Tài liệu ôn thi olympic Toán đại số

–

biên) –
–

THANH PHONG



thi
.
sinh viên

.

T

b

[6], [9], [10].
Oly

.

1


Ụ

Ụ
1

Ụ

Ụ

2

hươn 1
1

Ứ

6

Ứ .............................................................................................................6
.................................................................................................6
.....................................................................................................6
..........................................................................6
.....................................................................................................7
.....................................................................7
........................................................................................9

§2

Ủ

Ứ ...............................................................................11
..........................................................................11
.............................................................................................12
.........................................................................14

4.

................................................16

5.

..........................................................17

6

.........................................................................17
Ứ ......................................21

§3
1.

....................................21

2.

......................................................23

hươn

29

1

Ủ

N................29

.........................................................................................29
.........................................................29
.........................................................................................29
...............................................................................................29
5.

........................................................................................30
......................................................................31
2


................................................................................................ 31
2. Ma

.......................................................................................................31
.......................................................................31
.................................................................................................31
..................................................................................................31

6. Ma tr

...................................................................................................32
...................................................................................................32
................................................................................................ 33

§

...............................................................................35
1.

............................................................35
..........................................................36
Ừ

...................................................................................42
................................................................................42
........................................................42

NG Ủ

....................................................................................49
................................................................................................ 49
............................................49

3.

ng c a ma tr n .............................................................50
hươn

hươn
1

......................................................................................................55
Ứ

Ứ

61

........................................................................61
...........................................................................................61
.................................................................................61
Ứ ............................................................64
.............................................................................64
....................................................................................66
..........................................................................................67
............................................................70
...............................................72
.............................................................................................73
3


hươn

Ì

81
Ì

§1
1. H

.....................................81

n tính và khơng tuy n tính ................................................81

2. D ng ma tr n c a h

n tính ....................................................81

3. Nghi m c a h

..............................................................................82

4. H

n ........................................................................83
Ì

§

....................................83

.............................................................83
ử Gauss .....................................................................................86
.........................................................................................89
4. Sử d

nh lý v nghi m c

5. Sử d

ix

c. ......................................................91
gi i h

i x ng. ................94

.............................................................................................96
hươn
§1

Ơ
Ơ

E
E

TUY N TÍNH

..................................................................................108

1. Khái ni

.............................................................................108

c l p tuy n tính và ph

n tính ......................................................108

và s chi u c
4. Ma tr n chuy

108

...........................................................108

t

x1, x2 ,..., xn  sang  y1, y2 ,..., yn  ..............................109

5. Không gian con - H ng c a m t h

........................................................110

6. T ng và t ng tr c ti p.......................................................................................110
7.

.............................................................111

§2. ÁNH X TUY N TÍNH ..................................................................................116
1. Khái ni m ánh x tuy n tính .............................................................................116
2. Ma tr n c a ánh x tuy n tính...........................................................................117
3. Ảnh và h t nhân c
4. Giá tr
5. T

ng c u tuy n tính. ........................................................118
...................................................................................118

ng c

c ..............................................................................119
4


........................................................119

§3. CHÉO HĨA MA TR N VÀ ỨNG DỤNG ....................................................124
1. Chéo hóa ma tr n ..............................................................................................124
2. Ứng d ng c a chéo hóa ma tr n .......................................................................126
ng. .......................................................................................128
ỨC CỰC TIỂU ....................................................................................134

§

c c c ti u ................................................................................................134
n c c ti u ....................................................................134
3. Bài t p áp d ng .................................................................................................135
...........................................................................................136
hươn 6

