BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG…………………

LUẬN VĂN

Ứng dụng bài toán nội suy
Lagrange và khai triển Tatlor 1
Mu
.
cLu
.
c
Mo
.
’

-
ath´u
.
cnˆo
.
isuyTaylor 7
1.3 Ba`i toa´n nˆo
.
isuyNewton 7
1.3.1 Ba`i toa´n nˆo
.
isuyNewton 7
1.3.2 D
-
ath´u
.
cnˆo
.
isuyNewton 7
1.4 Ba`i toa´n nˆo
.
isuyHermite 8
1.4.1 Ba`i toa´n nˆo
.
isuyHermite 8
1.4.2 D
-
ath´u
.
cnˆo

.
i suy Lagrange . . . . . . . . . . . 13
2.1.1 Cˆong th´u
.
cnˆo
.
i suy Lagrange . . . . . . . . . . . . . . . . . . . 13
2.1.2 Mˆo
.
tsˆo
´
´u
.
ng du
.
ng 18
2.2 Mˆo
.
tsˆo
´
´u
.
ng du
.
ng cu
˙’
a c´ac cˆong th´u
.
cnˆo
.

˙’
u
.
´o
.
clu
.
o
.
.
ng v`a xˆa
´
pxı
˙’
h`am sˆo
´
38
3.1 U
.
´o
.
clu
.
o
.
.
ng h`am sˆo
´
38
3.1.1 U

´
phu
.
o
.
ng ph´ap kh´ac d¯ˆe
˙’
u
.
´o
.
clu
.
o
.
.
ng h`am sˆo
´
47
3.3 Xˆa
´
pxı
’
ha`m sˆo
´
theo d¯a th´u
.
cnˆo
.
isuy 50

’
ixa´cd¯i
.
nh gia´ tri
.
cu
’
amˆo
.
t ha`m
sˆo
´
f(x)ta
.
imˆo
.
td¯iˆe
’
m tu`y y´ cho tru
.
´o
.
c, trong khi d¯o´d¯iˆe
`
ukiˆe
.
nchı
’
m´o
.

.
i
mˆo
.
tsˆo
´
d¯ i ˆe
’
m x
1
,x
2
, ··· ,x
k
cho tru
.
´o
.
c.
V´o
.
inh˜u
.
ng tru
.
`o
.
ng ho
.
.

`o
.
ng la` ca´c d¯a th´u
.
cd¯a
.
isˆo
´
, tho
’
ama
˜
n ca´c d¯iˆe
`
ukiˆe
.
n
d¯ a
˜
cho. Ngoa`i ra, ta
.
inh˜u
.
ng gia´ tri
.
x ∈ R ma` x khˆong tru`ng v´o
.
i x
1
,x

.
.
cgo
.
i la` ha`m nˆo
.
i suy
cu
’
a f(x); ca´c d¯iˆe
’
m x
1
,x
2
, ···,x
k
thu
.
`o
.
ng d¯u
.
o
.
.
cgo
.
ila`ca´cnu´t nˆo
.

.
o
.
.
c gia´ tri
.
tu
.
o
.
ng d¯ˆo
´
i
chı´nh xa´c cu
’
a ha`m sˆo
´
f(x)ta
.
i x ∈ R tu`y y´ cho tru
.
´o
.
c. T`u
.
d¯ o´, ta co´ thˆe
’
tı´nh gˆa
`
n

´
. Do d¯o´, viˆe
.
c nghiˆen c´u
.
u ca´c ba`i toa´n nˆo
.
i suy la` rˆa
´
t co´ y´ nghı
˜
a.
O
.
˙’
ca´c tru
.
`o
.
ng phˆo
’
thˆong, ly´ thuyˆe
´
tvˆe
`
vˆa
´
nd¯ˆe
`
na`y khˆong d¯u

’
ng ha
.
n trong ca´c
phu
.
o
.
ng trı`nh d¯u
.
`o
.
ng ho˘a
.
cphu
.
o
.
ng trı`nh m˘a
.
tbˆa
.
c hai, trong ca´c d¯˘a
’
ng th ´u
.
cda
.
ng
phˆan th´u

c sinh gio
’
i ca´c cˆa
´
p.
Vı` vˆa
.
y, viˆe
.
c hı`nh tha`nh mˆo
.
t chuyˆen d¯ˆe
`
cho
.
nlo
.
cnh˜u
.
ng vˆa
´
nd¯ˆe
`
co
.
ba
’
n nhˆa
´
tvˆe

da
.
ng toa´n kho´ la` rˆa
´
tcˆa
`
n thiˆe
´
t. Ho
.
nn˜u
.
a, chuyˆen
d¯ ˆe
`
na`y cu
˜
ng co´ thˆe
’
la`m ta`i liˆe
.
u tham kha
’
o cho ca´c gia´o viˆen gio
’
iva` ca´c sinh viˆen
nh˜u
.
ng n˘am d¯ˆa
`

n sa´ ch chuyˆen
kha
’
o [2] ra d¯`o
.
i. D
-
ˆay v`u
.
a la` mˆo
.
t thuˆa
.
nlo
.
.
iv`u
.
ala`mˆo
.
t kho´ kh˘an cho nˆo
˜
lu
.
.
c tı`m kiˆe
´
m
4
nh˜u

cˆa
´
p na`o d¯ˆe
`
cˆa
.
pd¯ˆe
´
nvˆa
´
nd¯ˆe
`
na`y mˆo
.
t ca´ch tro
.
nve
.
n. Do d¯o´, luˆa
.
n v˘an khˆong qua´ d¯ˆe
`
cˆa
.
psˆauvˆe
`
ly´ thuyˆe
´
t ma` cˆo
´

.
ng du
.
ng thu
.
`o
.
ng g˘a
.
pcu
’
a cˆong th´u
.
cnˆo
.
i suy Lagrange va`
khai triˆe
’
n Taylor.
Luˆa
.
n v˘an da`y 56 trang, gˆo
`
m ca´c phˆa
`
nMu
.
clu
.
c, Mo

n.
Nˆo
.
i dung chu
.
o
.
ng na`y trı`nh ba`y mˆo
.
t ca´ch co
.
ba
’
n nhˆa
´
tvˆe
`
ca´c ba`i toa´n nˆo
.
i suy
cˆo
’
d¯ i ˆe
’
n, d¯o´ la` Ba`i toa´n nˆo
.
i suy Lagrange, Ba`i toa´n nˆo
.
i suy Taylor, Ba`i toa´n nˆo
.

