y = (x + y ) - 2(xy) = = c ) T a co ' ( x- y )2 = x 2-2 xy + y 2 = x 2 +?_xy + y 2-4 xy = (x + y) 2-4xy = ( -9) = 9. Suy ra x- y = ± 3. t Neu x- y = 3 thl P = x 2 -/ = (x- y)(x + y) = 3.(-9) = -27. Neu x- y = -3 thl P = x 2 - / = (x- y)(x + y) = (-3).(-9) = 27. Vi dl,l 9. nm X, y, z biet: 2 2 } Gidi. Ta c6 Suy ra X - 6x + y + I Oy + 34 = - ( 4z - I r. 2 2 } X - 6x + y + I Oy + 34 = -( 4z- I r 2 2 } (x -6x+9)+(y + 10y+25)=-(4z-lf <::::> (x- 3) 2 + (y + 5) 2 + (4z- 1) 2 = 0 Ta thay (x- 3) 2 ~ 0; (y + 5) 2 ~ 0; (4z- I ) 2 ~ 0 ma (x- 3) 2 + (y + 5) 2 + (4z- 1) 2 =0. l ( X - 3) 2 = 0 X = 3 nen (y+5) 2 <::::> y=~5 7=0 (4z-l)-=0 z=-. 4 NhtJ_n xet: Ta g<;>i phmmg phap giiti trong vf dy tren Ia phuong phap "Tong cac blnh phuong". N(>i dung cua phuong phap nay dt!a vao nh~m xet: A 2 ~ 0 ; B 2 ~ 0 ; C 2 ~ 0. Neu c6 A 2 + B 2 + C 2 = 0 thl A 2 = B 2 = C 2 = 0. Vi d1,110. Cho a+ b + c = 0, cht.xng minh rang a 3 + b 3 + c 3 = 3abc. II

12 Gidi. Tv a + b + c = 0, suy ra a + b = -c. Lap phuemg hai veta duqc (a+ b) 3 = (-cl Suy ra a 3 + b 3 + 3ab(a +b)= -c 3. Thay a+ b = -c vao d~ng thuc tren ta duqc a 3 + b 3 + 3ab(-c) = -c 3. Do d6 a 3 + b 3 + c 3 = 3abc. L1r11 y. Nen nha ket qua cua vf du nay d~ v~n dt_mg giai nhieu bai toan khac. Trong qua tiinh giai vf dl,l tren ta da khai tri~n (a+ b) 3 thanh a 3 + b -~ + 3ab(a +b) (I) ti~n lqi hem Ia khai tri~n thanh a 3 + 3a 2 b + 3ab 2 + b 3 (2) vi trong khai tri~n (I) c6 san (a+ b) d~ thay bang - c ra ket qua duac nhanh chong Vi dl;l 11. S6 a = I Ia so nguyen to hay hqp so? ( ~r * ) G tat.. T,? a co : 3 nen, ta d ~t = 3 n n E 1 "~ Do d6 a = I = (8 11 ) = ( )( ). So a Ia tfch cua hai so tl,f nhien Ian hem I nen a Ia hqp so. Vi dl;l 12. ChLrng minh dang thuc a 5 - b 5 - (a- b) 5 = Sab(a- b)(a 2 - ab + b 2 ) Gidi X et ' ve,- tra1 '. T : T = a s - bs - ( a - b)s. Xet ve phai P : = a 5 - b 5 - (a 5-5a 4 b + 10a 3 b 2 -!Oa 2 b 3 + 5ab 4 - b 5 ) = a 5 - b 5 - a 5 + 5a 4 b- 10a 3 b a 2 b 3-5ab 4 + b 5 = 5a 4 b- 10a 3 b a 2 b 3-5ab 4. P = Sab(a- b)(a 2 - ab + b 2 ) = 5ab(a 3-2a 2 b + 2ab 2 - b 3 ) Y~y T= P. = 5a 4 b- 10a 3 b a 2 b 3-5ab 4. Vi dl;l13. Cho (a+ b + c) 2 = 3(ab +be+ ca). Chung minh dng a= b =c. Gidi. Ta c6 (a+ b + c) 2 = 3(ab +be+ ca) <=> a 2 + b 2 + c 2 + 2(ab + be + ca) = 3(ab +be + ca) <=> a 2 + b 2 + c 2 - ab - be - ca = 0 12