TỔ HỢP

144

§1. CHỈNH HỢP – TỔ HỢP – HOÁN V ............................................................144
1. Ch nh h p..........................................................................................................144
2. T h p ...............................................................................................................144
3. Hốn v ..............................................................................................................145
§2. NH THỨC NEWTON – TAM GIÁC PASCAL ..........................................146
1. Nh th c Newton ...............................................................................................146
2. Tam giác Pascal ................................................................................................147
ỨNG MINH VÀ NGUYÊN LÝ QUY N P ..............148
ng minh tr c ti p và ph n ch ng ..........................................148
2. Nguyên lý qui n p .............................................................................................149
§4. NGUN LÍ DIRICHLET - NGUN LÍ CỰC H N ...............................152
1. Ngun lí Dirichlet (hay cịn g i là nguyên lí chu ng thỏ) .............................152

2. Nguyên lý c c h n ............................................................................................153
6 ................................................. Error! Bookmark not defined.
166

5


hươn 1

Ứ

,…

1
ửK

Ứ
,

K
.

1

h

n

hứ

nh n h

1

K

f ( x)  a0  a1 x  ...  an x n ,
ai  K , i  0,1,..., n

a0

do.
K
ủ

K[x].

hứ

f ( x)  a0  a1 x  ...  an x n

nh n h
an  0

f (x

ử an

h

K

f (x).

n

nh n h

ừ nh n h

hứ
n

m

i 0

i 0

f ( x)   ai x i ; g ( x)   bi x i

n  m vaø ai  bi , i  0,..., n .

nh n h

n

m

i 0

i 0

f ( x)   ai x i ; g ( x)   bi x i

6

n

n
deg( f


f ( x)  g ( x) 

max( m , n )

 a
i 0

i

mn

 bi  x i

f ( x).g ( x)   ck x k , ck 
k 0

nh

1

 ab

i  j k

i

j

0  f ( x), g ( x)  K [ x]

deg( f )  deg( g )
f ( x)  g ( x)  0
deg( f )  deg( g )
f ( x)  g ( x)  0
b) deg(fg)= deg(f )+deg(g).
a)

h

h

deg( f  g )  max{deg( f ),deg( g )}.
deg( f  g )  d eg( f )  deg( g ).

ư

nh

 K[x], g(x)
 K[x] sao cho f ( x )  g( x )q( x )  r( x ), deg(r)  deg( g)

 0.
f (x) cho

q(x), r(x
g(x).
nh n h

f (x), g (x)  K [x], K

q(x)  K [x] sao cho f (x) = q (x)g (x
f (x) trong K [x
h ng

n nh

nh n h 6
f (x
g(x
a) h (x
b) h (x)
c)

hứ

f (x)

ỏ
h (x
h (x) | f (x
h(x

g(x
ủ h

nh

a)

ủ h

f (x
g(x
f (x
g (x

nE

hứ

f ( x), g ( x)  K [ x] vaø deg( f )  deg( g )

=

g (x) hay g (x

f (x) | g (x) hay g ( x) f ( x).

ử 0  f ( x), g ( x)  K [ x]
UCLN ( f ( x), g ( x ))

nh n h
▪ h

f (x

g (x) 

UCLN ( f ( x), g ( x))  b1g ( x),
7

h (x) | g (x).
h (x).


b)

r ( x)  0

UCLN ( f ( x), g ( x))  UCLN ( g ( x ), r ( x)) .

.
a)

f ( x)  g ( x) q ( x).

r (x) =

UCLN ( f ( x), g ( x))  b1g ( x),
r ( x)  0

b)

b

g(x).
ử h( x)  UCLN ( f ( x), g ( x)),

f ( x)  g ( x) q ( x)  r ( x). G

h '( x)  UCLN ( g ( x), r ( x)) .

g (x
f (x

h( x ) | f ( x)

h( x) | g ( x) nên h( x ) | r ( x)

r (x). Suy ra h( x) | h '( x)
h '( x ) | h( x)
g (x

h(x

h(x
’(x

’(x

h( x)  h '( x). ■
nh

f ( x), g ( x)

u(x
hứn

v(x) sao cho f ( x)u ( x)  g ( x)v( x)  1.

nh.
ử f ( x), g ( x)

, UCLN ( f ( x), g ( x))  1.
ử deg( f )  deg( g )
UCLN ( f ( x), g ( x ))  1

n = deg(g
f ( x )u ( x )  g ( x ) v ( x )  1

u ( x ), v ( x ) sao cho

n = 0 hay g(x) = b0

u(x) =

v( x)  b01

ỏ

f ( x) u ( x)  g ( x) v( x)  1 .