.
i dung tro
.
ng tˆam cu
’
a luˆa
.
n v˘an. V´o
.
itˆa
`
m quan tro
.
ng
o
.
’
phˆo
’
thˆong, cˆong th´u
.
cnˆo
.
i suy Lagrange va`nh˜u
.
ng ´u
.
ng du
.
ng cu

da
.
ng va`mˆo
.
tsˆo
´
lu
.
o
.
.
ng ba`i tˆa
.
pd¯ˆe
`
xuˆa
´
t kha´ phong phu´. Nhiˆe
`
ud¯˘a
’
ng th´u
.
cdu
.
´o
.
ida
.
ng

c sinh gio
’
i quˆo
´
cgiava` quˆo
´
ctˆe
´
d¯ a
˜
d¯ u
.
o
.
.
c gia
’
ib˘a
`
ng
ca´ch a´p du
.
ng cˆong th´u
.
cnˆo
.
i suy na`y. Phˆa
`
n co`n la
.

nd¯o
.
ccu
˜
ng
d¯ u
.
o
.
.
c gi´o
.
i thiˆe
.
uo
.
’
phˆa
`
n cuˆo
´
i chu
.
o
.
ng.
Chu
.
o
.

o
.
ng na`y ta´ch riˆeng mˆo
.
t´u
.
ng du
.
ng cu
’
a ca´c cˆong th´u
.
cnˆo
.
i suy d¯ˆe
’
u
.
´o
.
clu
.
o
.
.
ng
va`xˆa
´
pxı
’

.
p, trong d¯o´ co´ nh˜u
.
ng ba`i trong ca´c d¯ˆe
`
thi cho
.
nho
.
c sinh gio
’
i quˆo
´
c
gia va` quˆo
´
ctˆe
´
.Mˆo
.
tsˆo
´
phˆa
`
ncu
’
a luˆa
.
n v˘an d¯a
˜

su
.
.
hu
.
´o
.
ng dˆa
˜
n khoa ho
.
cva` nhiˆe
.
t tı`nh cu
’
aTiˆe
´
n
sy
˜
Tri
.
nh D
-
a`o Chiˆe
´
n - Ngu
.
`o
.

.
c
trong suˆo
´
t th`o
.
i gian nghiˆen c´u
.
ud¯ˆe
`
ta`i. Chı´nh vı` vˆa
.
y ma` ta´c gia
’
luˆon to
’
lo`ng biˆe
´
t
o
.
n chˆan tha`nh va` sˆau s˘a
´
cd¯ˆo
´
iv´o
.
i Thˆa
`
y gia´o hu

n chˆan tha`nh d¯ˆe
´
n: Ban Gia´m
Hiˆe
.
u, Pho`ng d¯a`o ta
.
oD
-
a
.
iho
.
cva` sau D
-
a
.
iho
.
c, Khoa toa´n cu
’
a tru
.
`o
.
ng D
-
a
.
iho

gia´o du
.
cva` d¯a`o ta
.
otı
’
nh Gia Lai, Ban
Gia´m Hiˆe
.
u tru
.
`o
.
ng THPT Ia Grai d¯a
˜
cho ta´c gia
’
co
.
hˆo
.
iho
.
ctˆa
.
p, cu`ng v´o
.
i quı´ thˆa
`
y

ta´c gia
’
nghiˆen c´u
.
uva` hoa`n tha`nh luˆa
.
n v˘an na`y.
Trong qua´ trı`nh hoa`n tha`nh luˆa
.
n v˘an, ta´c gia
’
co`n nhˆa
.
nd¯u
.
o
.
.
csu
.
.
quan tˆam d¯ˆo
.
ng
viˆen cu
’
a ca´c ba
.
nd¯ˆo
`

´
tca
’
nh˜u
.
ng
su
.
.
quan tˆam d¯ˆo
.
ng viˆen d¯o´.
D
-
ˆe
’
hoa`n tha`nh luˆa
.
n v˘an na`y, ta´c gia
’
d¯ a
˜
tˆa
.
p trung rˆa
´
t cao d¯ˆo
.
trong hoc tˆa
.

i gian cu
˜
ng nhu
.
trı`nh d¯ˆo
.
hiˆe
’
ubiˆe
´
tnˆen trong qua´ trı`nh thu
.
.
chiˆe
.
n
khˆong thˆe
’
tra´nh kho
’
inh˜u
.
ng thiˆe
´
u so´t, ta´c gia
’
rˆa
´
t mong nhˆa
.

