NGttr. BlJI VAN TUYEN- NGUYEN DUC TRVONG TrQng tom kien thuc vo phuong ph6p gioi boi tqp ToAN8 T~PMQT (T ai bdn ld'n thft nhdt)...,.,.,,... NHA XUAT BAN GIAO Dl)C VI~T NAM
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9;;u nett diu---- - - ------- Cac em hqc sinh th{m men! Cac em di h9c, ai cung mong muon mlnh h9c gioi trong d6 c6 man Toan. Muon v~y, cac em can nam vci'ng If thuyet va biet each v~n dt.,mg de giai bai t~p. Lf thuyet th] nhieu va r(>ng nen can biet ch9n IQC dau Ia van de cd ban, tr9ng tam de hqc cho kiva ghi nho. Bai t~p c6 nhieu lo<;~i, v6 cung phong phu va da d<;~ng nen can phan loai cac d<;~ng va dua ra phudng phap giai cua tt.lng d<;~ng ay. Ngoai ra, m(>t dieu quan tr9ng nci'a Ia phai biet each suy lu~n khi giai toan. Cuon sach nay dlt<;1c bien so~n nham dap Ltng cac yeu cau tren day cua cac em h9c sinh. Sach du<;1c viet theo cac chudng va bai cua sach giao khoa. Moi bai gam ba phan: A. Trc;mg tam kien thuc Phan nay t6m tat cac kien thltc cd ban, tr9ng tam cua bai. NgUCJi ta thuc1ng n6i "C6 b(>t mai g(>t nen h6 ", b(>t d day Ia cac kien thltc Cd b~m. cua bai. C6 nam vci'ng dude thl mai c6 Cd Sd de giai bai t~p. B. Cac d~ng bai t~p va phlldng phap giai Qua trlnh "g(>t nen ho" ch fnh Ia qua trlnh luy$n t~p. trqng tam ova vao trqng tam kien thltc da neu d tren, cac tac gia da dua ra cac d<;~ng bai t~p de cac em luy$n giai. Trong moi d~ng c6 neu ngan 99n phudng phap giai va nhieu vf dl,l minh ho<;~. Dieu d6 giup cac em c6 kinh nghi$m giai bai t~p. d!nh huang du<;1c each giai, cac buoc can lam r6i thlfc hi$n theo cac buoc d6. Cac bai toan trong moi vf dl,l du<;1c llfa ch9n Ia nhci'ng bai t~p tieu bieu, ddn gian nhung khong tam thuang, chua dt,tng nhieu kien thltc, ki nang va phuong phap suy lu~n ma chuong trlnh doi hoi. 3
c. Bai t;jp tl/luy~n Phan nay gam m<?t so bai t~p. c6 lo~i trung blnh, c6 lo~i kh6, giup cc'ic em tl,f kiem tra, danh gia muc d9 thanh th~o cua mlnh. Cac em hay co gang tl,f giai. Neu g~p kh6 kh~n c6 the xem IC:Ii giai ho~c hllong dan a cuoi bai. Cac tac gia mong rang cuon sach nay Ia m<?t tai li~u c6 fch giup cac em tl,f hqc, tl,f dqc CO ket qua. Cuon sach cong cung cap cho cac thay giao, co giao nho'ng tll li~u tham khao de ch9n bai cho cac tiet luy~n t~p. boi dllang h9c sinh. Cac b~c phl,l huynh cong c6 the dung cuon sach nay de hllong dan con em h9c t~p. Rat mong nh~n Oli<;'C y kien d6ng g6p cua b~n OQC cho cuon sach. Cac y kien d6ng g6p xin glti ve dja chi : Cong ty CP Djch Vl,l xuat ban Giao dl,ic Ha N<?i - Nha xuat ban Giao dl,ic Vi~t Dong Da, Ha N<?i. Xin chan thanh cam on. Nam, 1878 Giang V6, qu~n cactac GIA 4