13 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} ~ - (2a + 2b + 2c - 2ab- 2bc- 2ca) = 0 2 ~ _!_[(a 2-2ab + b 2 ) + (b 2-2bc + c 2 ) + (c 2-2ca + a 2 )] = 0 2 ~_!_[(a- b) 2 + (b- c) 2 + (c- a) 2 ] = 0 2 ~(a- b) 2 + (b- c) 2 + (c- a) 2 = 0 ~(a - b) 2 = (b- c) 2 = (c- a) 2 = 0 (vl (a- b) 2 ~ 0; (b- c) 2 ~ 0; (c- a) 2 ~ 0). a - b = 0 ~ b- c = 0 ~ a = b =c. { c - a=o C. BAt T~P cac hang dtjng thuc < 1 >I (2) I (3) Chtrng minh r~ng gia tr! cua cac bi~u thuc sau khong ph1,1 thu<)c vao gia tr! cua bien: a) 5(x + 4) 2 + 4(x- 5) 2-9(4 + x)(x- 4); b) (x + 2y) 2 + (2x- y) 2-5(x + y)(x- y)- 10(y + 3)(y- 3) Tfnh gia tr! cua bi~u thuc bang each hqp 1f : a) 413(413-26) + 169; b) ( )(25 4-3) ; c) Khong tfnh gia trt Cl,l th~ cua m6i bi~u thuc, hay so sanh gia tr! cac bi~u thuc sau: a) A= va B = ; b) C =I+ 8( )( )(3x + 1) va D = (3 3 ) 5 + (3 5 ) 3 ; c) E = va F = ; d) M = 5x 2 + IOl- 2xy + 4x- 6y + 2 van= -I Cho M = I ; N = Tfnh gia tri cua bi~u thuc M - N

14 1.22. Rut gon roi tfnh gia trj cac bi~u thuc sau : I a) A= (3x- 2) + (3x + 2) + 2(9x - 4) tai x = -- ;. 3 b) B = (x + y -7) 2-2(x + y -7)(y- 6) + (y- 6) 2 tai x = 101; ') c) C = 4x~- 20x + 27 t~i x = 52, Cho X+ y = -9, tfnh gia trj cua bi~u th(rc : D = x 2 + 2xy + l- 6x - 6y Tlm x biet : 7 a) (5x- I r- (5x- 4 )(5x + 4) = 7 ; b) (4x- 1) 2 - (2x + 3l+ 5(x + 2) 2 + 3(x- 2)(x + 2) = Cho bi~u thuc A= (x 2 + x + l)(x 2 - x + l)(x 4 - x 2 +I). Cht!ng minh dmg bi~u thuc A lu6n lu6n c6 gia trj duang vai moi gia trj cua bien Cht!ng m inh dmg cac bi~u thuc sau I u6n lu6n c6 gia trj duang vai moi gia trj cua cac bien : a) M = 25x - 20x + 7; b) N = 9x-- 6xy + 2y + I ChUng minh rtmg gia trj cua cac bi~u thuc sau lu6n lu6n am vai mqi gia trj cua cac bien : a) P = 2x- x ; b) Q = -x - y- + Rx + 4y Tlm gia tr! nho nhat ci:1a cac bi~u thuc sau : a)a=x x+39 ; Tlm gia tri ian nhat cua cac bi u thuc sau : b)b=9x 2-12x. a) C = 4x- x 2 + I ; b) D = 3-10x 2-4xy- 4/ Cho day hlnh vu6ng (h.l) duai day : 1/inlt 1.1 1/inlt l.'l 1/inlt /.3 1/inlt / Hinlt I Hoi o hlnh 1.75 c6 bao nhieu hlnh vuong khong duqc to dam?

