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Phuomg trinh, bat phuang trinh la nhOng noi dung can ban trong chuong trinh toan ph6 thong.. Co dugc ky nang t6t trong viec giai phuang trinh, bSt phuang trinh se khong nhung gop phan qu

THI TRirOfNG CHUYEN

^ O N THI THPT QUOC GIA

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Phuomg trinh, bat phuang trinh la nhOng noi dung can ban trong chuong trinh toan ph6 thong Co dugc ky nang t6t trong viec giai phuang trinh, bSt phuang trinh se khong nhung gop phan quan trong dS hinh thanh va phat trign nang lire giai quyet van de ciia hoc sinh ma con giup cac em dat k6t qua t6t trong nhtjng ky thi quan trong nhu: thi vao truang Chuyen, thi dai hoc, thi hoc sinhgioi cac d p

Vai muc dich ay, chung toi bien soan cuon sach nay nham cung cap cho ban doc mot he thdng bai tap phong phu, da dang vai nhi§u bai mai la va cac phuang phap giai hieu qua ve phuang trinh, bdt phuang trinh

Noi dung ciia cu6n sach dugc trinh bay thanh ba chuong:

Chmmg 1 de cap den phuang trinh bat phuang trinh dang da thuc va huu ty; ChiroTig 2 de cap den phuang trinh, bat phuang trinh v6 ty;

ChiroTig 3 va Chirong 4 theo thu tu de cap den phuang trinh, bat phuang trinh

Chung toi hy vgng rang cuon sach "Phuang trinh, bat phiromg trinh va

phuang phdp gidi" se thuc su huu ich cho cac em hoc sinh cung nhu cac thay,

CO day Toan a truang pho thong

Du da hit sue c6 ging trong qua trinh bien soan, nhung bg sach kho tranh

khoi nhung thieu sot nhat dinh Cac tac gia chan thanh cam an y kien dong gop cua cac thay giao, c6 giao va cac em hoc sinh gan xa de Ian tai ban bg sach se dugc hoan thien hon

Mgi y kien dong gop cho tac gia xin quy ban dgc gai ve:

nhasachhongan@hotmail.com

C A C T A C GIA

1

ChUctng 1

PHLfdNG TRINH, BAT PHtTdNG TRINH HlTU TI

§1 TAM THlTC, PHirONG TRINH, BAT PHlTOfNG TRINH BAG HAI 1) DIU cua tarn thipc bac hai

Tom tit ly thuyet

7.7 Dinh ly ve ddu cua tarn thuc

Cho tarn thuc bac hai f{x)^ax^+bx + c,a^Q Dat A = b^ - 4ac Khi do: NSu A<0 thi af (x)>0 vai moi xeR;

NSu A = 0 thi af (x)>0 voi moi x^-2a

moi xe{x^;x2), trongdo x, <X2 la hai nghiem ciia /(x)

7.2 Dieu kien khong doi ddu cua tarn thuc

Cho tam thuc bac hai f{x) = ax^+bx-\-c,a* 0 Dat A = b^ -4ac Khi do

1.3 Gid tri Ian nhdt, gid tri nhd nhdt cua tam thuc

Cho tam thuc bac hai f(x) = ax'^+bx + c,a^ 0 Ta c6

Vdfi a < 0, / (x

2a

Vi du

Tim gia tr

i lo

m nhd

t

cuabiduthuc A = 9xy

+ I0yz

+

1 Izx

LcigiaL Tha

y z

= \-x-y

vao ^ tac

6

A = 9xy + \(iyz +

nzx^9xy + z(\Qy +

\\x)^9xy + (\-x-y){\Qy +

+llx + 10>;-12x

v

-^

• =

llx'+

(ll-12

>;

)x-I0/

+10>

+ A = 0

»-+176

;/ + 121-44^

>0

' ll

ciia A

la

495

148 0-

2y

2 •

(Di thi dgi hoc

khoi B 2008)

X

2fx

^+

6xv) 2(

x^

+6xy) 2/^+12

^

Vai>

;^0,da

tr =

-tac6

P^-^

-^ = -

2x

v + 3

/ t^+2t

+ 3

Dodo P[t'+2t + 3)

= 2t^+\2tc>{P

-2y+2{P-6)t

+ 3P = 0.

