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 gia i 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 . trQng tam

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.

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 gia i 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

4

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 Nam, 1878 Giang V6, qu~n

Dong Da, Ha N<?i.

Xin chan thanh cam on.

cAcTAC GIA

    

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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) -3x2(5x3- 4x2 + 3x- 1).

Gidi

a) _!_x 3(x2 - 6x -10) = _!_x 5 - 3x4 - 5x3. 2 . 2

b) -3x2(5x3- 4x2 + 3x- 1) = -15x5 + 12x

4- 9x3 + 3x

2.

5

Vi d~;~ 2. ThtJC hi~n cac phep tfnh

a) (x + 8)(x - 5) ;

b) (2x-'- 1)(3x2 -7x + 5).

Gidi

a) (x + 8)(x - 5) = x2- 5x + 8x- 40

= x2 + 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)= x2 +(a+ b).x + ab.

b) (2x- 1)(3x2- 7x + 5) = 6x3

- 14x2 + lOx- 3x2 + 7x- 5

= 6x3- 17x2 + 17x- 5.

Vi d~;~3. Tim h~ s6 cua x3• trong ket qua cua phep nhan (x2

- x)(x2 +X- 1).

Gidi

(x2- x)(x2 + x- 1) = x4 + x3

- x2- x3

- x2 + x.

= x4- 2x2 + x.

V~y h~ s6 cua x3 la 0.

D{lng 2. RUT GQN BIEU THUC VA TfNH GIA Tlq CUA BIEU THUC

6

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(x2- 4x - 5)- 5x2

.

? 2 2 Giai . A= 8x(x- 2)- 3(x - 4x- 5)- 5x .

A= 8x2- 16x- 3x2 + 12x + 15- 5x2

.

A= -4x + 15.

Vi d~;~ 2. Rut g<;>n- bi~u thuc

B = 2(x- 5)(x + 1) + (x- 3)(x + x2).

Gidi. B = 2(x- 5)(x + 1) + (x- 3)(x + x2)

B = 2(x2- 4x- 5) + (x2+ x3

- 3x- 3x2)

B = 2x2- 8x- 10 + x2 + x3

- 3x - 3x2

B=x3 -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.

Gidi . A= (x + 5)(2x- 3)- 2x(x + 3)- (x- 15)

A=2x2 -3x+ 10x-15-2x2 -6x-x+ 15=0

Ket qua Ia da thuc 0, da thuc nay khong c6 b~c.

Vi dt,~4. Cho bieu thuc B = 5x2(3x- 2)- (4x + 7)(6x2- x)- (7x- 9x\

Rut gon r6i tfnh gia tri cua bieu thuc B v6i X = -2. . . 4

B = 5x2(3x- 2)- (4x + 7)(6x2- x)- (7x- 9x3

)

B = 15x3- 10x2

- (24x3- 4x2 + 42x2

- 7x)- (7x- 9x3)

B = 15x3- 10x2

- 24x3 + 4x2- 42x2 + 7x- 7x + 9x3

B = -48x2

V6i x =-! thl B = -48.( -%J = -48. 1~ = -27.

Vi dt,~ s. Cho bieu thuc

C = x(x + x3) + (x- 1 )(x2 + x3

) + 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.

Gidi

C = x(x + x3) + (x- l)(x2 + x\+ I

C = x2 + x4 + x3 + x4- x2

- x3 + 1

C = 2x4 +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 .

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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.

Gidi

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.

Gidi . B = 10- 5x(x- 1,2) + 2x(2,5x- 3)

B = 10 - 5x2 + 6x + 5x2- 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?

Gidi

C = x(x - y) + y(x + y) - (x + y)(x - y) - 2l C = x2

- xy + xy + /- (x2 - xy + xy -/)- 2/

8

2 2 2 2 2 C = x - xy + xy + y - x + xy - xy + y - 2y

C=O.

V~y v6i m9i g ia 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 thanh ve kia ho~c bien d6i ca hai ve cung bang m(>t bi~u thvc.

Vi dt,~ 1. Chvng minh d~ng thvc 3 2 2 3 4 4 (x - y)(x + x y + xy + y ) = x - y .

Gidi

X , .- , . T ( )( 3 ? 2 3) e t ve tra1 = x - y x + x-y + xy + y . 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~ng thtrc da cho Ia dung.

Vi dt,~ 2. Ch(rng minh d~ng thtrc

(x + y)(x + y + z) - 2(x + 1 )(y + 1) + 2 = x2 + l . Gidi

Xet ve tni i 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 tra iT dung bang ve phai P nen d~ng thtrc da cho Ia dung.