.
o
.
.
c hoa`n thiˆe
.
nho
.
n.
Quy Nho
.
n, tha´ng n˘am 2008
Ta´c gia
’
6
Chu
.
o
.
ng 1
C´ac b`ai to´an nˆo
.
i suy cˆo
˙’
d¯ i ˆe
˙’
n
Trong chu
.
o

o
.
ng sau, d¯o´ la`: Ba`i toa´n nˆo
.
i suy Lagrange, Bai toa´n nˆo
.
i suy Taylor, Ba`i
toa´n nˆo
.
i suy Newton va` Ba`i toa´n nˆo
.
i suy Hermite. L`o
.
i gia
’
i cho ca´c ba`i toa´n na`y la`
ca´c d¯a th´u
.
cnˆo
.
i suy tu
.
o
.
ng ´u
.
ng ma` ch´u
.
ng minh chi tiˆe
´

.
imo
.
i i = j, i, j =1, 2, ···,N.Ha
˜
yxa´c
d¯ i
.
nh d¯a th´u
.
c L(x) co´bˆa
.
c degL(x) ≤ N −1 va` tho
’
aca´c d¯iˆe
`
ukiˆe
.
n
L(x
i
)=a
i
, ∀i =1, 2, ···,N
.
1.1.2 D
-
ath´u
.
cnˆo

la` d¯a th ´u
.
c duy nhˆa
´
t tho
’
ama
˜
nd¯iˆe
`
ukiˆe
.
ncu
’
a ba`i toa´n nˆo
.
i suy Lagrange va` ta go
.
i
d¯a th´u
.
c na`y la` d¯a th´u
.
cnˆo
.
i suy Lagrange.
7
1.2 B`ai to´an nˆo
.
i suy Taylor

.
n
T
i
(x
0
)=a
i
, ∀i =0, 1, ··· ,N − 1.
1.2.2 D
-
ath´u
.
cnˆo
.
i suy Taylor
D
-
ath´u
.
c
T (x)=
N −1

i=0
a
i
i!
(x − x
0

i suy Newton
1.3.1 Ba`i toa´n nˆo
.
i suy Newton
Cho ca´c sˆo
´
thu
.
.
c x
i
,a
i
, v´o
.
i i =1, 2, ··· ,N.Ha
˜
y xa´c d¯i
.
nh d¯a th´u
.
c N(x) co´bˆa
.
c
degN(x) ≤ N −1 va` tho
’
ama
˜
nca´c d¯iˆe
`

x
x
1

t
x
2

t
1
x
3
···

t
i−2
x
i
dt
i−1
dt
2
.dt
1
.dt; i =1, 2, ··· ,N.
khi d¯o´, d¯a th´u
.
c
N(x)=
N

N −1
(x
1
, ···,x
N −1
,x)
la` d¯a th´u
.
c duy nhˆa
´
t tho
’
ama
˜
nd¯iˆe
`
ukiˆe
.
ncu
’
a ba`i toa´n nˆo
.
i suy Newton va` ta go
.
id¯a
th ´u
.
c na`y la` d¯a th´u
.
cnˆo

0
  
i lˆa
`
n
,x

=

x
x
0

t
x
0

t
1
x
0
···

t
i−2
x
0
dt
i−1
dt

n
,x

=
= a
0
+ a
1
R(x
0
,x)+a
2
R
2
(x
0
,x
0
,x)+···+ a
N −1
R
N −1

x
0
, ···,x
0
  
N −1 lˆa
`

0
)
i
i!
≡ T(x).
Vˆa
.
y, v´o
.
i x
i
= x
0
, ; ∀i =1, 2, ···,N, thı` d¯a th´u
.
cnˆo
.
i suy Newton chı´nh la` d¯a th´u
.
c
nˆo
.
i suy Taylor.
1.4 Ba`i toa´n nˆo
.
i suy Hermite
1.4.1 Ba`i toa´n nˆo
.
i suy Hermite
Cho ca´c sˆo

.
c H(x) co´bˆa
.
c
degH(x) ≤ N − 1 va` tho
’
ama
˜
nca´c d¯iˆe
`
ukiˆe
.
n
H
(k)
(x
i
)=a
ki
, ∀i =1, 2, ···,n; ∀k =0, 1, ···,p
i
− 1
1.4.2 D
-
ath´u
.
cnˆo
.
i suy Hermite
Ky´ hiˆe

j
; i =1, 2, ··· ,n
Go
.
i d¯oa
.
n khai triˆe
’
n Taylor d¯ˆe
´
ncˆa
´
pth´u
.
p
i
− 1 − k,v´o
.
i k =0, 1, ··· ,l; l =
0, 1, ···,p
i
− 1, ta
.
i x = x
i
cu
’
a ha`m sˆo
´
1

(x=x
i
)
(x − x
i
)
l
l!
.
khi d¯o´, d¯a th´u
.
c
H(x)=
n

i=1
p
i
−1

k=0
a
ki
(x − x
i
)
k
k!
W
i

i suy Hermite va`tago
.
id¯a
th ´u
.
c na`y la` d¯a th´u
.
cnˆo
.
i suy Hermite.
Nhˆa
.
n xe´t 1.2.
V´o
.
i n = 1, thı` i =1va` p
1
= N. Khi d¯o´, ta co´
W (x)=(x − x
1
)
N
;
W
1
(x)=
W (x)
(x − x
1
)


k=0
a
k1
(x − x
1
)
k
k!
≡ T (x).
Vˆa
.
y, v´o
.
i n = 1, thı` d¯a th´u
.
cnˆo
.
i suy Hermite chı´nh la` d¯a th´u
.
cnˆo
.
i suy Taylor.
Nhˆa
.
n xe´t 1.3.
V´o
.
i k = 0, thı` p
i

khi d¯o´, d¯oa
.
n khai triˆe
’
n Taylor
T

1
W
i
(x)

0
(x=x
i
)
=
1
W
i
(x
i
)
=
1
N

j=1,j=i
(x
i

.
i suy Hermite chı´nh la` d¯a th´u
.
cnˆo
.
i suy Lagrange.
Trong tru
.
`o
.
ng ho
.
.
ptˆo
’
ng qua´t, viˆe
.
cbiˆe
’
udiˆe
˜
nd¯ath´u
.
c Hermite kha´ ph´u
.
cta
.
p. Du
.
´o

ch´u
.
ad¯a
.
o ha`m bˆa
.
c nhˆa
´
t.
Nhˆa
.
n xe´t 1.4.
Nˆe
´
u p
i
= 2, v´o
.
imo
.
i i =1, 2, ···,n, thı` khi d¯o´ k = 0 ho˘a
.
c k =1.
+V´o
.
i k = 0, ta co´
T