{[[W+bz0FkV43GmRt7u4DpvuYxd]]} DAISO Chztdng I. PHEP NHAN va. PHEP CHIA cac DA THUc l. NHAN DON THlfC VOl DA THlfC 2. NHAN DA THlfC VCJI DA THlfC A. TRONG TAM KI~N THUC Oic quy de nhan dcm thuc vm da thuc va nhan da thuc vm da thuc : A.(B + C) = A.B + A.C (A + B)(C- D) = A.C- A.D + B.C- B.D. B. cac DANG BAI TAP VA PHUONG PHAP GIAI Dt;~ng 1. LAM TfNH NHAN PhYdngphapg~i------------------------------------~ Ap dl;lng quy tac nhan ctcm thuc vm cta thuc va nhan cta thuc vm cta thuc. Luu y quy tac dau cua phep nhan va thu gqn cac hl;lng tir dong dl;lng. Vi d1;1 1. Lam tfnh nhan 1 1 2 a) '2 x (x - 6x - 10) ; b) -3x 2 (5x 3-4x 2 + 3x- 1). a) _!_x 3 (x 2-6x -10) = _!_x 5-3x 4-5x 3. 2. 2 b) -3x 2 (5x 3-4x 2 + 3x- 1) = -15x 5 + 12x 4-9x 3 + 3x 2. 5
Vi d~;~ 2. ThtJC hi~n cac phep tfnh a) (x + 8)(x - 5) ; b) (2x-'- 1)(3x 2-7x + 5). a) (x + 8)(x - 5) = x 2-5x + 8x- 40 = x 2 + 3x- 40. Nh(m xh. B<;1n nen nh6' tfch hai da thuc d<;10g (x + a)(x +b) c6 ket qua nhu sau d~ nhlim nhanh ra ket qua : (x + a)(x +b)= x 2 +(a+ b).x + ab. b) (2x- 1)(3x 2-7x + 5) = 6x 3-14x 2 + lox- 3x 2 + 7x- 5 = 6x 3-17x 2 + 17x- 5. Vi d~;~3. Tim h~ s6 cua x 3 trong ket qua cua phep nhan (x 2 - x)(x 2 +X- 1). (x 2 - x)(x 2 + x- 1) = x 4 + x 3 - x 2 - x 3 - x 2 + x. V~y h~ s6 cua x 3 la 0. = x 4-2x 2 + x. D{lng 2. RUT GQN BIEU THUC VA TfNH GIA Tlq CUA BIEU THUC PhLiang phap giai --------------------., Thl;(c hi~n cac phep nhan don thuc v6'i da thuc, da thuc v6'i da thuc, bo dau ngo~c, thu g<.m cac h<;lng tir d6ng d<;!ng. Thay gia ti1 CUa cac bien vao bieu thuc da rut gqn r6i thl;(c hi~n cac phep tfnh. Vi d~;~ 1. Rut g<;>n bi~u thuc A= 8x(x- 2)- 3(x 2-4x - 5)- 5x 2.? 2 2 Giai. A= 8x(x- 2)- 3(x - 4x- 5)- 5x. A= 8x 2-16x- 3x 2 + 12x + 15-5x 2. A= -4x + 15. Vi d~;~ 2. Rut g<;>n- bi~u thuc B = 2(x- 5)(x + 1) + (x- 3)(x + x 2 ). 6