Vi dt,~ 3. Cho ab = I . ChLrng minh d~ng thvc

a(b + I) + b(a + I ) = (a + I )(b + I). Gidi

• 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----------------.

10

• 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

Gidi

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

x-1=4

X= 5.

Vi d~;~ 2. Tim x biet

Gidi

2 2 (x+ 1)(3x +x-2)-x (3x+4)=5.

(x + 1)(3x2 + x- 2)- x2(3x + 4) = 5

3x3 + x2- 2x + 3x2 + x - 2- 3x3

- 4x2 = 5

-x- 2 = 5

-X= 7

X =-7. Vi d~;~ 3. Tim x biet

GiJi

3(x - 2)(x + 3)- x(3x + 1) = 2.

3(x- 2)(x + 3)- x(3x + 1) = 2

3(x2 +x-6)-x(3x+ 1)=2 2 2 3x + 3x - 18 - 3x - x = 2

2x- 18 = 2

X= 20: 2

X= 10.

C. BAI TAP Tl,J LUY~N

1. Lam tinh nhan

a) -4x\x2- 3x + 2)

b) -~ x2(5x3 + 10x2 -15x). 5

2. Lam tinh nhan

a) (2x + 7)(3x- l )

b) (5x2- 4x)(2x2 + 9x- 3).

3. Tinh gia tJi cua bi~u thuc A v6i X = 999

A=x6 -x\x-l)-x4(x+ l)+x\x-l)+x2(x+ 1)-x(x-1)+ 1.

4. Cho bi~u thuc A= x(l + x)- x2(1- x) + x\x2

- 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

(x - 3)(x+x2)+2(x-5)(x+ 1)-x3 = 12.

6*. Cho ~ = I. Chll'ng minh rang a b

2 2 2 2 2 (x + y )(a + b ) = (ax + by) .

HUONG DAN - DAP s6

1. a) -4x5 + 12x4- 8x3

;

2. a) 6x 2 + l9x - 7 ; b) 10x4 + 37x3- 51x2 + 12x.

3. 1000.

4. A= x5 +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 la ba htmg d~ng thuc dau tien.

1. (A + B)2 = A 2 + 2AB + B2

2. (A - 13)2 = A 2 - 2AB + B2

3. (A- B)(A +B)= A2- B2

4. (A + B)3 = A 3 + 3A 2B + 3AB2 + B3

5. (A- B)3 = A3- 3A2B + 3AB2

- B3

6. (A+ B)(A 2 - AB + B2) = A3 + B3

7. (A- B)(A2 + AB + B2) = A3

- B3.

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.

12

Vi d1,1 1. Tfnh :

a) (4x + 7)2;

b) ( 6x- ~y r c) (3x2

- 5xy3)(3x2 + 5xy\

Gidi

a) (4x + 7)2 = (4x)2 + 2.4x.7 + 72 = 16x2 + 56x + 49.

b) ( 6x- ~y r = (6x)2

- 2.(6x)(~y J + (~y r = 36x2

- 4xy + il· c) (3x2

- 5xy3)(3x2 + sxl) = (3x2)2

- (5xl)2 = 9x4- 25x2

/.

    

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Vi d1,1 2. Tfnh :

a) (2x2 + 5y)3;

b) (3x3- 4xy)3

;

c) ( 6x + ~)(36x2 - 3x + ~)

d) (x - 5y2 )(x 2 + 5xy2 + 25l ).

Gidi

a) (2x2 + 5y)3 = (2x2)3 + 3.(2x2

)2

. (5y) + 3.(2x2)(5y)2 + (5y)3

= 8x6 + 60x4y + 150x2/ + 125l.

b) (3x3- 4xy)3 = (3x3

)3

- 3.(3x3)2 (4xy) + 3(3x\(4xy)2

- (4xy)3

= 27x9- 108x7y + 144x5

/- 64x3l.

c) ( 6x + ~)(36x 2 - 3x + ~ ) = (6x)3

+ (~J = 216x3

+ ~ · d) (x- 5/)(x2 + 5x/ + 25/) = x3

- (5/)3 = x3- 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) 8x3- 12x2y + 6x/-l.

Gidi

2 1 2 2 1 ( 1 )2

( 1 )2

a) 25x -5xy+"4:-Y =(5x) -2.5x.2

y+ 2y = 5x-2y

b) 8x3- 12x2y + 6x/-l = (2x)3

- 3.(2x{y + 3.2x./- y3 = (2x- yl

Vi d1,1 4. Di~n cac don thuc thfch hqp vao 6 tr6ng

a) (X - ~ r = X 2 - D + x12 ;

13

Gidi

( 1 )

2 2 1 a) X-- =X -0 + - .

x x2

Dt;mg 2. RUT GQN BIEU THUC VA TfNH GIA T~ CUA BIEU THUC

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.