1
W

(l)
(x=x
i
)
(x − x
i
)
l
l!
=
1
W
i
(x
i
)
−
W

i
(x
i
)
W
2
i
(x
i
)
(x −x

.
i k = 1, ta co´
T

1
W
i
(x)

(p
i
−1−k)
(x=x
i
)
= T

1
W
i
(x)

(0)
(x=x
i
)
=
0

l=0

2
i
(x
i
)
(x − x
i
)=
1
W
i
(x
i
)
.
11
Khi d¯o´, ta co´
H(x)=
n

i=1
1

k=0
a
ki
(x − x
i
)
k

(x)

(1)
(x=x
i
)
+a
1i
(x − x
i
)W
i
(x)T

1
W
i
(x)

(0)
(x=x
i
)

=
n

i=1
W
i

i
)
1
W
i
(x
i
)

=
n

i=1
W
i
(x)
W
i
(x
i
)

a
0i

1 −
W

i
(x

a
0i
−

a
0i
W

i
(x
i
)
W
i
(x
i
)
− a
1i

(x −x
i
)

.
Ngoa`i ra, trong phˆa
`
n ba`i toa´n nˆo
.
i suy Lagrange, ta d¯a

Do d¯o´
L
i
(x
i
) ≡ 1, ∀i = 1,n.
Vˆa
.
y
W
i
(x)
W
i
(x
i
)
=
n

j=1,j=i
(x − x
j
)
2
(x
i
− x
j
)

i
)
=2L
i
(x)L

i
(x)=2L

i
(x
i
).
Do d¯o´, d¯a th´u
.
cnˆo
.
i suy Hermite trong tru
.
`o
.
ng ho
.
.
p na`y co´ da
.
ng
H(x)=
n


.
tva`i minh ho
.
a cho viˆe
.
cvˆa
.
ndu
.
ng ca´c cˆong th´u
.
cnˆo
.
i suy (do ta´c
gia
’
sa´ng ta´c)
12
Ba`i toa´n 1.1. Cho d¯a th´u
.
c P (x) bˆa
.
c 4, tho
’
ama
˜
nca´c d¯iˆe
`
ukiˆe
.

Ap du
.
ng cˆong th´u
.
cnˆo
.
i suy Taylor (v´o
.
i N = 3), ta tı`m d¯u
.
o
.
.
c
P (x)=x
4
+2ax
2
+ a (a>0)
Suy ra:
P

(x)=4x
3
+4ax ;
P

(x)=12x
2
+4a ;

n
P
(n)
≤ 0;
P (2008) > 0,P

(2008) ≥ 0,P

(2008) ≥ 0, ···,P
(n)
(2008) ≥ 0.
Ch´u
.
ng minh r˘a
`
ng ca´c nghiˆe
.
m thu
.
.
ccu
’
a P (x) thuˆo
.
c (2007; 2008).
Gia
’
i.
´
Ap du

u x ≥ b thı` P(x) khˆong co´ nghiˆe
.
m x ≥ b.
V´o
.
i a = 2007, a´p du
.
ng cˆong th´u
.
cnˆo
.
i suy Taylor, ta co´
P (x)=P (a)+
−P

(a)
1!
(a −x)+
P

(a)
2!
(a −x)
2
+ ···+
(−1)
n
P
(n)
(a)

ng du
.
ng cu
˙’
a cˆong th´u
.
c
nˆo
.
i suy
Chu
.
o
.
ng na`y trı`nh ba`y mˆo
.
tsˆo
´
´u
.
ng du
.
ng cu
’
a ca´c cˆong th´u
.
cnˆo
.
i suy, trong d¯o´d¯ˆe
`

’
hˆe
.
phˆo
’
thˆong chuyˆen toa´n.
Vˆa
´
nd¯ˆe
`
´u
.
ng du
.
ng cˆong th´u
.
cnˆo
.
i suy trong u
.
´o
.
clu
.
o
.
.
ng va`xˆa
´
pxı

.
o
.
.
c trı`nh ba`y o
.
’
chu
.
o
.
ng sau.
2.1 Mˆo
.
tsˆo
´
´u
.
ng du
.
ng cu
˙’
a cˆong th ´u
.
cnˆo
.
i suy La-
grange
2.1.1 Cˆong th´u
.

thı` tˆo
`
nta
.
i duy nhˆa
´
tmˆo
.
td¯ath´u
.
c P (x) v´o
.
ibˆa
.
c khˆong vu
.
o
.
.
t qua´ n−1, tho
’
ama
˜
n
P (x
j
)=a
j
; ∀j =1, 2, ···,n. (2.1)
D

cgo
.
i la` d¯a th´u
.
cnˆo
.
i suy Lagrange ho˘a
.
c cˆong th´u
.
cnˆo
.
i suy
Lagrange. Ca´c sˆo
´
x
1
,x
2
, ···,x
n
d¯ u
.
o
.
.
cgo
.
ila`ca´c nu´t nˆo
.