. B = 2(x- 5)(x + 1) + (x- 3)(x + x 2 ) B = 2(x 2-4x- 5) + (x 2 + x 3-3x- 3x 2 ) B = 2x 2-8x- 10 + x 2 + x 3-3x - 3x 2 B=x 3-11x-IO. Nh(m xet. Khi tfnh tfch 2(x - 5)(x + 1) be;tn nen tfnh tfch (x- 5)(x + 1) tru6c r6i nhan 2 v6i ket qua. Vi dt,~ 3. Rut g<;m bieu thuc A= (x + 5)(2x- 3)- 2x(x + 3)- (x- 15) r6i cho biet b~c cua da thuc ket qua.. A= (x + 5)(2x- 3)- 2x(x + 3)- (x- 15) A=2x 2-3x+ 10x-15-2x 2-6x-x+ 15=0 Ket qua Ia da thuc 0, da thuc nay khong c6 b~c. Vi dt,~4. Cho bieu thuc B = 5x 2 (3x- 2)- (4x + 7)(6x 2 - x)- (7x- 9x\ Rut gon r6i tfnh gia tri cua bieu thuc B v6i X = -2... 4 B = 5x 2 (3x- 2)- (4x + 7)(6x 2 - x)- (7x- 9x 3 ) B = 15x 3-10x 2 - (24x 3-4x 2 + 42x 2-7x)- (7x- 9x 3 ) B = 15x 3-10x 2-24x 3 + 4x 2-42x 2 + 7x- 7x + 9x 3 B = -48x 2 V6i x =-! thl B = -48.( -%J = -48. 1 ~ = -27. Vi dt,~ s. Cho bieu thuc C = x(x + x 3 ) + (x- 1 )(x 2 + x 3 ) + 1. Rut g<;m bieu thuc c r6i cht!ng to rang v6i hai gia tr1 d6i nhau cua X thl bieu thuc C c6 cung m<)t gia trj. C = x(x + x 3 ) + (x- l)(x 2 + x\+ I C = x 2 + x 4 + x 3 + x 4 - x 2 - x 3 + 1 C = 2x 4 +I. Ket qua chi chua luy thua chdn cua bien X nen v6i hai gia tr1 d6i nhau cua X thl bieu thuc c c6 cung m<)t gia tr1. 7
Dt;~ng 3. CHUNG MINH GIA TR_l CUA BIEU THUC KHONG PHl) THUQC V Ao GIA TRl CUA CAC BIEN. PhLiung phap giai -------------------, Bien d6i bi~u thuc da cho th~mh m t bi~u thuc khong chua bien. Vi d~;~ 1. Chtrng minh rang gia tr! cua bi~u thuc sau khong phl;l thu c vao gia tr! cua cac bien : A= (2x- 3)(x + 7)- 2x(x + 5)- x. A= (2x - 3)(x + 7) - 2x(x + 5) - x A = 2x 2 + 14 x - 3 x - 21-2x 2 - lox - x A= -21. V~y gia tr! cua bi~u thuc A kh6ng phl;l thu c vao bien X. Clu1 y : D~ ki~m tra ket qua, b~n c6 th~ thay X = 0 vao bi~u thuc da cho r6i thl,l'c hi~n cac phep tfnh. Neu ket qua trimg vm ket qua tren thl dung. Ch~ng h<:m, v6'i X = 0 thl A= -3.7 = -21. Ket qua nay trung v6'i ket qua tren. Vf d~;~2. Cho bi~u thuc B = 10-5x(x- 1,1) + 2x(2,5x- 3). Chtrng minh dmg gia tr! cua bi~u thuc nay luon luon khong d6i.. B = 10-5x(x- 1,2) + 2x(2,5x- 3) B = 10-5x 2 + 6x + 5x 2-6x B= 10. V ~y gia tr! cua bi~u thuc B khong phl;l thu c vao bien, luon luon c6 gia tr! la 10. VI dl,l 3. Cho bi~u thuc C = x(x- y) + y(x + y)- (x + y)(x- y)- 2l. V <Ji mqi gia tr! CUa X va y thi gia tr! CUa bi~u thuc C la m<)t s6 am hay m<)t s6 duong? 8 C = x(x - y) + y(x + y) - (x + y)(x - y) - 2l C = x 2 - xy + xy + /- (x 2 - xy + xy -/)-2/
