Chia biểu thức chứa nhiều căn

7/25/2019 Chia biu thc cha nhiu cn

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VNCASIOer Team______CHIA BIU THC CHA NHIU CN

I. PHNG PHP CHIA BIU THC DNG u v f

1. C s

Phng php c sng to bi Bi Th Vit.

Gi s ta cn chia2 2

1 1

g h fA u v f

u v f

trong g, h, f, u1, u2, v1, v2l cc a

thc ca x. Xt1 1

g h fB

u v f

, ta c: 1 1

2 2

1 12

A B gu hv f

u v f

v 1 1

2 2

1 12

A B gv hu

u v ff

Mt khc ta li thy:

1 1 1

1 1 1 1

2 2 2 2

1 1 1 1 1 1 1 1

1 1

2 2

g u v f h f u h f A B A B gu hv f gv huf f

u v f u v f f u v f u v f

g h fA

u v f

Vy:2

2

A Bu

v 2

2

A Bv

f

2. Cch thc tin hnh

Nhp1 1

g h f

u v f

, cho 1000X (hoc 100) v gn kt qu vo A. Sa biu thc

thnh1 1

g h f

u v f

(i du trc cn), bm v gn kt qu vo B.

Tnh 22

A Bu v 22

A Bvf

, t 2 kt qu p dng phng php xp xtruy


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ngc li c u2, v2theo x

Nu f v ngha ti 1000X (100), ta c th thay bng

1

100X hoc

1

1000X .

Do nu nhng gi tr ny u khng thay c th cch ny v dng.

3. V d

Thc hin php chia: 4 2 3 2

2

10 19 7 13 1

1 1

x x x x x x xA

x x

Nhp biu thc trn, cho 1000X ta c kt qu 2001495393 A , i du trc

2 cn ta c kt qu 490639,1897 B

Tnh

2

10009930162

A Bu , dng phng php xp xsuy ra 3 2

2 7 16u x x x

Tnh

22

21000003 3

2 1A Bv X

X X(cng l xp x).

Vy kt qu l 3 2 2 27 16 3 1A x x x x x x

II. PHNG PHP CHIA BIU THC NHIU CN

1. C s

Thc cht, y ch l m rng ca phng php trn, khi ta thay g v u1trong php

chia trn thnh biu thc cha 1 hoc nhiu cn khc nhau (v khc f ) thay v a

thc.

Thut ton: gi s ta chia 1 12 1 1

1 1 1

......

...

n n

n n n

n n

g h f h f A u w f w f

u v f v f


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t

1 1 1 1

1 1 1 1 1

2 1 1 1 1

...

...

...

n n n

n n n

n n n

g h f h f G

u v f v f U

u w f w f V

, ta c: n n nn n n n

n n n

G h fV w f A

U v f

, lc ny

ta s tm c2

n nn

n

A Bw

f

vi n n nn

n n n

G h fB

U v f

Bng cch tng t, t

1 1 2 2 1

1 1 1 2 2 1

2 1 1 2 2 1

...

...

...

n n n n n

n n n n n

n n n n n

g h f h f h f G

u v f v f v f U

u w f w f w f V

, ta c

1 11

12

n nn

n

A Bw

f

, vi 1 1 1

1

1 1 1

n n n

n

n n n

G h fA

U v f

v 1 1 1

1

1 1 1

n n n

n

n n n

G h fB

U v f

. C nh vy ta

tm c tt c cc h s ca cn.

Cui cng 1 12 1 1

1 1 1

......

...

n n

n n

n n

g h f h f u w f w f

u v f v f

, c th thay gi tr tm ra

d dng.

Ta nhn thy 1 2 1... n nA A A A A , do ta ch cn quan tm vic tnh ton 1B ,

2B , ..., nB m thi.

2. Cch thc tin hnh

y ta ch ly 2 dng c th gp trong thi l dng 2 cn v 3 cn, cc dng cn

li thao tc tng t

Vi dng2 2 2

1 1 1

a b f c g d fga b f c g

a b f c g

(trong a, b, c, d, a1, b1, c1, a2,

b2, c2l cc a thc ca x), u tin ta nhp biu thc, richo 1000X (hoc 100)


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v gn kt qu vo A. Quay li sa biu thc thnh1 1 1

a b f c g d fg

a b f c g

(i du

trc g ), bm v gn kt qu vo B.

Tnh2

2

A Bc

g

, t p dng phng php xp xtruy ngc li c c2theo x

Li sa biu thc thnh1 1 1

a b f c g d fg

a b f c g

(i du trc f ), bm lu kt

qu vo B, v ta tm c2

2

A Bb

f

Cui cng2 2 2

1 1 1

a b f c g d fga b f c g

a b f c g

d dng tm c qua phng

php xp x.

Vi dng 3 cn1 1 1 1

a b f c g d h u fg v gh w hf

a b f c g d h

ta lm tng t, u

tin tnh gi tr biu thc ny ti 1000X (100), lu kt qu vo A (c nh lun).