a P(x). Thˆe
´
thı`
degP (x) ≤ 1va` P (x
1
)=a
1
; P (x
2
)=a
2
.
+V´o
.
i n = 3, d¯a th´u
.
cd¯o´la`
P (x)=a
1
(x − x
2
)(x − x
3
)
(x
1
− x
2
)(x
1

)(x
3
− x
2
)
. (2.4)
Ro
˜
ra`ng degP (x) ≤ 2va` P(x
1
)=a
1
,P(x
2
)=a
2
),P(x
3
)=a
3
.
()T`u
.
cˆong th´u
.
cnˆo
.
i suy Lagrange, ta co´
D
-

t qua´ n − 1 d¯ ˆe
`
uco´thˆeviˆe
´
tdu
.
´o
.
ida
.
ng
P (x)=
n

j=1
P (x
j
)
n

i=1,i=j
x − x
i
x
j
− x
i
. (2.5)
Nhˆa
.

’
ng ha
.
n (2.4).
Gia
’
su
.
’
r˘a
`
ng, trˆen m˘a
.
t ph˘a
’
ng to
.
ad¯ˆo
.
Oxy cho 3 d¯iˆe
’
m A(x
1
; y
1
),B(x
2
; y
2
),C(x

.
ng cong y = P (x), trong
d¯ o´ la` d¯a th´u
.
cv´o
.
i degP (x) ≤ 2, tho
’
ama
˜
n
P (x
1
)=y
1
(nghı
˜
a la` d¯u
.
`o
.
ng cong qua d¯iˆe
’
m A);
P (x
2
)=y
2
(nghı
˜

o
.
ng trı`nh cu
.
thˆe
’
la` y = P (x), tro`n d¯o´ P(x)co´
da
.
ng (2.4) va`ca´chˆe
.
sˆo
´
a
j
chı´nh la` y
j
,j=1, 2, 3.
+V´o
.
i degP (x)=2,d¯ˆo
`
thi
.
y = P (x) la` parabol d¯i qua 3 d¯iˆe
’
m A, B, C.
+V´o
.
i degP (x) = 1, d¯ˆo

`o
.
ng th˘a
’
ng d¯i qua 3 d¯iˆe
’
m A, B, C,
cu`ng phu
.
o
.
ng v´o
.
i tru
.
c hoa`nh.
V´o
.
i ca´c minh ho
.
a trˆen ta thˆa
´
yr˘a
`
ng, cˆong th´u
.
cnˆo
.
i suy Lagrange chı´nh la` ”ca´c
gˆo

c trong m˘a
.
t ph˘a
’
ng to
.
ad¯ˆo
.
.
D
-
o´ la` ”ca´i gˆo
´
c” nhı`n du
.
´o
.
i go´c d¯ˆo
.
hı`nh ho
.
c.
Du
.
´o
.
i d¯ˆay, v´o
.
imˆo
.

.
c P (x)co´ degP (x) ≤ n −1 cho tru
.
´o
.
c, ca´c sˆo
´
a
j
trong (2.2) d¯u
.
o
.
.
c thay
bo
.
’
i P(x
j
), v´o
.
i j =1, 2, ···,n.
Bˆay gi`o
.
ta thu
.
’
d¯i tı`m mˆo
.

−
n

i=1
(x − x
i
). (2.6)
D
-
ath´u
.
c na`y d¯u
.
o
.
.
c khai triˆe
’
ndu
.
´o
.
ida
.
ng
P (x)=S
1
x
n−1
− S

x
3
+ ···+ x
n−1
x
n
;
···
S
n
= x
1
x
2
···x
n
(2.8)
Bo
.
’
i (2.7), ta thˆa
´
yr˘a
`
ng degP (x) ≤ n − 1.
Ngoa`i ra, t`u
.
da
.
ng (2.6), ta co´

thˆa
´
yr˘a
`
ng vˆe
´
pha
’
icu
’
a (2.9) la` d¯a th´u
.
cco´hˆe
.
sˆo
´
d¯ ´u
.
ng tru
.
´o
.
c x
n−1
la`
n

j=1
x
n


j=1
x
j
. (2.11)
D
-
˘a
’
ng th´u
.
c (2.11) la` mˆo
.
td¯˘a
’
ng th´u
.
c liˆen quan d¯ˆe
´
n phˆan th´u
.
c, thu
.
`o
.
ng g˘a
.
p trong
chu
.

x
1
− x
2
+
x
2
2
x
2
− x
1
= x
1
+ x
2
hay
x
2
1
− x
2
2
x
1
− x
2
= x
1
+ x

1
− x
3
)
+
x
3
2
(x
2
− x
3
)(x
2
− x
1
)
+
x
3
3
(x
3
− x
1
)(x
3
− x
2
)

ng minh r˘a
`
ng v´o
.
i3sˆo
´
nguyˆen bˆa
´
t ky` kha´c nhau t`u
.
ng d¯ˆoi mˆo
.
t, sˆo
´
sau d¯ˆay cu
˜
ng la` mˆo
.
tsˆo
´
nguyˆen:
a
3
(a − b)(a − c)
+
b
3
(b − c)(b − a)
+
c

sa´ng ta´c d¯u
.
o
.
.
c kha´ nhiˆe
`
u ba`i tˆa
.
p phong phu´. Ngoa`i
ra, ta co`n co´ thˆe
’
so sa´nh S
2
,S
3
, , S
n
o
.
’
hai vˆe
´
cu
’
a (2.5) d¯ˆe
’
tı`m thˆem nh˜u
.
ng d¯˘a

i degP (x)  n − 1.
17
Bˆay gi`o
.
, ta tiˆe
´
ptu
.
c tı`m kiˆe
´
m thˆem ca´c d¯˘a
’
ng th ´u
.
c theo mˆo
.
thu
.
´o
.
ng kha´c.
V´o
.
i n sˆo
´
phˆan biˆe
.
t x
1
,x