15 1.31. Cht!ng minh rang neu 2(a 2 + b 2 ) =(a+ b) 2 thl a= b Cho (a - b) 2 + (b - c) 2 + (c - a) 2 = 4(a 2 + b 2 + c 2 - ab - be - ca). Chung minh dmg a= b = c. Tim CaC SO X, y, Z biet dmg: X+ y + Z = 1,5 (1) va x 2 + / + z 2 = 0,75 (2). Cho biet (a+ b + c + 1)(a- b- c + I)= (a- b + c- I )(a+ b- c- I) Cht!ng minh ding a = be Cho hai so nguyen X va y, moi soia t6ng cac blnh phvong cua hai so nguyen nao d6. ChUng minh rang tfch xy c6 th~ viet dv6i d<;mg tong cua hai blnh phvong Cht!ng minh ding tfch cua bon so nguyen lien tiep c(mg v6'i I Ia m(>t so chfnh phvang Tim ba s6 nguyen lien tiep biet t6ng cac blnh phvang cua hai so dau bang b'inh phvang cua s6 cu6i. Cac h&ng d6ng thuc (4), (5), (6), (7) nm x biet : a) (x + 4)(x 2-4x + 16)- x(x- 5)(x + 5) = 264; b) (x- 2) 3 - (x- 2)(x 2 + 2x + 4) + 6(x- 2)(x + 2) = Tim gia trj cua bi~u thuc : a) A= x 3-15x x- 124 t~i x = 35; b) B = x x x + 16 tai x = -26; c) C = (x - 4y )(x - 2xy + 4y )(x + 2xy + 4y ) Tim gia trj cua cac bi~u thuc sau : a) M = 3(x 2 + /) - (x 3 + /) + I biet x + y = 2. b)n=8x 3 -!2x 2 y+6x/-/+ i2/-i2xy+3/+6x-3y+ II v6i2x-y= Chting minh rang: a) P = chi a het cho 1350 ; b) Q = chia het cho Cho a, b, c Ia cac so thoa man dieu ki n a= b +c. Cht!ng minh n"mg : a 3 +b 3 a+b ---= 15

16 1.43. Chtffig minh d~ng thuc : (a+ b + c) 3 - a 3 - b 3 - c 3 = 3(a + b)(b + c)(c +a). Ap dyng : Thu gqn bi~u thuc A= (a+ b + c) 3 - (a+ b- c) 3 - (a- b + c) 3 - (-a+ b + c) 3. Binh phuong cua do thuc va luy thua cua nhj thcrc Thu gqn roi tfnh gia tr~ cac bi~u thuc sau : a) (a+ b + c) 2 +(a+ b- c) 2-2(a + b) 2 vai c = -10. b) (a+ b + c) 2 +(-a+ b + c) 2 +(a- b + c) 2 +(a+ b- c) 2 vm a 2 + b 2 + c 2 = I I Cho cac so a, b, c khac 0, thoa man a + b + c = abc va = 2. a b c Ch 1111g, mm. h, rang: I =. 2 a b c Chtffig minh d~ng thl"rc : (x + 4) + x + y = 2(x + xy + y ) Xac d!nh cac h~ so a, b d~ da thuc sau Ia blnh phuang cua m t da thuc A =x x-x+ax+. 2 b Tim x, biet : a) (x + 3) 4 - (x - 3) 4-24x 3 = 108 ; b) (x + 2) 5 - (x - 2) 5 = Cho (a+ b + c) = 4(a + b +c)+ 2(ab +be+ ca). Chtffig minh rang a= b = c = Cho bi~u thuc Tfnh: M =(a+ 1) 2 + (2b + 1) 2 + (3c + 1) 2 + 2(2ab + 3ac + 6bc) + 2(a + 2b + 3c) van= (a+ 2b + 3c + 1/_ Tfnh hi~u M- N. a) (x + I ) 5 ; 4 c)(x-1) Tfnh t6ng cac h~ so cua tat d cac h<:mg til trong khai tri~n cac nh! thuc sau : a) (4x- 5)x; b) (5x- 3y) Tim so du trong phep chia : a) cho I 5 ; Tim so du khi chia cho b) cho 15.