Sau tm h s trc cn no, ta ch vic i du trc cn trong biu thc

trn, ri lu kt qu vo B, sau tnh2 ...

A Bvi ... l cn m ta ang tm h s.

Chng hn tm h s f , ta i thnh1 1 1 1

a b f c g d h u fg v gh w hf

a b f c g d h

ri tnh2

A B

f

Nu c 1 cn no v ngha ti 1000X (100), ta c th thay bng 1

100X hoc


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1

1000X . Do nu nhng gi tr ny u khng thay c th cch ny v dng.

3. V d

a) Thc hin php chia:

3 2 2 3 32

4 5 3 1 4 2 3 1 2 3 ( 1)( )

2 1

x x x x x x x x x x x xf x

x x x x

u tin tnh gi tr f(X) ti 100X : 21029,14963 A

Ta tm h s xtrc, do i du trc xta c biu thc:

3 2 2 3 32

4 5 3 1 4 2 3 1 2 3 ( 1)

2 1

x x x x x x x x x x x x

x x x x

, bm lu

kt qu vo B: 19029,14963 B

Suy ra 1002

A BX

X

, chnh l h s ca x

tm h s 1x , ta li sa biu thc thnh:

3 2 2 3 32

4 5 3 1 4 2 3 1 2 3 ( 1)

2 1

x x x x x x x x x x x x

x x x x

, lu tip kt

qu vo B: 20968,85037 B

Suy ra 32 1

A B

X

, vy h s ca 1x l 3

Ta nhn thy phn cn li ca kt qu chia bng ( ) 3 1f x x x x , do tatnh

23 1 19999 2 1A X X X X

Vy kt qu php chia l 22 1 3 1x x x x


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b) Thc hin php chia:2

( )

( 2) 1 1 2

f x

x x x x x x , trong

4 3 2 2 3 3

3 4 3 2

( ) 4 8 4 1 4 1 2 1

2 ( 1) (2 3) ( 1) 2 4 1

f x x x x x x x x x x x x x x

x x x x x x x x x x x

V f(x) rt di, nn ta s ct thnh 2on, tnh ring tng on ri cng li. on th

nht l 4 3 2 2 3 34 8 4 1 4 1 2 1A x x x x x x x x x x x x x ,

on th 2 l

3 4 3 22 ( 1) (2 3) ( 1) 2 4 1B x x x x x x x x x x x

(do ta s dng 2 bin A, B lu 2gi tr tng ng tnh cca mi on).

u tin ta tnh:

4 3 2 2 3 3

100

3 4 3 2

100

2

4 8 4 1 4 1 2 1

919053330,2

2 ( 1) (2 3) ( 1) 2 4 1

9998063419

109484,6565( 2) 1 1 2

x

x

x x x x x x x x x x x x x

A


B

A BC

x x x x x x

tm h s x, ta sa v tnh cc php tnh:

4 3 2 2 3 3

100

3 4 3 2

100

2

4 8 4 1 4 1 2 1

918853350,2

2 ( 1) (2 3) ( 1) 2 4 1

9800985755

107484,6565( 2) 1 1 2

x

x


A


B

A B Dx x x x x x


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tm h s 1x , ta li thc hin cc php tnh:

4 3 2 2 3 3

100

3 4 3 2

100

10

2

4 8 4 1 4 1 2 1

1091114752

2 ( 1) (2 3) ( 1) 2 4 1

1,000198367.10

109524,856( 2) 1 1 2

x

x


A


B

A BM

x x x x x x

V tm h s ca 1x , ta tnh:

4 3 2 2 3 3

100

3 4 3 2

100

2

4 8 4 1 4 1 2 1

899155531,6

2 ( 1) (2 3) ( 1) 2 4 1

9797065504

87552,85602( 2) 1 1 2

x

x


A


B

A BY

x x x x x x

T h s ca tng cn ln lt l:

+ H s x: 1002

C D

XX

+ H s 1x : 22 1

C M

X

+ H s 1x : 299002 1

C YX X

X

Do phn cn li khng cha cn s bng:


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2

( )2 1 ( 1) 1

( 2) 1 1 2

f xx x x x x x

x x x x x x

, hay chnh l

2

2 1 ( 1) 1 10001 1C X X X X X X X

Vy kt qu chia l: 2 1 2 1 ( 1) 1x x x x x x x

Ch : do biu thc qu di, nn b nh my dng lu cc php tnh tnh s

nhanh chng b y, do trnh phi nhp li 2 biu thc A, B (b xa mt do b

nh y), ta phi tnh lin tip D, M, Y nh trn ri cui cng mi tm 3 h s ca

cn. Cc bn c th tnh lun2

C D

X

ngay sau khi tnh c D tm h s ca x,

nhng khi biu thc A nhp chc chn s b xa do b nh c hn, dn n

phi nhp li v tn rt nhiu thi gian.

Vic biu thc cng knh cng khin cho thao tc chy chut sa 2 biu thc A, B

kh chm chp, do cc bn cn phi ht sc ch trnh sa nhm, cng nhbit cch bm sao cho vic sa li c mau chng.

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