Voi P^2, phuomg trinh c6 nghiem khi vachi khi

Ket hop lai ta c6 gia tri nho nhdt cua P la - 6 , gia tri Ion nhdt cua P la 3

Vi du 3 Cho a,b^O Tim gia tri nho nhdt cua bi^u thuc

P = a' +b' +^ + -

- a a L&igidL Xem P nhu la mot tam thuc bac 2 doi voi bien b

Taco P = b' +2b— + -K +

D4U bang xay ra khi

b = 2a

-a' = 4a' 2

Vi du 4 Cho cac so duong a, b, c thoa man a + b + c = 3 Chung minh rang

Dat / ( a ) = (2c + l)a^ +(2c^ - 5 c - 4 ) a + ^ > 0 Tachung minh / ( a ) > 0

Ta CO / ( a ) la mot tam thuc bac hai c6 he so ciia a' la 2c +1 > 0, va lai c6

A = ( 2 c ' - 5 c - 4 ) ' - 1 8 ( 2 c + l ) = ( 2 c - l ) ' ( c ' - 4 c - 2 ) < 0 do 0 < c < 3

3 1

Tir d o/( a) >

0 Da

u ban

g xa

y ra khi a = — ;6 = l:c = —

2 2

Vi d

u 5.

Cho 4 s6 thu

c a,6,c,£/tho

a man : + 6^ = 1; c - <i = 3

Chung min

h rSn

g F

= ac + bd-cd<

3)c - 3b

{a + b + 3)

= (a

- 6) + 2a

1 tre n [-72;V2]

ta c

6 /(r ) < 9 +

6V2

Do d

o F <

, a =

— , 6

cY-a(b + c) +

>0

8

<;=>(Z) + c) -2-{b + c) + — + >0

b + c

+ • 12a > 0, luon dung vi > 36

Vi du 7 Cho hai s6 x,y thoa man -2xy-2x + 4y-7 = 0 Tim gia tri

cua x \ihi y dat gia tri Ion nhit

(Di thi tuyin sinh THPT Chuyen Qudng Ngai 2013)

ti

<::>x^ - 2{y + \)x + y^ +Ay-1 = 0

De ton tai gia tri cua x thi phuomg trinh tren phai c6 nghiem, do do

Khi y = A thay vao phuong trinh ta c6 x = 5

4 Cho cac so thuc a,b,c,d sao cho \ <2 \k a + b + c + d-6

Tim gia tri lom nhat cua P = a^ + b^ +c^ +d^

(Di thi tuyin sinh THPT Chuyen Long An 201 A)

5 Cho a,b la cac s6 thuc thoa man a + b = a^ -ab + b^

Tim gia tri Ion nhat ciia +b^

- x +1) +

x^ + y^ + z^ <ll

Hir on

g d an giai ba

i ta

p p ha

n 1.

1

1 Ta

<=

>6 x^

+7 /- 8x -8

>; + 8xv +

/-; +

l =

0 ,

ta c6

A'

= 16(:

i;

-1)'-6

(7 /- 83

; +

1) =-26/

+16>

^-Do do

z l

a ha

i nghie

m cua phuong

trinh

- (

5 x)/

+ x

^ 5x +

8 =

0

Phuong trinh c6 nghiem

A' >

0 <

= > (

5 x)^

- 4(x

^ 5x +

+10x-7

u v

oi y,z

ta c6 dieu phai chun

g minh _ '

T'

3 Vo

i >

; = 0,tac

6 x;

tO, P = l

' ' ^^r^^^i ^il^

ml

V a i a = 2,6 = 2,c = 1,^/ = 1 hoac cac hoan v i t h i P = 10

Vay gia tri Ian nhdt ciia P la 10

y.rs

1

5 Tir a + b = a^-ab + b" =(a + b)^-3ab>(a + bf -^{a + bf =-^(a + bf

Dodo {a + bf-4(a + b)<0<^0<a + b<4

T a c o +b' =(a + b)[a^-ab + b^)^{a + bf <16

c + l ) + 3|2x-l|=-(2x-l)'+3|2x-l|-

3

= 2x- lp+

-3|2 x-l

3^' t

-3 3'

3 1 Da — — <

u ' = ' xa

= z +

l Tac

o a,b,ce[-2;2],

a + b + c =

(x2; + co)

• Tap nghiem S cua bat phuong trinh / ( x ) > 0 ducrc xac dinh boi bang sau

13

Lai d

o X, + ^2 =

Vay a

<0 hoac

0

L&igidL Taco 2x +

\4p^-7 = 0^x =

-lp^

Vi -7/?