ử
f (x), g (x

ỏ
deg( f )  deg( g )

deg(g) = n

f ( x)  g ( x)q( x)  r ( x),deg(r )  deg( g ) neáu r ( x)  0.

n, n > 0.
q (x
r (x) sao cho
r ( x )  0 thì g(x

r ( x )  0 , suy ra 1  UCLN ( f ( x), g ( x)) UCLN ( g ( x), r ( x))

’(x), ’(x) sao cho
g ( x)v '( x)  r ( x)u '( x)  1 hay f ( x)u '( x)  g ( x)(v '( x)  q( x)u '( x))  1
u ( x)  u '( x);

deg(r )  deg( g )  n

v( x)  v '( x)  q ( x)u '( x)

f ( x)u ( x)  g ( x )v ( x )  1.

ử
UCLN ( f ( x ), g ( x ))

u(x), v(x) sao cho f ( x)u ( x)  g ( x)v( x)  1
f (x

UCLN ( f ( x), g ( x))  1. ■

8

g (x


ụ

6.

hứ

ụ 1.

x 2017 cho ( x  2)3 trên

ử x 2017  ( x  2)3 q( x)  ax 2  bx  c

[ x].

(*).

4a  2b  c  22017.

Thay x

2017 x 2016  3( x  2) 2 q( x)  ( x  2)3 q '( x)  2ax  b

Thay x = 2

(**).

4a  b  2017.22016 .

(**)

a  2017.2016.2 2014 .

x=2

b  2017.2015.22016 ; c  (1  1007.2017)22017.
2017.2016.22014 x2  2017.2015.22016 x  (1  1007.2017)22017.
ụ 2.

UCLN ( x m  1, x n  1)  x d 1 , d  UCLN (m, n); m, n 
d  UCLN (m, n)

*

m  dm ', n  dn '.

m ', n '

x m  1   x d   1; x n  1   x d   1 . Suy ra x m  1 x n  1
m'

n'

xd  1

xm  1

xn  1.

u, v 

d  UCLN (m, n)

x m  1, x n  1

h( x )

h( x) | x mu  x nv  x nv ( xd 1)
h( x) | ( x d  1) .
ụ

xd  1

sao cho um  vn  d

h( x) | ( x mu  1)  ( x nv  1) hay

UCLN ( xm  1, xnv )  1 nên UCLN (h( x), xnv )  1

f ( x)  [ x]

a)

f ( xn )

b)

a

m, n 

*

x 1
*

ỏ

f ( xn )

cho ( xn  an )m .
9

ử

f ( xn )

( x  1)m

x n  1.
f ( xn )


ử f (x

c)

f ( x), h( x)  [ x]

x2  x  1

f ( x)  g ( x3 )  xh( x3 ).

x 1

g ( x)

ỏ

h(x

x  1.

hay

f (1n )  f (1)  0

x 1

f ( xn )

a)
f (x

f ( x)  ( x  1) g ( x)

x  1,

f (x)

f ( xn )  ( xn  1) g ( xn )

x n  1.

f ( xn )

( x  1)m

f ( xn )

b)

f ( m1) ( x n )

f (an )  f '(an )  ...  f ( m1) (an )  0.

( x  1).

f ( x), f '( x),..., f ( m1) ( x)

an

x  a 

n m

, thay x

, 2

c)

x 2  x  1.
g (1)  h(1)  0

g ( x)

( x  1)m1

f '( x n )

xn

( x n  a n )m .

f ( xn )

1
3
 i
2
2

, 2

x2  x  1 nên f ()  f (2 )  0.

f (x

g (1)  2h(1)  0

g(1) = h(1) =

x  1.

h(x

.
2
x100  2 x 51  1 cho x  1.