),
v´o
.
i degω

(x)=n − 1.
V´o
.
imˆo
˜
i j ∈{1, 2, , n}, ta co´
ω

(x
j
)=
n

i=1,i=j
(x
j
−x
i
).
Bˆay gi`o
.
,v´o
.
imˆo
˜

˜
i j ∈{1, 2, , n}, (2.10) la` mˆo
.
t d¯a th´u
.
cva` degω
j
(x)=
n − 1. D
-
ath´u
.
c na`y co´ tı´nh chˆa
´
t
ω
j
(x
k
)=0,v´o
.
i k = j;
ω
j
(x
k
)=1,v´o
.
i k = j.
Bˆay gi`o

2
, , x
n
, thı` P (x)=a
n
ω(x).
Do d¯o´, v´o
.
imˆo
˜
i j ∈{1, 2, , n}, ta co´
P

(x
j
)=a
n
ω

(x
j
)
hay
ω

(x
j
)=
P


n
ω(x)
(x − x
j
)P

(x
j
)
=
n

i=1,i=j
x − x
i
x
j
− x
i
. (2.15)
Bˆay gi`o
.
, ta ha
˜
y tı`m mˆo
.
t´u
.
ng du
.

x
n−1
+ + a
1
x + a
0
, a
n
=0,n ≥ 2, co´
18
n nghiˆe
.
m thu
.
.
c phˆan biˆe
.
t x
1
,x
2
, , x
n
.
V´o
.
i n gia´ tri
.
phˆan biˆe
.

j
ω
j
(x)
Bo
.
’
i (2.15), ta co´
x
k
=
n

j=1
x
k
j
ω(x)
(x − x
j
)ω

(x
j
)
= a
n
n

j=1

’
a x
n−1
la`
a
n
n

j=1
x
k
j
P

(x
j
)
.
So sa´nh ca´c hˆe
.
sˆo
´
cu
’
ad¯ath´u
.
c x
k
, ta d¯u
.

)
=
1
a
n
, v´o
.
i k = n − 1. (2.17)
2.1.2 Mˆo
.
tsˆo
´
´u
.
ng du
.
ng
Phˆa
`
n tro
.
ng tˆam cu
’
a phˆa
`
n na`y tˆa
.
p trung va`o viˆe
.
ca´pdu

c, khu vu
.
.
cva` quˆo
´
ctˆe
´
.
Ba`i toa´n 2.1. Xa´c d¯i
.
nh d¯a th´u
.
cbˆa
.
c hai nhˆa
.
n gia´ tri
.
b˘a
`
ng 3; 1; 7,ta
.
i x b˘a
`
ng −1;
0; 3 tu
.
o
.
ng ´u

+ f(0)
(x − 3)(x +1)
(0 − 3)(0 + 1)
+f(3)
(x + 1)(x − 0)
(3 + 1)(3 −0)
= x
2
−x +1.
19
Ba`i toa´ n 2.2. Cho a
1
,a
2
, , a
n
la` n sˆo
´
kha´c nhau. Ch ´u
.
ng minh r˘a
`
ng nˆe
´
u d¯a th´u
.
c
f(x) co´bˆa
.
c khˆong l´o

n
− a
2
) (a
n
− a
n−1
)
=0.
Gia
’
i. Theo cˆong th´u
.
cnˆo
.
i suy Lagrange thı`, mo
.
id¯ath´u
.
c f(x) co´ bˆa
.
c khˆong l´o
.
n
ho
.
n n − 1d¯ˆe
`
uviˆe
´

) (a
1
− a
n
)
+ f(a
2
)
(x − a
1
)(x − a
3
) (x −a
n
)
(a
2
− a
1
)(a
2
− a
3
) (a
2
− a
n
)
+ + f(a
n

o
.
’
vˆe
´
tra´i b˘a
`
ng 0, co`n hˆe
.
sˆo
´
cu
’
a x
n−1
o
.
’
vˆe
´
pha
’
i la`:
T =
f(a
1
)
(a
1
− a

u pha
’
ich´u
.
ng minh.
Ba`i toa´ n 2.3. Ch´u
.
ng minh r˘a
`
ng nˆe
´
u d¯a th´u
.
cbˆa
.
c hai nhˆa
.
n gia´ tri
.
nguyˆen ta
.
iba
gia´ tri
.
nguyˆen liˆen tiˆe
´
pcu
’
abiˆe
´

.
ng cˆong th´u
.
cnˆo
.
i suy Lagrange cho d¯a th´u
.
cbˆa
.
c hai f(x)v´o
.
ibasˆo
´
nguyˆen
k −1, k, k + 1, ta co´
f(x)=f(k −1)
(x − k)(x −k −1)
2
+ f(k)
(x − k + 1)(x − k − 1)
−1
+f(k +1)
(x − k)(x − k +1)
2
.
D
-
˘a
.
t m = x − k, ta co´

.
i A
i
(i =1,2, , n) la`
phˆa
`
ndu
.
trong phe´p chia d¯a th´u
.
c f(x) cho x −a
i
.Ha
˜
y tı`m phˆa
`
ndu
.
r(x) trong phe´p
chia f(x) cho (x − a
1
)(x − a
2
) (x −a
n
).
20
Gia
’
i. Go

t x = a
i
(i =1, 2, , n)va`d¯ˆe
’
y´r˘a
`
ng A
i
= f(a
i
). Thˆe
´
thı`, ta co´ r(a
i
)=A
i
(i =1, 2, , n).
Nhu
.
vˆa
.
y,tabiˆe
´
td¯u
.
o
.
.
c ca´c gia´ tri
.