^ + < + ^ ne

'

Gia s

u a

la mot nghie

a + — >

-S — + -

i vo

i ye

u cku

bai toan

voi

2<-7/7'+

-<

3<

p'+

—<

4,nentaco—

</

?'<

—

2 4 14

14

Vi d

y 3 Chun

Bat dang thiic can chung minh tra thanh

Vay nghiem ciia bat phuomg trinh la 6 < x < 8

Vi du 5 Tim m sao cho bat phuomg trinh dung vai moi x e ^

2m<0

x^ +{m + 7)x

—z

<

1

ax -4x + a-3

3 Ch

o f[x)

= x^+ax + l

voi 3<a<

—

Giai hit

phuong trinh

+ m-\>0

5 Ti

m m

de bat phuon

1 K

i hie

u

Huang dan giai ba

w + 2)x + 2m<

0 (1)

+(7w + 7)

x + 7m<0

w-^>

0

;A2=

(m-7)^

>0

Ta

CO phuon

g trin

h (m + 2) x + 2

) x + 7

e v

6 nghiem

Voi m^2\m^l,

m>0 nghie

m cu

a (l) l

a 5, =(w;2) hoa

c S^-[2•,m)•,

'nghiem cua (2) l

a ^2

=(

-m

7) hoac

^3

=(

-7

w) R

Voi m<0

nghiem cua (l) la

5,

=(

w;

2), nghie

m cu

a (2) l

a -{-l;-m)

16

Ta CO iS", n <S'2 ?t 0 nen he luon c6 nghiem Vay w < 0

2, Ta tim diSu kien dS ox^ - 4x + a - 3 9^ 0, Vx e R

Taco ca^-4x + a-3^0yxsR^

'/'{x)-x'] + a[f{x)-x] =[f{x)-x][f{x)

+ x + a] e

l

^(x^ +(a-\)x + \)[x'

+{a + \)x +

a + 2)

Bat g{x) = x'+{a-l)x

+ lc6 A^=a'-2a-3

| ne

n

^^, dod

6 h{x)

= x^

+{a+ \)x + a +

2>0,\fxeR,

tudo f(f{x))-x>0

l-a -V a'-

2a -3

<=>g(x)>0<=>x^ +

-l)

x + l>

Oo

x>

2

l-a + V a'-

fm -l>

0 A<0

x + 4>0,VxGl

A,=

m)' -16

t phuon

g trin

h c6 tap nghie

m la

R

3) Mot s6 dang phu 'O'ng trinh hCeu ti dipa v§ b^ic hai

Dang t6ng quat: au^ (x) + bu{x) + c = 0

3.1 Dang trung phuong ox^ + + c = 0

Cdch gidi: Dat = ^ > 0 dk dua wh phuong trinh bac hai theo t

3.2 Dang (x + a)"^ + (x + Z?)'* = c

Cdch gidi: Dat x + = t se thu dugc phuong trinh trung phuong theo /

3.3 Dangnghich dao ox'* + bx^ + cx^ ±bx + a ^ 0, a ^0

Cdch gidi: Chia ca hai ve cua phuong trinh cho x^ 9^ 0 thu dugc phuong trinh

trinh tren se thu dugc phuong trinh bac hai theo t

3.4 Dang h6i quy ax"^ + bx^ + cx'^ + dx + e = 0, voi — =

k - ~ -) k~

Dat x + —= bieu dien x + ^ theo t, thay vao phuong trinh tren se thu

dugc phuong trinh bac hai theo t

3.5 Dang (x + a)(x + fe)(x + c)(x + c/) = e, voi a + <i = 6 + c

CflcA ^ifl/; Viet phuong trinh duoi dang

((x + a)(x + ^))((x + 6)(x + c)) = e

«> ^x^ +{a + d)x + ad^{x^ +{b + c)x + bc^ = e

D?lt x^ +(a + d)x = r thu dugc phuong trinh bac hai theo /

19

3.6 Dan

g (

x +

a)(x + b)(x +

c)(x

+ d)-ex ,\(n ad

+ c)) = ex^

<=>(x^

+(a + (i)