2 2 2 2 2 C = x - xy + xy + y - x + xy - xy + y - 2y C=O. V ~y v6i m9i gia trj cua X va y thl gia trj cua bi ~ u thtrc c 1u6n 1u6n bang 0, khong phai 1a s6 am va cgng kh6ng phai Ia s6 duong. D~ng 4. CHUNG MINH DANG THUC Phudngphapg~i -------------------------------------, Bien d6i mot ve th anh ve kia h o~c bien d6i ca hai ve cung bang m(>t bi ~ u thvc. Vi dt,~ 1. Chvng minh d ~ n g thvc 3 2 2 3 4 4 (x - y)(x + x y + xy + y ) = x - y. X,.-,. T ( )( 3? et ve tra1 = x - y x + x-y + xy 2 + y 3). 4 3 2 2 3 3 2 2 3 4 4 4 T = X +X y + X y + xy - X y - X y - xy - y =X - y. Ta thay ve trai T dung bang ve phai P nen d ~ n g thtrc da cho Ia dung. Vi dt,~ 2. Ch(rng minh d ~ n g thtrc (x + y)(x + y + z) - 2(x + 1 )(y + 1) + 2 = x 2 + l. Xet ve tnii T = (x + y)(x + y + 2) - 2(x + I )(y + 1) + 2 ~? = x~ + xy + 2x + xy + y- + 2y - 2(xy + x + y + 1) + 2 2 2 = x + xy + 2x + xy + y + 2y- 2xy - 2x - 2y - 2 + 2 2 2 = X + y. Ve trait dung bang ve phai P nen d~ n g thtrc da cho Ia dung. Vi dt,~ 3. Cho ab = I. ChLrng minh d ~ n g thvc a(b + I) + b(a + I) = (a + I )(b + I). Xet ve trai T = a(b + I) + b(a + 1) = ab + a + ab + b = a + b + 2 (vi ab = I). Xet ve phai P = (a + I)(b + 1) = ab + a + b +I = a + b + 2 (vi ab = I). 9
D{lng 5. TIM GIA T~ CUA x THOA MAN DANG THUC CHO TRUde PhLidng phap giai----------------. Thl;I'C hi~n cac phep nhan da thuc r6i thu g<;m ve di;llig ax = b. Suy ra x = _!: (neu a :;e 0). a Vi d~;~ 1. Tim x biet 2 2 (x + 1 )(x + 2x - 1)- x (x + 3) = 4. 2 2 (x+ 1)(x +2x-1)-x (x+3)=4 3 2 2 3 2 X + 2x - X + X + 2X - 1 - X - 3x = 4 Vi d~;~ Vi d~;~ GiJi 2. Tim x biet x-1=4 X= 5. 2 2 (x+ 1)(3x +x-2)-x (3x+4)=5. (x + 1)(3x 2 + x- 2)- x 2 (3x + 4) = 5 3x 3 + x 2-2x + 3x 2 + x - 2-3x 3-4x 2 = 5 3. Tim x biet -x- 2 = 5 3(x - 2)(x + 3)- x(3x + 1) = 2. 3(x- 2)(x + 3)- x(3x + 1) = 2 3(x 2 +x-6)-x(3x+ 1)=2 2 2 3x + 3x - 18-3x - x = 2 2x- 18 = 2 -X= 7 X =-7. X= 20: 2 X= 10. 10
C. BAI TAP Tl,J LUY~N 1. Lam tinh nhan a) -4x\x 2-3x + 2) b) -~ x 2 (5x 3 + 10x 2-15x). 5 2. Lam tinh nhan a) (2x + 7)(3x- l) b) (5x 2-4x)(2x 2 + 9x- 3). 3. Tinh gia tji cua bi~u thuc A v6i X = 999 A=x 6 -x\x-l)-x 4 (x+ l)+x\x-l)+x 2 (x+ 1)-x(x-1)+ 1. 4. Cho bi~u thuc A= x(l + x)- x 2 (1- x) + x\x 2-1). Chll'ng minh rang v6i hai gia trt doi nhau cua X thl bi~u thuc A c6 hai gia tr! doi nhau. s. Tim x biet 6*. Cho ~ = I. Chll'ng minh rang a b (x - 3)(x+x 2 )+2(x-5)(x+ 1)-x 3 = 12. 2 2 2 2 2 (x + y )(a + b ) = (ax + by). HUONG DAN - DAP s6 1. a) -4x 5 + 12x 4-8x 3 ; 2. a) 6x 2 + l9x - 7 ; b) 10x 4 + 37x 3-51x 2 + 12x. 3. 1000. 4. A= x 5 +X. Bi~u thuc A chi chua luy thila le cua X nen v6i hai gia tji doi nhau cua X thl bi~u thuc A c6 hai gia tr! doi nhau. 5. X= -2. 6. Dat ~ = r = k suy ra X = ka. y = kb. a b ' ' Thay X, y vao hai ve roi SO sanh. 11