(x −a
2
)(x − a
3
) (x −a
n
)
(a
1
− a
2
)(a
1
− a
3
) (a
1
− a
n
)
+ A
2
(x − a
1
)(x −a
3
) (x −a
n
)
(a

n
− a
n−1
)
=
n

i=1
A
i

j=1,j=i
x −a
j
a
i
−a
j
.
Ba`i toa´n 2.5. (Vˆo d¯i
.
ch Chˆau
´
A Tha´i Bı`nh Du
.
o
.
ng, 2001)
Trong m˘a
.

.
pnˆe
´
umˆo
.
t trong hai tha`nh phˆa
`
nto
.
ad¯ˆo
.
cu
’
ad¯iˆe
’
md¯o´ la` sˆo
´
h˜u
.
utı
’
, tha`nh
phˆa
`
n kia la` sˆo
´
vˆo tı
’
. Tı`m tˆa
´

nho
.
.
p na`o ca
’
.
Gia
’
i. Ca´c d¯a th´u
.
ccˆa
`
n tı`m la` ca´c d¯a th´u
.
cbˆa
.
c1v´o
.
ihˆe
.
sˆo
´
h˜u
.
utı
’
.
Thˆa
.
tvˆa

tca
’
ca´c hˆe
.
sˆo
´
cu
’
a f(x)d¯ˆe
`
ula`
sˆo
´
h˜u
.
utı
’
.
Vı` vˆa
.
y, nˆe
´
u d¯a thu´c co´ mˆo
.
thˆe
.
sˆo
´
vˆo tı
’

td¯iˆe
’
mhˆo
˜
nho
.
.
p.
Dˆe
˜
da`ng thˆa
´
yr˘a
`
ng ca´c d¯a th´u
.
cbˆa
.
c 0 (khi d¯o´, no´ d¯u
.
o
.
.
cbiˆe
’
udiˆe
˜
nb˘a
`
ng d¯u

yr˘a
`
ng ca´c d¯a th´u
.
cbˆa
.
c1v´o
.
ihˆe
.
sˆo
´
h˜u
.
utı
’
(khi d¯o´, no´ d¯u
.
o
.
.
cbiˆe
’
u
diˆe
˜
nb˘a
`
ng mˆo
.

´
p theo, xe´t d¯a th´u
.
c co´ bˆa
.
c n ≥ 2co´hˆe
.
sˆo
´
a
i
∈ Q
f(x)=a
0
+ a
1
x + a
2
x
2
+ + a
n
x
n
.
21
Khˆong mˆa
´
t tı´nh tˆo
’

ng trı`nh f(x)=r va` af(x)=ar tru`ng nhau,
v´o
.
i a la` sˆo
´
nguyˆen (r la` sˆo
´
h˜u
.
utı
’
). Ho
.
nn˜u
.
a, nˆe
´
u ta kı´ hiˆe
.
u
g(x)=a
n−1
n
f

x
a
n

thı` g(x) la` d¯a th´u

uva`chı
’
nˆe
´
uphu
.
o
.
ng trı`nh
g(x)=a
n−1
n
r co´ mˆo
.
t nghiˆe
.
mvˆotı
’
,cu
˜
ng thˆe
´
,nˆe
´
uva`chı
’
nˆe
´
uphu
.

nguyˆen, v´o
.
i a
n
=1,a
0
=0.
Bˆay gi`o
.
,go
.
i r la` sˆo
´
nguyˆen tˆo
´
d¯ u
’
l´o
.
nd¯ˆe
’
cho
r>max {f(1),x
1
,x
2
, , x
k
},
v´o

.
trung gian, tˆo
`
nta
.
i ı´t nhˆa
´
t
mˆo
.
tsˆo
´
s ∈ (1,r) sao cho
f(s) − r =0.
Gia
’
su
.
’
s ∈ Q, ta viˆe
´
t s = p/q,v´o
.
i p, q la` hai sˆo
´
nguyˆen tˆo
´
cu`ng nhau. Thay
va`o d¯˘a
’

, no´i ca´ch kha´c, d¯ˆo
`
thi
.
cu
’
a f(x)d¯i
qua mˆo
.
t ”d¯iˆe
’
mhˆo
˜
nho
.
.
p”.
Ba`i toa´n 2.6. Tı`m tˆa
´
tca
’
ca´cc˘a
.
pd¯ath´u
.
c P (x) va` Q(x) co´bˆa
.
cbav´o
.
ica´c hˆe

b) Nˆe
´
u P(1) = 0 ho˘a
.
c P (2) = 1, thı` Q(1) = Q(3) = 1;
c) Nˆe
´
u P(2) = 0 ho˘a
.
c P (4) = 0, thı` Q(2) = Q(4) = 0;
d) Nˆe
´
u P (3) = 1 ho˘a
.
c P(4) = 1, thı` Q(1) = 0.
Gia
’
i. Gia
’
su
.
’
kı´ hiˆe
.
u a
k
= P (k), b
k
= Q(k), v´o
.

3
b
4
khˆong thˆe
’
b˘a
`
ng sˆo
´
na`o trong ca´c sˆo
´
0000, 0110, 1001, 1111, vı` ca´c d¯a th´u
.
c P ( x)
va` Q(x ) co´ bˆa
.
c 3. M˘a
.
t kha´c, sˆo
´
a
1
a
2
a
3
a
4
khˆong thˆe
’

1
= 0, vˆo lı´.
T`u
.
d¯ o´, theo d¯iˆe
`
ukiˆe
.
n c) ta thˆa
´
yd¯iˆe
`
ukiˆe
.
n ba`i toa´n tho
’
ama
˜
nv´o
.
iva`chı
’
v´o
.
i
7c˘a
.
psˆo
´
(a

c
2
c
3
c
4
tu
.
o
.
ng ´u
.
ng va`o
ca´c d¯a th´u
.
c R(x), tho
’
ama
˜
n ca´c d¯˘a
’
ng th ´u
.
c P(k)=c
1
,v´o
.
i k =1, 2, 3, 4.
Khi d¯o´ ta nhˆa
.