1.
ư n

n

1

§1, 6)

2. (USAMO 1976) Cho f (x), g (x), h (x), s (x

f ( x5 )  xg ( x5 )  x2h( x5 )  ( x4  x3  x2  x  1)s( x)
f (x

x - 1.

ư n
n Thay x
f (1) = 0.

10

- 2x + 2.

f (x


3. (

f ( x), g ( x)  [ x]

[3]

f ( x 2010  2009)  x g ( x 2010  2009)

x 2  x  1.

x  2010.

f ( x), g ( x)

ư n

ỏ

n

§1, 6).
x100  1

4.

x 45  1 trên

[ x].

x5 – 1.

Ủ

§2

1

h

hứ

nh

nh n h

1

ửK

ửc

Ứ

f ( x)  a0  a1 x  ...  an x n  K [ x].

cK

f (c)  a0  a1c  ...  anc n  0 .

f ( x)

a0  a1 x  ...  an x n  0 trong K.

f (x) trong K

nh
f (c).

1

c  K , f ( x)  K[ x].

f ( x) cho x  c

Khi chia f ( x) cho x  c
ử
K

f ( x )  ( x  c)q( x )  r, q( x )  K[ x]
Thay x = c

f (x)

f (c)  r .

f ( x) cho x  c
1.

f (c). ■

f ( x)

f ( x)

a, b  K , f (a )  f (b) a  b.
11

x  c.

nh

2.
.

a khơng
• ơ

n

f ( x) cho x  c

n
Cho f ( x )  a0  a1 x  ...  an x  K [ x ].

q( x )  b0  b1 x  ...  bn1 x n1
q ( x)

r
an
c

2

h

...

an 1

a1

bn 1  an bn 2  an 1  cbn 1

a0

b0  a1  cb1 r  a0  cb0

hứ

nh n h

ử c

f ( x)  ( x  c)m g ( x), g ( x)  K[ x]
nh

r  f (c).

*

m

f ( x)  K [ x]

g(c)0.

3. Cho f ( x)  K[ x]

f (x).■
2.

ử f(x
f (x) - g(x

x
ỏ
K. Suy ra f (x) – g (x
n

f (x) – g(x
nh

4.

n trên K
n

n+1

n

f (x) = g(x).■
x1 , x2 ,..., xk

f ( x)  an xn  axn1  ...  a0  [ x]
m1  m2  ...  mk  n

m1 , m2 ,..., mk

f ( x)  an ( x  x1 ) m ( x  x2 ) m ...( x  xk ) m .
1

12

2

k


f(x

( x  x1 )m , ( x  x2 ) m , ..., ( x  xk ) m .
1

2

( x  x1 )m , ( x  x2 ) m , ..., ( x  xk ) m
1

k

2

k

( x  x1 )m ( x  x2 ) m ...( x  xk ) m .

f(x

1

2

k

f ( x)  q( x)( x  x1 )m ( x  x2 ) m ...( x  xk ) m ,
1

2

so

k

.■

f (x
nh

5.

)

f ( x)  an x n  an1 x n1  ...  a1 x  a0  K [ x], an  0
x1 , x2 ,..., xn  K
x1  x2  ...  xn 

an 1

an

x1 x2  x1 x3  ...  xn 1 xn 

an  2
an

...
x1 x2 ... xn 

ử f(x

n

(1) n a0
an

x1 , x2 ,..., xn ,

f ( x)  an ( x  x1 )( x  x2 )...( x  xn )  an x n  an ( x1  x2  ...  xn ) x n1  ...  (1)n an .x1x2 ...xn .
an 1

 x1  x2  ...  xn  a
n

an 2

 x1 x2  x1 x3  ...  xn 1 xn  a
n


...