- ~ ~,,, }, 4, ~. NHUNG HANG DANG THLfC DANG NHO A. TRONG TAM KI~N THUC Bang hang d~ng thuc dang nh6 va nhiing t1ng dt,mg, d~c bi~t thuc dau tien. 1. (A + B) 2 = A 2 + 2AB + B 2 2. (A - 13) 2 = A 2-2AB + B 2 3. (A- B)(A +B)= A 2 - B 2 4. (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3 5. (A- B) 3 = A 3-3A 2 B + 3AB 2 - B 3 6. (A+ B)(A 2 - AB + B 2 ) = A 3 + B 3 la ba htmg d~ng 7. (A- B)(A 2 + AB + B 2 ) = A 3 - B 3. B. CAC D~NG BAI TAP VA PHUONG PHAP GIAI D{lng I. van" DVNG cac HANG DANG THUC DE TfNH Phudngphapg~i--------------------------------------- Xem bieu thuc da cho thu c d<;tng hang d~ng thuc nao thi v~n d1,1ng hang d~ng thuc ay de khai trien ra va nguqc l<;ti. Vi d1,1 1. Tfnh : a) (4x + 7) 2 ; r b) ( 6x- ~y c) (3x 2-5xy 3 )(3x 2 + 5xy\ a) (4x + 7) 2 = (4x) 2 + 2.4x.7 + 7 2 = 16x 2 + 56x + 49. r r b) ( 6x- ~y = (6x) 2-2.(6x)(~y J + (~y = 36x 2-4xy + il c) (3x 2-5xy 3 )(3x 2 + sxl) = (3x 2 ) 2 - (5xl) 2 = 9x 4-25x 2 /. 12
{[[W+bz0FkV43GmRt7u4DpvuYxd]]} Vi d1,1 2. Tfnh : a) (2x 2 + 5y) 3 ; b) (3x 3-4xy) 3 ; c) ( 6x + ~)(36x 2-3x + ~) d) (x - 5y 2 )(x 2 + 5xy 2 + 25l ). a) (2x 2 + 5y) 3 = (2x 2 ) 3 + 3.(2x 2 ) 2. (5y) + 3.(2x 2 )(5y) 2 + (5y) 3 = 8x 6 + 60x 4 y + 150x 2 / + 125l. b) (3x 3-4xy) 3 = (3x 3 ) 3-3.(3x 3 ) 2 (4xy) + 3(3x\(4xy) 2 - (4xy) 3 = 27x 9-108x 7 y + 144x 5 /- 64x 3 l. c) ( 6x + ~)(36x 2-3x + ~ ) = (6x) 3 + (~J = 216x 3 + ~ d) (x- 5/)(x 2 + 5x/ + 25/) = x 3 - (5/) 3 = x 3-125l. Vi d1,1 3. Viet cac da thuc sau duai d<;tng binh phuong hay l~p phuong cua m t t6ng hay hi~u 2 1 2 a) 25x - 5xy + - y 4 b) 8x 3-12x 2 y + 6x/-l. 2 2 a) 25x 2-5xy+"4:-Y 1 2 =(5x) 2-2.5x. 1 y+ ( 2y 1 ) = ( 5x-2y 1 ) 2 b) 8x 3-12x 2 y + 6x/-l = (2x) 3-3.(2x{y + 3.2x./- y 3 = (2x- yl Vi d1,1 4. Di~n cac don thuc thfch hqp vao 6 tr6ng r a) (X - ~ = X 2 - D + x12 ; 13