(x)=(−1/6)x
3
+(3/2)x
2
− (13/3)x +4;
R
4
(x)=(−2/3)x
3
+5x
2
− (34/3)x +8;
R
5
(x)=(−1/2)x
3
+4x
2
− (19/2)x +7;
R
6
(x) =(1 /3)x
3
−(5/2)x
2
+ (31/6)x − 2.
Nhu
.
vˆa
.

1
(x)), (R
5
(x); R
1
(x)), (R
6
(x); R
4
(x)).
Ba`i toa´n 2.7. (Vˆo d¯i
.
ch My
˜
- 1975)
D
-
ath´u
.
c P (x) bˆa
.
c n tho
’
ama
˜
nca´c d¯˘a
’
ng th´u
.
c P (k)=

k − i
=
n

k=0

i=k
(x − i)
C
k
n+1
(−1)
n−k
(n − k)!k!
=
n

k=0
(−1)
n−k
n +1− k
(n + 1)!

i=k
(x −i).
Suy ra
P (n +1)=
n

k=0

0
+ c
1
x + c
2
x
2
+ + c
n
x
n
co´ gia´ tri
.
h˜u
.
utı
’
khi x
h˜u
.
utı
’
.Ch´u
.
ng minh r˘a
`
ng, tˆa
´
tca
’

cnˆo
.
i suy Lagrange v´o
.
i a
k
= k (k =0, 1, 2, , n), ta co´
f(x)=
(−1)
n
f(0)
n!
(x − 1)(x − 2) (x − n)+
(−1)
n−1
f(1)
1!(n − 1)!
x(x − 2) (x − n)
+
(−1)
n−2
f(2)
2!(n − 2)!
x(x − 1)(x − 3) (x − n).
Theo gia
’
thiˆe
´
t, f(0), f(1), , f(n) la` nh˜u
.

’
aca´clu
˜
yth`u
.
acu
’
a x d¯ ˆe
`
u la` nh ˜u
.
ng
sˆo
´
h˜u
.
utı
’
.D
-
ˆo
`
ng nhˆa
´
t d¯a th´u
.
co
.
’
hai vˆe

.
cnˆo
.
i suy Lagrange ta
.
i n +1d¯iˆe
’
m a
k
(k =0, 1, 2, , n)h˜u
.
utı
’
tu`y y´ va` kha´c nhau, thı` cu
˜
ng d¯i d¯ˆe
´
nkˆe
´
t qua
’
trˆen. Do d¯o´,
ta co´ kˆe
´
t qua
’
sau:
Nˆe
´
u d¯a th´u

h˜u
.
utı
’
.
Ba`i toa´ n 2.9. Cho p la` mˆo
.
tsˆo
´
nguyˆen tˆo
´
va` P (x) ∈ Z[x] la` d¯a th´u
.
cbˆa
.
c s tho
’
a
ma
˜
nca´c d¯iˆe
`
ukiˆe
.
n
1) P(0) = 0, P (1) = 1.
2) P(n) ho˘a
.
c chia hˆe
´

.
.
cla
.
i, s  p −2. Khi d¯o´, theo cˆong th´u
.
cnˆo
.
i suy Lagrange, ta co´
P (x)=
p−2

k=0
P (k)
p−2

j=0;j=k
(x − j)
p−2

j=0;j=k
(k −j)
=
p−2

k=0
P (k)
p−2

j=0;j=k

k
p−1
.
Theo gia
’
thiˆe
´
t thı` p nguyˆen tˆo
´
,nˆenC
k
p−1
≡ (−1)
k
(mod p). Do d¯o´
P (p − 1)=(−1)
p
p−2

k=0
P (k)(modp).
24
Nˆe
´
u p = 2, thı` 1 = P(1) = (−1)
2
P (0) (mod p), vˆo ly´.
Nˆe
´
u p ≥ 3, thı` p le

t luˆa
.
n trˆen.
Vˆa
.
yd¯iˆe
`
u gia
’
thiˆe
´
t s  p − 2 la` sai. Ta co´ d¯iˆe
`
u pha
’
ich´u
.
ng minh.
Ba`i toa´ n 2.10. Tı`m tˆa
´
tca
’
ca´c d¯a th´u
.
c P (x) co´bˆa
.
c nho
’
ho
.

ica´cnu´t nˆo
.
i suy x
k
= k ta co´, mo
.
i
d¯a th´u
.
c P(x) co´ bˆa
.
c nho
’
ho
.
n n d¯ ˆe
`
u co´ da
.
ng
P (x)=
n−1

k=0
P (x
k
)
(x − x
0
) (x −x

)
(x −0) (x − (k −1))(x − (k + 1)) (x −(n − 1))
(k −0) (k − (k − 1))(k − ( k + 1)) ( k −(n − 1))
.
Do d¯o´
P (n)=
n−1

k=0
P (x
k
)
(n − 0) (n − k + 1)( n −k −1) 1
k!(−1)
n−k−1
(n −k −1)!
=
n−1

k=0
(−1)
n−k−1
.C
k
n
P (k)
Suy ra
n

k=0


k=0
a
k
n−1

i=0,i=k
x − x
i
k −i
v´o
.
i a
k
∈ R.