14

• 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)(9x2- 3xy + l)- (3x- y)3

- 27x2y.

Gidi

a) 2 (7x + 4) - (7x + 4)(7x- 4)

= 49x2 + 56x + 16- (49x2

- 16)

= 49x2 + 56x + 16 -49x2 + 16

= 56x + 32.

b) (x + 2y)3- 6xy(x + 2y)

c)

= x3 + 6x2y + 12xl + sl- 6x2y- 12xl

= x3 + sl. (3x + y)(9x2

- 3xy + l) - (3x - y)3 - 27x2y

= 27x3 + l- (27x

3- 27x

2y + 9xl- y3

)- 27x2y

= 27x3 + l- 27x

3 + 27x2y- 9xl + y3

- 27x2y

= 2y3

- 9xl.

l

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

Gidi

A= 5(x + 3)(x- 3) + (2x + 3)2 + (x- 6)2

A= 5(x2- 9) + (4x2 + l2x + 9) + (x2

- 12x + 36)

A= 5x2- 45 + 4x2 + 12x + 9 + x2

- 12x + 36

A= 10x2.

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 = x2 + l . Gidi

2 2 2 2 B = x + y = x + y + 2xy - 2xy

2 B = (x + y) - 2xy

B = 152- 2.(- 100) = 425.

Vi d~;~ 4. Tfnh nhanh gia tr! cua bieu thuc :

a)C=392 +78.61 +612;

b) D = 502 -49.51.

Gidi

a) C = 392 + 78.61 + 61 2

C=(39+61)2

c = 1002 = 10000.

b) D = 502 -49.51

D = 502- (50 -1)(50 + 1)

D = 502- (502

- 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,~ 1. Cht.Ing minh gia tti cua bieu thuc sau khong phl,l thu¢c vao gia tr! cua bien:

Gidi

2 3 A= (3x + 2)(9x - 6x + 4)- 3(9x - 2).

A= (3x + 2)(9x2- 6x + 4)- 3(9x3

- 2)

A= 27x3 + 8- 27x3 + 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 ?

Gidi

B = (x + 1)3- (x- 1)(x2 + x + 1) - 3x(x + 1)

B = (x + 1)3- (x- 1)(x2 + x + 1)- 3x(x + 1)

B = x3 + 3x2 + 3x + 1 - (x3- 1)- 3x2

- 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

(x + y)2- (x- y)2 = 4xy.

Gidi

Bien d6i ve trai T ta duqc

T = (x + y)2

- (x - y)2

= x2 + 2xy + l- (x2

- 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) .

Gidi

• Bien d6i ve tnii T ta duqc :

T = 3(x2 + / + z2)- (x - y)2

- (y- z)2 - (z- x)

2

= 3x2 + 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

= x2 + / + z2 + 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) + z2

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.

Nh A ' B A h ' k •' , ( )2 2 2 2 2 2 2 CJn xet. ~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

Gidi

(2x + 1)(1 - 2x) + (2x- 1)2 = 22

1 - 4x2

+ 4x2- 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) = a2 - b2 d~ tfnh

(2x + 1 )( 1 - 2x) thl b Ht 2x chu khong phai la 1.

Vi dl,l 2. Tim X biet rang

(x- 5)2 + (x- 3)(x + 3)- 2(x + 1 )2 = 0.

Gidi

(x- 5)2 + (x- 3)(x + 3)- 2(x + 1)2 = 0

(x2- lOx+ 25) + (x2

- 9)- 2(x2 + 2x + 1) = 0

x2- lOx + 25 + x2

- 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.

18

Gidi

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 = 4a2 + 8a + 4- 4a2

= 8a + 4 = 4(2a + 1) : 4.

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 + B2 = (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 = x2 - 2x + 3 luon luon duang v6'i

mot x.

Gidi

P = x2- 2x + 3

= x2- 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- x2- 10 luon luon am v6'i

mQi gia tr! ciia x.

Gidi

Q = 6x- x2- 10 = 6x- x2

- 9- 1

= -(x2 - 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 + B2 =(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

20

Vi dt.Il. Tim gia trj nho nhat cua bi~u thuc P = x2

+ lOx+ 28.

Gidi

P = x2

+ 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 = 5x2- lOx.

Gidi

Q = 5x2- lOx

= 5(x2

- 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

Gidi

P = x- x2- 1.

P = x- x2

- 1

2 1 3 =x-x ----

4 4

= -(x2-x+±)-!

3

4