( 1 ) 2 2 1 x x2 a) X-- =X -0 + -. Dt;mg 2. RUT GQN BIEU THUC VA TfNH GIA T~ CUA BIEU THUC 14 PhvungphapgMi------------------------------------- V~n dl;lng cac hang d~ng thuc dang nh6 de khai trien de luy thl.ra, khai trien de tfch roi rut g<;m. Thay de gia tri CUa bien X vao bieu thuc da rut g<;m roi thl;i'c hi~n cac phep tfnh. Vi d1;1 1. Rut g<;m de bieu thuc : a) (7x + 4) 2 - (7x + 4)(7x- 4); b) (x + 2y) 3-6xy(x + 2y) ; c) (3x + y)(9x 2-3xy + l)- (3x- y) 3-27x 2 y. a) 2 (7x + 4) - (7x + 4)(7x- 4) = 49x 2 + 56x + 16- (49x 2-16) = 49x 2 + 56x + 16-49x 2 + 16 = 56x + 32. b) (x + 2y) 3-6xy(x + 2y) c) = x 3 + 6x 2 y + 12xl + sl- 6x 2 y- 12xl = x 3 + sl. (3x + y)(9x 2-3xy + l) - (3x - y) 3-27x 2 y = 27x 3 + l- (27x 3-27x 2 y + 9xl- y 3 )- 27x 2 y = 27x 3 + l- 27x 3 + 27x 2 y- 9xl + y 3-27x 2 y = 2y 3-9xl.
Vi d~;~2. Cho bieu thuc A= 5(x + 3)(x- 3) + (2x + 3) 2 + (x - 6l Rut gon r6i tfnh gia tri cua bieu thuc A v6i X = -.!..... 5 A= 5(x + 3)(x- 3) + (2x + 3) 2 + (x- 6) 2 A= 5(x 2-9) + (4x 2 + l2x + 9) + (x 2-12x + 36) A= 5x 2-45 + 4x 2 + 12x + 9 + x 2-12x + 36 A= 10x 2. vm x = - ~ thi A= w.( - ~)' = 10. ; 5 = ~- Vi d~;~3. Cho biet X+ y = 15 va xy = - 100. Tfnh gia tti cua bieu thuc B = x 2 + l. 2 2 2 2 B = x + y = x + y + 2xy - 2xy 2 B = (x + y) - 2xy B = 15 2-2.(- 100) = 425. Vi d~;~ 4. Tfnh nhanh gia tr! cua bieu thuc : a)c=39 2 +78.61 +61 2 ; b) D = 50 2-49.51. a) C = 39 2 + 78.61 + 61 2 C=(39+61) 2 c = 100 2 = 10000. b) D = 50 2-49.51 D = 50 2 - (50-1)(50 + 1) D = 50 2 - (50 2-1) D=l. 15
D~ng 3. CHUNG MINH GIA TRf CUA BIEU THUC KHONG PHl) THUQC V Ao cacbien Ph~dngphapgmi---------------------------------------.. V~n dt,mg cac hang d~ng thuc de bien d6i bieu thuc da cho thanh m t bieu thuc khong chua bien. Vi dt,~ cua bien: 1. Cht.Ing minh gia tti cua bieu thuc sau khong phl,l thu c vao gia tr! 2 3 A= (3x + 2)(9x - 6x + 4)- 3(9x - 2). A= (3x + 2)(9x 2-6x + 4)- 3(9x 3-2) A= 27x 3 + 8-27x 3 + 6 A= 14. V~y gia tr1 cua bieu thuc A khong phl,l thu c vao gia tr! cua bien. Vi dt,~ 2. Gia tr! cua bieu th(rc sau c6 phl,l thu c vao gia tr1 cua bien khong? B = (x + 1) 3 - (x- 1)(x 2 + x + 1) - 3x(x + 1) B = (x + 1) 3 - (x- 1)(x 2 + x + 1)- 3x(x + 1) B = x 3 + 3x 2 + 3x + 1 - (x 3-1)- 3x 2-3x B=2. V~y gia tti cua bieu thuc B khong phl,l thu c vao bien. D~ng 4. CHUNG MINH DANG THOC 16 Ph~dng phap giai-------------------------, v ~n dl,lng cac hang d~ng thuc de bien d6i m t ve thanh ve kia ho~c bien d6i ca hai ve cung bang m t bieu thuc. Vi dt,~ 1. Cht.Ing minh dfing thuc Bien d6i ve trai T ta duqc (x + y) 2 - (x- y) 2 = 4xy. T = (x + y) 2 - (x - y) 2 = x 2 + 2xy + l- (x 2-2xy + /) =4xy.