17 Chuyende'3 ", " ' "? PHAN TICH DA THUC THANH NHAN Tlf A. KIEN THUC CAN NHO 1. Phan tich da thuc th~mh nhan tu (hay thua so) Ht bien d6i d ~ thuc d6 thimh m(>t tfch cua nhfing da thuc khac. 2. Cac phuang phap thuemg dung : - D~t nhan tu chung - Dung hang dltng thuc - Nh6m cac h~ng tu - Phoi hqp nhieu phuang phap. C6 khi ta phai dung nhfmg phuang phap d~c bi t khac (xem chuyen de 1 ). KIEN THUC BO SUNG 1. Dc;mg tong quat cua cac h&ng d&ng thuc a 2 - b 2 =(a- b)(a +b) a 3 - b 3 =(a- b)(a 2 + ab + b 2 ) Ht : An- Bn =(A- B)(An-1 + An ABn-2 + Bn-1). 2. DQng tong quat cua h&ng d&ng thuc a 3 + b 3 =(a+ b)(a 2 - ab + b 2 ) Ia: An+ Bn =(A+ B)(An-1 -A n An ABn-2 + Bn-1) v6i n!e. 3. Ap d.. mg vao tinh chat chia het A 11 - Bn : A - B v6'i n E N va A :;:. B. An : A+ B v6'i n leva A :t:- -B. A 2 k - B 2 k : A v6'i k E N va A :1:- ±B. - "", B. MQT SO VI Dl) Vi d1,114. Tfnh gia trt cua bi~u thuc sau bang each hqp If: A= (x - 3) 2 - (8x + 3)(3 - x) +x(x - 3) t~i x = CDTOAN8TAP1 2A

18 Gidi. Ta c6 A= (x- 3) 2 - (8x + 3)(3- x) + x(x- 3) = (x- 3) 2 + (8x + 3)(x- 3) + x(x- 3) = (x- 3)(x x x) = 10x(x- 3). Vm x = 103 thl A= (103-3) = Nh{m xit : Phan tich da thuc thfmh nhan tl.r c6 nhi~u t!ng dt,mg. M t trong cac t1ng dt,mg d6 Ia tinh gia tr! cua bieu thuc. Phuang phap phan tich trong vi dl,l nay Ia phuang phap d~t nhan tl.r chung. Vi dl.l 15. Phan tich da thuc thanh nhan tl.r B = (x 2 + 9) 2-36x 2 Gidi. Ta c6 B = (x 2 + 9) 2-36x 2 = (x 2 + 9) 2 - (6x) 2 = (x x)(x x) = (x- 3) 2 (x + 3l Vi dl_l16. Tim cac c~p s6 nguyen t6 (x; y) sao cho x 2 - l = 2/ G tat. T / 2 l 2 2? o a co x - = y Suy ra x 2 - I Ia s6 chan, do d6 x 2 Ia s6 h~ va x Ia s6 I e. Tu d~ bai, suy ra (x- l)(x + l) = 2/. Vl X Ia sole nen X- l va X+ l Ia hai s6 chan lien tiep. Do d6 tfch (x- l)(x + I) : 4, 2 suy ra 2y : 4. V.;ty / : 2 va y Ia s6 chan, suy ray= 2. Thay y = 2 vao d~ bai ta duqc x 2 - I = 8 <=> x 2 = 9 rna X Ia s6 nguyen t6 nen X= 3. V.;ty c~p s6 nguyen t6 can tim Ia X= 3; y = 2. Nhij.n xet: Phuang phap phan tfch da thuc thanh nhan tl.r trong vf dl,l 15 va 16 Ia phuang phap dung hang d~ng thuc. Phan tich da thuc thanh nhan tl.r con giup ta v.;tn dl,lng duqc cac tfnh chat chia het tlr d6 tim duqc cac s6 thoa man di~u kien cho tru6c. Vi dl.l 17. Phan tfch da thuc thanh nhan tl.r C = ax - ay - bx + by+ (y- xl Gidi. Ta c6 C = ax - ay - bx + by + (y - x) 2 = a(x - y) - b(x - y) + (x - y) 2 = (x - y)(a- b + x - y). Cdnh bao : Viet (y- x) 2 = -(x- y) 2 la sai vi (y - x) 2 = (x - y) 2. Vi dl_l 18. Cho x, y, z Ia cac s6 hfru ti thoa man di~u ki~n xy + yz + zx = 1. Cht!ng minh rang s6 m = (x 2 + l)(/ + l )(z 2 + I) Ia blnh phuang cua m t s6 hfru ti. 18 CDTOAN8TAP1 28