(1) n a0

...
x
x
x

 1 2 n
an


■

13


3.

hứ
nh n h

h
1

n
ử K

ư n

f (x)  K [x

trong K [x
f (x) = g ( x )h ( x ), g (x), h (x)  K [x
nh

K [x
g (x) hay h (x

1.
.

f (x

g(x
R[x] sao cho
0  deg( fg )  deg( f )  deg( g )

f ( x) g ( x)  1

R
deg( f )  deg( g )  0 , hay f ( x)

ử

f (x

f ( x)

R

.■
ụ

[ x]

x+

nh

[ x] .

6.

a)
b)

f ( x)  ax  b  K [ x], a  0 .

a)

f ( x)  0

f ( x)

g ( x), h( x)  K [ x] , deg(g) + deg(h) =
ử f ( x )  g ( x ) h( x )
deg( g ),deg( h)  deg( f ).
deg(g) = 1; deg(h) =
deg(g) = 0; deg(h) = 1.

deg(g) = 1; deg(h) =

h(x

deg(g) = 0; deg(h) =

g(x

f ( x)

.■

f(x
ử f (x

K
 f (x)= g(x)h(x); g(x), h(x)K[x

f (x

 f (x)

f(x)

g(x

K.
K. ■

f(x)
14

h(x


 x ,

nh

.
6

 x

2,

 x

ử f (x
x–c

2, 1), f (x

f ( x )  ( x  c)g( x ),deg(g)  0, g( x )  [ x],
f (x
.■

f (x
nh

2

z  a  ib, b  0

f ( x)  [ x]

z  a  ib

x 2  2ax  a 2  b 2 .

f ( x)

ử f ( x)  an xn  an1xn1  ...  a0

z

f ( x)

f ( z)  an z n  an1z n1  ...  a1z  a0  0.





n

an z  an1 z

n 1

 ...  a1 z  a0  0

f (x

z

f (x

( x  z)( x  z)  x2  2ax  a 2  b2 , a, b  . ■
[ x],

6

2, 3

 x .

f (x

deg(g)  0, g( x )  [ x].
f (x
z

 x

ử f (x
c
f (x
2

xc
2, 3) suy ra f (x

g ( x)  ( x  z)( x  z)  x2  ( z  z) x  z  z,
15

f ( x)  ( x  c) g ( x),

f (x
2, 1), f (x


  0. Do f (x

g(x
f ( x)  k g ( x), k  \ {0}.
  0. ■

f (x
h

4.

h

ủ

hứ

h

h

ỏ

f ( x )  n x n  n1 x n1  ...  0 , n  0, i  , i  1, n.

1
1
f ( x )  (an x n  an1 x n1  ...  a0 )  g( x ),
b
b
ai  , i.

b
f (x



g (x

ann  an1n1 ...  a0  0, an  0 hay  an   an1  an 
n

Suy ra an

n 1

g(x

...  a0 ann1  0.

x n  an1 x n1 ...  a0 ann1  0

(*).

f (x
nh

f ( x )  an x n  an1 x n1  ...  a0 , n  0



.

p
, UCLN( p, q)  1
q
n.



0

f (x
n

 p
 p

 p
f    an    an1  
q
q
q

n 1

 p
 ...  a1    a0  0
q

Hay a0 q n   p(an p n1  an1 p n2 q  ...  a1q n1 ),
v an p n  q(an1 p n1  ...  a1 pq n2  a0 q n1 ) .
p
q

a0 q n

q

a an p n

an.. ■
16

UCLN (p, q) = 1 nên p

a0

nh
f ( x )  x n  an 1 x n 1  ...  a0 , n  0



a0

1 



a0

p
, UCLN ( p, q)  1
q
q

(1), 1  

-1).

f (x



p
a0.


f ( x )  ( x  )g( x )
g( x )  [ x].
f (x
f (1)  (1  )g(1)
f (1)  (1  )g(1)
1 
f (-1). ■
1 

f
▪

n h

h

ủ

hứ



f (1); f -

như

f ( x )  x n  an1 x n1 ...  a0 , ai 



f (x) không?
ỏ

a0

f (1) f (1)
;
1  1 

f (x) không?
hứ

5.

h

n

ư n

h
[x

Eisenste
h

n Eisenstein.

f ( x )  an x n  an1 x n1 ...  a0 , an  0 (n  1)

n
2

0

quy trong [x].
ụ

6

n h

hứ

ụ 1. Cho m

f (x) = x5 – x +

[ x].

m
.