Ta thay ve tnii dung b{mg ve ph<h nen d~ng thuc da cho la dung. Vi dt.i 2. Chti'ng minh d~g thuc 2 2 2 2 2 2 2 3(x + y + z ) - (x - y) - (y- z) - (z- x) = (x + y + z). Bien d6i ve tnii T ta duqc : T = 3(x 2 + / + z 2 )- (x - y) 2 - (y- z) 2 - (z- x) 2 = 3x 2 + 3/ + 3i - (x - y) 2 - (y - z) 2 - (z- x) 2 2 2 2 2 2 2 2 2 2 = 3x + 3y + 3z - x + 2xy- y - y + 2yz- z - z + 2xz- x = x 2 + / + z 2 + 2xy + 2xz + 2yz. Bien d6i ve phai P ta duqc : P = (x + y + z) 2 = [(x + y) + z] 2 = (x + y) 2 + 2z(x + y) + z 2 2 2 2 2 2 2 = x + xy + y + xz + yz + z 2 2 2 = x + y + z + 2xy + 2xz + 2yz. Ta thay ve tnii T dung bang ve phai P. V~y d~ng thuc da cho la dung. A ' Nh B A CJn xet. h ' k ', ( )2 2 2 2 2 2 2 ~n nen n o et qua x + y + z = x + y + z + xy + xz + yz de v~n dl;lng tinh nhanh ket qua. Dc;mg 5. TIM x THOA MAN DANG THUC Phvungphapgmi--------------------------------------. v ~n dl;lng cac hang d~ng thuc dang nh6' de khai trien ra roi thu gqn v~ d~ng ax= b. Suy ra X = b neu a::;:. 0, Vx E R neu a= b = 0, kh6ng c6 X neu a = 0, b::;:. 0. a Vi dl_ll. Tim X biet rang 2 (2x + 1)(1-2x) + (2x- 1) = 22. 2. TIKTToAN 8/1-A 17
(2x + 1)(1-2x) + (2x- 1) 2 = 22 1-4x 2 + 4x 2-4x + 1 = 22-4x + 2 = 22-4x = 22-2 -4x = 20 X= 20: (-4) X =-5. Cdnh bao! Khi v~n dt,mg hang ding thuc (a + b)(a - b) = a 2 - (2x + 1 )( 1-2x) thl b Ht 2x chu khong phai la 1. b 2 d~ tfnh Vi dl,l 2. Tim X biet rang (x- 5) 2 + (x- 3)(x + 3)- 2(x + 1 ) 2 = 0. (x- 5) 2 + (x- 3)(x + 3)- 2(x + 1) 2 = 0 (x 2 - lox+ 25) + (x 2-9)- 2(x 2 + 2x + 1) = 0 x 2 - lox + 25 + x 2-9- 2x~- 4x- 2 = 0-14x+14=0-14x=-14 X= 1. D[Jng 6. CHUNG MINH CHIA HET Phvdngphapgmi---------------------------------------. V~n dt,mg cac hang ding thuc dang nha de bien d6i s6 da cho ve d~g a = k.b (k :;t: 0). Luc d6 a: k. Vi d1,1. Chting minh dng hi~u cac binh phuang cua hai so ch~n lien tiep thl chia het cho 4. Gqi hai s6 chan lien tiep la 2a va 2a + 2 (a E Z). Hi~u cac blnh phuang cua chung la : (2a + 2) 2 - (2a) 2 = 4a 2 + 8a + 4-4a 2 = 8a + 4 = 4(2a + 1) : 4. 18 2. moan st1 B