x +

fl<ij(x^ +(b

+ c)x + bc^^ex^

Cdch 1.

Dat x

^ +

a + b-\-c +

d

x + ad-t th

u tru

e tiSp

Xet X ?t 0, chi

i t

X

ad

x + {a + d) +

— X +

x +

a

X

Dat = ?, th

X +inx + k

X +nx + k

Cdch giai:

Xet x = 0 Khon

Xet X ^ 0 Vid

Dat x

+ — = t,

phuong trinh tra thanh

—

^ + —

^ =

c,

tu do dua

phuong

X

t +

m t + n

trinh bac hai the

o

Cdch gidi: Gia sii v(x) ^ 0 Chia ca hai vg cho v^(x) ^ 0, r6i dat = f d6

v(x)

dugc phucmg trinh bac hai theo t

3.10 Dang au^(x)v^(x) + b{u{x) + v{x)f +c = 0, vai u(x)-v(x) = k

Cdch gidi: Viet phuong trinh da cho ve dang

aw^ (x)v^ (x) + 46w(x)v(x) + c + 6yt^ = 0

Dat u{x)v(x) -1, thu dugc phuong trinh bac hai theo t

3.11 Dang x'=ax^+bx + c

Cdch gidi: Ggi a la s6 thuc thoa man 6^ = 4(a + 2a)(c + «^)

(Day la mot phucmg trinh bac ba d6i vai a nen luon ton tai a) j Khido: x'=ax'+Z>x + c ^x' +2ax' ={a + 2a)x^ +bx + (c + a')

<^(^x^ +af ={a + 2a)x^ +bx + (^c + a^y

NSu a + 2a ;t 0, khi do vl phai (a + 2a)x^ + 6x + (c + ) la mot tam thuc bac hai CO A = b^-4{a + 2a)[c + a^) = Q

NIU <3 + 2a < 0 thi FP < 0, > 0, do do ding thuc khong xay ra

NSu fl + 2a = 0=>6 = 0.Tac6: (x^ + a)^ =(c + a^), tu do tim dugc x

LoigidL

DiSukie

n: +Ax

+ 2^Q, +lx +

2^Q

Ro rang x = 0 khong pha

= 1

<:>9/ +

12 = (/

+ 4)(

r + 2)

or-3/-

4 = 0

<»

t = -\

t =

4

Va

i f = -

l tac6

x + — = -l<

=>

x^

+x + 2 = 0, v6nghie

<::

>x

4x + 2 = 0

ciia phuang

trinh lax = 2 ± V2

Vi du

-(De thi tuyin sink

i

x'+4-3x + - -2 = 0

Dat ?

= X

, kh

<»

x — = 2

«x

2x-2 = 0

^-«x =

Wai t = l<^x = l<i:>x^

-x-2 = 0^

x =-

x =

2

Vay nghiem

cua phuang

trinh la

x =

1 ± V3 , x = -l;

x = 2 •

- {

Vi du

+ x + 2) =

12

(Di thi tuyin sink

THPT Chuyen

DH Vinh

2008)

L&igidL Qat +x + l = t,(t >0) phuang trinh da cho tra thanh: /(r +1) = 12

» r^+^-12 = 0»r = 3,dor>0

''x = l

Vay nghiem cua phuang trinh la x = -2; x = 1,

Vi du 4 Giai phuang trinh jc" + - 8JC -12 = 0

(Di thi tuyin sinh THPT Chuyen DH Vinh 2006)

Vay nghiem ciia phuang trinh la x = -1 ± V?