17


ử f ( x)  g ( x)h( x), g ( x), h( x)  [ x].

a5  a  0(mod5)

m 5

a
deg( g ), deg(h)  2.

f (x

g ( x)  x  bx  c  [ x] c
2

Suy ra x15  x25  x1  x2  2m  0

ử

.G

g, h

 x1  x2  b

 x1 x2  c
 x 5  x  m  0, i  1, 2
i
 i

x 1 , x2

x1  x2 ; x15  x25

( x1  x2 )5  x15  x25 (mod 5)

.

 x1  x2 

5

 x1  x2 (mod5) .

suy ra x15  x25  x1  x2 (mod5) hay m

[ x].

f (x
ụ 2.

n

f ( x)  ( x  1)

2 n 1

x

n2

c x  x  1.
2

1

3
   i
.
2
2
f ( )  0 . V

x2  x  1
x2  x  1
2 n 1

x2  x  1

f (x

n

n2

1
 1
3
3
f ( )    i
  i

2 
2 
2
 2

 (2n  1) 
 (2n  1)  
 ( n  2)2 
 ( n  2)2 
 cos 
  i sin 
  cos 
  i sin 
0
3
3
3
3









 (2n  1) (n  2)2


 
 do
3
3




f ( x)  ( x  1)2 n1  xn2
ụ 3. (

c x2  x  1.

2012, [5]) Cho m, n
xm  xn  1

cho 3.

18

x2  x  1

mn  2


m  3q  r; n  3q1  r1 , q, q1  , r , r1  0;1;2.
xm  xn  1  ( x3q  1) xr  ( x3q  1) xr  xr  x r  1
1

xm  xn  1

1

x2  x  1

r , r1  0;1;2

x 2  x  1.

1

xr  xr  1

o

1

xr  xr  1

x2  x  1

1

r  2, r1  1.

r  1, r1  2

mn  2  9 q q1  3(qr1  q1r )  r r1  2

1. Cho f (x

f

f

f

ư n

n

f (a )  f (b) (a  b)

2. Cho f (x
a, b, c sao cho f (a) = b, f (b
ư n

n

f (c) = a.

f (a )  f (b) ( a  b).

3.
f (2014) = 2011.
ư n

n

f (2017) =

ử

a, b  K , f (a)  f (b) a  b.

4. Cho f (x
f (x

ư n

f

n Sử

5.

a, b  K , f (a)  f (b) a  b.
a, b

ư n

n

f

f ( x)  axn1  bxn  2n1

ử

a = n, b= - 2n -

6
xp  x  a

ư n

[ x].

n

§2, 6).
19


7.

f ( x)  x2012  x1006  1

2011, [4])
x 2  x  1.

ư n

n

§2, 6).

8.
ư n

n T

n

§2, 6),

x 3 m  x 3 n 1  x 3 p  2

9.
m, n, p.
ư n

x 2  x  1.

x2n  xn  1

n

n

x2  x  1

n

§2, 6).

10

n

a) ( x  1) n  x n  1

x 2  x  1.

b) ( x  1) n  x n  1

x 2  x  1.

ư n
11.

n

§2, 6).

f ( x)  xn  a1xn1  ...  an1x  1

ử

n

f (2)  3n.