Dl;lng 7. CHUNG MINH GIA TRJ CUA M(H BIEU THUC LUON LUON DlfdNG (HAy AM) VOl MQI GIA TR.j: CUA BIEN PhLidng phap giai--------------------, Mu6n chll'ng minh gia tri ciia m t bieu thuc luon luon duang v6'i mqi gia tr! ciia bien, ta v~n dt,mg cac h~ng d~ng thuc A 2 ± 2AB + B 2 = (A ± B) 2, de bien d6i bieu thuc v~ d~ng [f(x)] 2 + k v6'i k > 0. Mu6n chll'ng minh gia tr! ciia m t bieu thuc luon luon am v6'i mqi gia tr! ciia bien, ta bien d6i bieu thuc v~ d~ng -[f(x)] 2 + k (v6'i k < 0). Vi d1,1 1. Chll'ng minh gia tr! ciia bieu thuc P = x 2 - mot x. 2x + 3 luon luon duang v6'i P = x 2-2x + 3 = x 2-2x + 1 + 2 = (x- 1) 2 + 2. Vi (x - 1 ) 2 ;::: 0 v6'i mqi x nen (x - 1 ) 2 + 2 > 0 v6'i mqi x. Vi d1,1 2. Chll'ng minh gia tr! ciia bieu thuc Q = 6x- x 2 - mqi gia tr! ciia x. 10 luon luon am v6'i Q = 6x- x 2-10 = 6x- x 2-9- 1 = -(x 2-6x + 9) - 1 = -(x- 3) 2-1. VI -(x- 3) 2 :S 0 v6'i mqi x nen Q < 0 v6'i mqi x. Dl;lng 8. TIM GIA TR.j: NHO NHAT, GIA TR.j: LON NHAT CUA BIEU THUC 2 P(x) = ax + bx + c PhLidng phap giai-------------------, Mu6n tim gia trj nho nhat ciia bieu thuc P(x), ta v~n dt,mg cac h~ng d~ng thuc A 2 ± 2AB + B 2 =(A± B) 2 de bien d6i P(x) v~ d~ng [f(x)] 2 + k (k Ut h~ng s6). Vi [f(x)] 2 ~ 0 nen P(x) ~ k. Do d6 gia tr! nho nhat cua P(x) 1a k (ta phai tim x de f(x) = 0). Ta viet min P(x) = k. Mu6n tim gia tr! Ian nhat ciia bieu thuc P(x), ta bien d6i P(x) v~ d~ng -[f(x)] 2 + k (k Ia h~ng so). VI -[f(x)] 2 :S 0 nen P(x) :S k. Do d6 gia tr! 16'n nhat ciia P(x) Ia k (ta phai tlm x de f(x) = 0). Ta viet max P(x) = k. 19
Vi dt.il. Tim gia trj nho nhat cua bi ~ u thuc P = x 2 + lox+ 28. P = x 2 + lox+ 28 = x 2 + I Ox + 25 + 3 = (x + 5) 2 + 3. Vi (X + 5) 2 ~ 0 nen (X+ 5) 2 + 3 ~ 3 (dau "=" xay ra khi Va chi khi X = -5). V~y min p = 3 khi va chi khi X= -5. Vi dt.i 2. Tim gia trj nho nhat dta bi~u thuc Q = 5x 2 - lox. Q = 5x 2 - lox = 5(x 2-2x) = 5 [ x 2-2x + I - I] = 5 [ (x - I ) 2 - I] = 5(x - I ) 2-5. VI 5(x- 1) 2 ~ 0 nen 5(x- 1) 2-5 ~ -5 (dau "=" xay ra khi va chi khi X = 1). V~y min p = -5 khi va chi khi X= 1. Vi dt.i 3. Tim gia trj 16'n nhat cua bi ~ u thuc P = x- x 2-1. P = x- x 2-1 2 1 3 =x-x ---- 4 4 = -(x 2 -x+±)-! 3 4 20