(De tuyin sinh THPT Chuyen Lam Son Thanh Hoa 2013)

L&igidL *) Dieu kien x^-5

Phuang trinh da cho tuomg duang vai

Vay phucm

g trin

h c6 hai nghie

m x = 1±V

x + 5 ) =82

(Di tuyin sink THPT

Chuyen Lam San Thanh

H6a 2013

)

LM giaL

Dat ? = + 3x + 4 T

a c

6

(^ -1 )'+

x + 6 = 0 (v6nghi?m )

^ +3

x + 2 = 0

o x = -l x = -2

Vi d

u 7 Bil

6 nghiem Chun

THPT Chuyen

Ha Tinh

XQ + OXQ

+ 6X 0

+ OXQ

1 =

0 <=

> XQ + OX

Q +

6 + — + ^ =

+ b = 2-t^

Do do ( 2 - f = (at + bf <{a'+b'){t'+!)<:> a'+b'>

— + 4 4

luon dung v i \t\>2

Vay ta c6 di6u phai chung minh

V i du 8 Chung minh rang phuomg trinh sau khong c6 nghi?m nguyen

(x-y)(x-2y)(x-3y)(x-4y) + /+2 = z \ (De thi tuyin sink THPT Chuyen Phan Boi Chdu Nghe An 20\4)

L&igiaL Ta c6 (x-y)(x-2y){x-3y)(x-4y) + y' +2 = z^

<=> ((X -y){x- 4y)){{x - 2y){x - 3y)) + /+2 = z'

^[x^-5yx + 4y^)[x^-5yx + 6y^) + y*+2^z\

Bat t = x^-5xy + 5y\Tac6

(t'-/)(t'+/) + /+2 = z'<^t'+2 = z'<=>(z-t')(z + t') = 2

Khong m4t tinh tong quat gia su z > 0, khi do ta c6 • ^ ' ^ ^ ^ ,

Vay phuong trinh da cho Ichong c6 nghiem nguyen

V i du 9 Giai phuong trinh = ~x^ - 4x + 3

x(2xl) "-^

-(Di thi tuyin sinh

THPT Chuyen Quang Trung

= 0

(DJ thi

tuyin sinh THPT Chuyen

Lam Som Thanh H6a

x - 8)

^ = 16

i la nghiem ^

• ^

Voi X ^ 0 , chi

6 /^-5

? + 4 = 0

Vay nghiem ciia phuong trinh la x, ^ = ^ - +

3 Taco x'-4yf3x-5^0^x' =4yf3x + 5

x'-2x + - -0

x^+2x +

- =

0

V2-

6 r '-2 r-3 = 0

«

Voi / = -1 , ta

CO

x^ 2x = -1

<:> X =

1

Voi ? = 3,tac

6 x^-2

x = 3o x^- 2x -3 = 0<:^

t = -l t = 3

6 Dieukie

n x^±2 Da

t fl = ^

^; 6 = ^

^

x+2

x-1 x + 1

1

<»(x-l)(x-2) = (x + l)(x +

2)<»x

=

3-<»(x-l)(x-2) = 3(x +

l)(x +

«2x'+12x +

4 = 0o

24)(x^ +

1 Ix + 24 ) = 4

Voi X = 0, khong phai la nghiem

Voi X 0 , tir phuong trinh (x^ +\4x + 24)(x^ + l l x + 24) = 4x^ chia cahai v l

^~ 2

29

Vod ^ = -tac

6 12x'+ll

x + = -<=>12x'+llx-

-2 = 0o

x = -

^^

^^

Voi ' = taco 12x^+ll

x + ^ = -|<::>12x^+ll

x + 3 = 0(v6nghi?m)