ư n

n

n

ử x1 , x2 … x n

f(x

f ( x). Suy ra f ( x)  ( x  x1 )( x  x2 )...( x  xn ),

x1 , x2 ,..., xn

12. Cho

qui trên

:

a) f ( x)  ( x  x1 )( x  x2 )...( x  xn )  1
b) f ( x)  ( x  x1 ) 2 ( x  x2 ) 2 ...( x  xn ) 2  1

ư n

n

ử f (x) =g (x) h(x

deg (g), deg (h) >

13

:
20

n

n

f (2)   (2  xi ).
i 1


a ) f ( x)  x p 1  ...  x  1 với p là số nguyên toá.
b) f ( x )  3 x 4  5 x 3  4 x  1

ư n

n

Ứ

§3
nh

1.

hứ

h

nh h

n h m

hứ

ử

,

.
ụ 1.

ỏ

n
f ( x  1)  f ( x)  2 x, x.

n

f ( x)  an x n  an1 x n1  ...  a1 x  a0 , ai 
D

f ( x  1)  f ( x)  2 x, x

f (0) = a0.

f (1)  f (0), f (2)  f (0)  2, ..., f (n  1)  f (0)  2  4  ...  2n  f (0)  n( n  1) .

§2, 4),

n

n

f ( x)  x 2  x  a0 .
f ( x)  x 2  x  a0 ,

a0  .

ụ2

Thay x =

f (x

ỏ

xf ( x  2)  ( x  20) f ( x), x 

x hay f ( x)  x f1 ( x).

f (x

f

Thay f ( x)  x f1 ( x)
( x  2) f1 ( x  2)  ( x  20) f1 ( x), x 

Nên f1 (2)  0,

(1)

f1 ( x)  ( x  2) f 2 ( x) . Thay f1 ( x)  ( x  2) f 2 ( x)

21

(*)


( x  4) f 2 ( x  2)  ( x  20) f 2 ( x), x 

f ( x)  x( x  2)( x  4)...( x  18)q( x), x 
q(x) = q( x - 2), x 

Thay f (x

f ( x)  ax( x  2)( x  4)...( x  18),
ụ .

q(x) = a
ỏ

a

f (x

( x  2)

2

f (x) cho
f (x) cho ( x  3)

- 3x +14.

3

x

deg f  3.
deg f  3

f ( x)  a( x  3)3  3x  14.

( x  3)3  (( x  2)  1)3  ( x  2)3  3( x  2)2  3( x  2)  1  ( x  2)2 ( x  5)  3x  7
f ( x)  a( x  2)2 ( x  5)  3x(a  1)  7a  14
chia f(x) cho ( x  2)2

3x(a  1)  7a  14  3x hay a = 2.

x

f ( x)  2( x  3)3  3x  14.
ụ4

f(x

ỏ

f ( x)  c 

f ( x) f ( x  1)  f ( x 2 )(*)

=

=

f (x) =

ỏ

f (x) =
ử f (x)



n

(*) ta suy ra  , , ,...
2

4

f (x

f (x

 0

f(x

  1

Thay x    1

  1

2

0

2

f (x

  1

2

22

1.

 1


  1

 1

  1

2

  1

 4sin 2

2



2




2



1

2

3
2

1

  cos   i sin 

 cos(   )  i sin(   ) 

 1 nên 4sin 2

  1




2

  1


3

f (x

 2  2cos 2  2  2cos



5
.
3



 1

2

2

 4sin 2

2

f (x
f (x

5
3

f ( x)  axt ( x  1)s
f ( x)  axt ( x  1)s

t = s.
ỏ

f (x) = 0; f ( x)  xt ( x  1)t , t 
nh

nh

2

2
3
3



2.



hứ

1.

h

hươn

a0 , a1 ,..., an

h

n

1

b0 , b2 ,..., bn
f ( x)

f (ai )  bi , i  0,..., n.
.

g( x)  ( x  a0 )( x  a1 )...( x  an )
n

n

g '( x )   ( x  a j )
i 0 j 0
j i

fi ( x ) 

n

g '(ai )   (ai  a j ), i  0,1,..., n
j 0
j i

n xa
g( x )
j
, i  0,1,..., n .

( x  ai )g '(ai ) j 0 ai  a j
j i

fi ( x )

n

0 khi k  i
fi (ak )  
.
1 khi k  i
23

t
a