19 Gidi Ta c6 x 2 + l = x 2 + xy + yz + zx = x(x + y) + z(x + y) = (x + y)(x + z) l + l = l + xy + yz + zx = y(y + x) + z(y + x) = (y + z)(y + x) 2 2 z + l = z + xy + yz + zx = y(x + z) + z(z + x) = (z + x)(z + y). Do d6 m = (x + y)(x + z)(y + z)(y + x)(z + x)(z + y) = [(x + y)(y + z)(z + x){ VI X, y, z Ht cac so hfru ti nen X + y, y + z va z +X Ia nhfrng so hfru ti, tich cua chung Ia so hfru ti. Do d6 m Ia blnh phuong cua m(>t so hfru ti. Vi d~,il9. Phan tich da tht!c thanh nhan tu M = (5x- IO)(x 2 - l)- (3x- 6)(x 2-2x + l) Gidi. M = (5x- IO)(x 2 - l)- (3x- 6)(x 2-2x + l) = 5(x- 2)(x- l)(x + l)- 3(x- 2)(x- r/ = (x - 2)(x - l )[5(x + I) - 3(x - 1 )] = (x - 2)(x - l )(2x + 8) = 2(x - 2)(x - 1 )(x + 4) Vi d1,1 20. Phan tfch da thuc thanh nhan tu N = 5x 4 + 3x 3 y- 20x 2 l- l2x/ Gidi. N = 5x 4 + 3x\- 20x 2 l- l2x/ = x\sx + 3y)- 4xl(5x + 3y) = x(5x + 3y)(x 2-4l) = x(5x + 3y)(x- 2y)(x + 2y). Vi d1,1 21. Phan tich da tht!c thanh nhan tu P = 6x 2 - rsol + 60y - 6 Gidi. P = 6x 2 -!SOl+ 60y- 6 = 6(x 2-25l +loy- 1) = 6[x 2 - (25l- loy+ I) = 6[x 2 - (5y- 1) 2 ] = 6(x- 5y + l)(x + 5y- 1). Nhtj,n xet : Khi phai phoi hqp nhieu phuong phap de phan tfch da tht!c thanh nhan tu duqc tri~t de ta thuang dung phuong phap d~t nhan tu chung tru6c (neu c6 the), cac phuong phap khac sau. M6i phuong phap c6 th~ dung nhieu ian. C. BAt TAP Phuong phap dc;lt nhan tu chung Phan tfch da thuc thanh nhan tu a) 8xn+ 2 /- l2xn/; b) (a- b) 2 - (b- a)(a-,3b); c) a(a- b) 2 (a+ b)- (b- a) 2 (a 2-5ab + b 2 ). 19

20 1.56. ChLrng minh ning : 2. a) : 50; b) : Tim m(>t so biet rang blnh phuang cua n6 bang 4 Ian l~p phuang cua so ay Tfnh gia tr! cua bieu thuc sau bang each hqp If : A= ChLrng minh rang so Ia tfch cua hai so nguyen lie!f tiep. n n Cho biet x 3 = 2p + 1 trong d6 x Ia so tl! nhien, pia so nguyen to. Hay tim x. Phuong phap dung h&ng d6ng thuc Phfm tfch da thuc thanh nhan til a) (x- 9)(x- 7) + 1 ; b) x 6 -l; Phan tfch da thuc thanh nhan til 2 a) (x + 2y- 3) - 4(x + 2y - 3) + 4; Phan tfch da thuc thanh nhan tl:t c) (x + x - I) + 4x + 4x. b) (x- y) (x- y)(x- y- I). (x 2 + / - 17) 2-4(xy- 4) Tim X, biet : a) x x x = 0; nm cac c~p so nguyen (x; y) sao cho x = /. b) (x + 2) 3 + (x - 3 ) 3 = r<~... b b2r<~...' minh,,. 1':\T*thln n n bn.. \....uo X + y = a + ; X + y = a +. \....utmg rang VOl n E 1 'I X + y = a +. Phuong phap nh6m cac hqng tv Phan tfch da thuc thanh nhan til a) x 3 + 2x 2-3x- 6; b) 2a 2 c 2-2abc + bd- acd; Phan tfch da thuc thanh nhan tl:t a) ab(x 2 -/)- xy(a 2 - b 2 ); Phan tfch da thuc th~mh nhan til : 2 2 b)x +(a-b)xy-aby. 2 c) 12x - 3xy- 8xz + 2yz. abc + ab + be + ca + a + b + c Phan tfch da thuc thanh nhan til: A= x 35 + x 34 + x x 2 + x ChLrng minh rang: x(x + 1) 4 + x(x + 1) 3 + x(x + 1) 2 + (x + 1) 2 = (x + 1) Tfnh gia tf! cua bieu thuc: A= x Ch 0 h,. h, ox+ y + z =, c ung mm rang: x + x z + y z- xyz + y =. 20