0,

taco

X

— = 0<»x^-

2 = 0«

x =

Vdir = 2,tac6x —

= 2 ox

^-2 x-2 = 0<::>

16<»f'+6r'-7 = 0<=>f'=l (v

Taco /'=

1.2 Dinh ly Bezout ^ j ^ y;^ ^ =

Cho P(x) = aQx" + + a„_,x + a„,('3o ^ 0 , « > 1) va XQ K h i do XQ

la nghiem ciia P(x) khi va chi khi P{x) = (x - Xo)S(x), trong do Q{x) la da

Bo so 6o,6j, 6„ do dugc xac dinh bai 6o = ^c^* - ^o^*-i +cii^,k = 1,2, ,«

Chu y rang Z?„ = P(XQ), nen XQ la nghiem cua P(x) khi va chi khi Z>„ = 0

1.4 Dinh ly ve nghi?m hihi ti

Gia su s6 huu t i r = —(p e Z , o eN*,(p,flr) = l ) la nghiem ciia da thuc he s6

nguyen P{x)^aQx"-^a^x"'^+ + a„_^x->ra„,{aQ^Q,n>\) K h i do p\a„

" + + +

<3„_,

x + a„

deu la so nguyen

b) Mo

" +a,x"

"' +

+ a„_,x + a

„

deu lauac ciia a

„

7.5 Dinh

ly ton tai nghiem

thuc

a) Mo

b) Gi

a s

u a,b{a<b)

la hai s

6 thu

c v

a P(

x) la da thuc thoa

man

P{a).P{b) <

0 K

hi d

o P(

x) c6 nghiem

thuoc khoang

(a;b)

1.6 Dinh

ly phdn tich da

-(

^^

+PiX + qi),

a da

y pf

-4^, <0,k +

2l =

n N

oi eac

h khac, mo

i d

a thu

c dku

phan tich

dugc

thanh cac thua s6 bac

"' +

+fl„

_,x+

da thirc.

Gia su

a 0 va

x) = a(

x XjXx

0,l,

ki P(

x) c6 dau nguac

L&i gi&L

Dat P(

x) = x'+

l

Ta c6 c6 P(l)P(-l) = l.(

-3

)<

0 ne

n

Matkhac v o i m o i x < X Q , P{x) = + x-I < xl + -1 = 0 vayai moi X > X Q ,

P{x) = x^ +x-\>xl+Xo-l = 0, nen phuang trinh da cho Idiong c6 nghiem

Vi du 2 Cho da thuc P{x) = x' +x'-9x' +ax^ +bx + c

BigtrSng P(x) chiahStcho ( x - 2 ) ( x + 2 ) ( x + 3 ) Hay tim P(x)

L&igidL Taco P{x) chiahgt cho ( x - 2 ) ( x + 2 ) ( x + 3) nen

P(2) = 0; P(-2) = 0; P ( - 3 ) = 0 T u do thay vao ta c6 a = - 1 ; 6 = 20; c = - 1 2

Da thuc can tim la P(x) = x^ + x^ - 9x^ - x^ + 20x - 1 2

Vi du 3 Tim m de phuang trinh ~x ^ - x + m + - = 0c6 3 nghiem phan

x , ^ > 1 4 <=>|An| > 1

Vi du 4 T i m m de phuang trinh x^ - 3mx^ + 9A: - 7 = 0 (1)

CO 3 nghiem phan biet lap thanh cdp s6 cong

m

CO

-2m^ + 9m -

5 2 -1-Vl5

_ , , , -l->/l5

-m + l)

x - 2 = 0 c

i do

ta c6 :

gix) = {x-x^){x -

){X - X^

)

jCj +

^2 + JC3 =

3m

X^X2 + X2XT +

-m -1 =

4 + \/2.3

m <=

> m = —

m thi

y tho

a man Va

6 4 nghie

m pha

Bai 2 Ti

o 2

a + 6

6 +19

c = 0 Chun

Bai 4 Cho P{x) la da thiic xac dinh tren [a; b] va n dilm x^, , ,x„ e [a;6] Chiing minh r&ig t6n tai c e [a; b] sao cho P{c) = — [ ) + P(x^) + + )

n

Bai 5 Cho da thuc vai he so thirc ^

> P ( x ) = x"+a„_ix""'+ + a,jc + ao, Q{x) = x^ +X + 2

Bik ring da thiic P{x) c6 « nghiem thuc phan biet va da thuc

khong CO nghiem thuc Chung minh ring P(2)> —

Huang dan giai bai tap phan 1

l.Taco jc'*-(3m + 2)x^+3m + l = 0

Tir do de phuong trinh da cho c6 4 nghiem phan biet deu nho hon 2 khi va chi

khi (2) CO hai nghiem phan biet khac ±1 va nho hem 2

0<3m + l < 4 f 1

<=> \

Dat t^x^,t>0 thi (1) tro thanh: /(t) = - 2(m +1)? + 2m +1 = 0

Goi < ?2 la 2 nghiem cua /(O = 0, khi do cac nghiem cua (1) Idn lugt la:

<^ X2 - x^ - x^ - X2 = x^ - x^ <=> t2

-it

5m = 4(

m + l)

<»

5m = 4m+ 4

Z?x + c

„ „

) + +

gix,) = P{x,) [Pix,) +

P(x,) + + P(x„)

g{x„ [P{x,) +

P{x,) + +

Do d

o g(x,) + g

(x 2) +

+ g(x„) =

o d

o t6

n tai

c

nSm giua

0, t

a c

6

P{c) = -[P{x,) +

P{x,) + + P{x„)\

5 D

o P{x)

CO «

nghiem thirc pha

M,(/

= 1,2, ,«)

<;:^x'+

X +

2

-X , 9^0,Vxe]

R

o A < 0 <=> < 2 - X,, (/ = 1 , 2 , n ) ,

Dod6P(2) = ( 2 - x , ) ( 2 - x , ) ( 2 - x „ ) > i ,

2) Phiro'ng phap giai

2.1 Phuang phdp phan tich

Vi du 1 Giai cac.phuang trinh sau

Lei giai 1) Nhan thay phuang trinh c6 cac nghiem x = 2, x = 3 Tu do ta phan

tich thanh nhan tii

^x = 2 , x^- 8x^+21x-18 = 0<:^(x-2)(x^-6x + 9) = 0 o

2) Ta nhSm x-yjl la nghiem Khi do, phan tich dugc:

o (3x

+ 2)[9x^

- 21x-18) =

3

x = -3

x = —

2

4) T

a C

O X = V

x ^/

-3 =

0 3

-V3 x'

- 2V3x' + 2.3

x +

X V3

= 0

" x = V3

<=>(x-V3

)(x'-2V3x + l) =

x^

+j

c + 6) = 0<»

phuang trinh

2x^ + lOx = 1

2 <

^ + 5x

- 6 = 0 «

0 ox

-^ + 5mx + 2m^ =

0 c

6 3

nghiem phan biet

Lai giai

Ta c

6 x

^ (3 + 2w)x

-^ + 5/nx + 2w^ =

0

X =

2m

x'-3x-

m = 0(2)

De phuan

g trin

h c

6 3 nghiem

phan biet kh

i v

a ch

i kh

i (2) c

uiifi,^ -(Ism

m^O;m^ —

4 -9

Vi d

u 3 Gia

i phuan

g trin

h (x

^ + 3x

- 4)

^ + 3(x^

+ 3

x 4) =

+ 3(x

^ +

4) = x + 4

x-( x - l ) x-( x + 4)] +3x-(x-\)x-(x + 4) = x + 4

o ( x + 4 ) ( ( x - l ) ' ( x + 4) + 3 ( x - l ) - l ) = 0

(x + 4)(;c' + 2x' - 4x) = 0 <=> X ( x + 4)(x' + 2x - 4) = 0 , ^ , ^ ^ y

Vay nghiem ciia phuong trinh la x-0;x 4;x \±^

Vi du 4 Giai hk phuong trinh (x +1)(x + 2)(x + 3)(x + 6) > 35x'

L&igidL Taco (x + l)(x + 2)(x + 3)(x + 6)>35x'

>35x' (x + l)(x + 6)][(x + 2)(x + 3)

Vi dy 5 Giai bdt phuong trinh (4x + l)(l2x - l)(3x + 2)(x +1) < 4

x +

+ 2015x'''"'

Phuang trin

h da cho tr

a than

h (x -1 )^

5 = 1 (1 )

Taco x

5 = x + 2x

2015x''"' - (

l + x + x' + + x'°''), ha

^+

+ x^""") ) = 1

«(x-l) 2015x'''''-

-m =

Bai 2 Ti

40