Đa thức và hàm hữu tỷ

Chu
.
o
.
ng 2
D
-
ath´u
.
cv`ah`amh˜u
.
uty
’
2.1 D
-
ath´u
.
c ..................... 44
2.1.1 D
-
ath´u
.
c trˆen tru
.
`o
.
ng sˆo
´
ph´u
.
c C ....... 45

2.1.2 D
-
ath´u
.
c trˆen tru
.
`o
.
ng sˆo
´
thu
.
.
c R ....... 46
2.2 Phˆan th´u
.
ch˜u
.
uty
’
............... 55
2.1 D
-
ath´u
.
c
Dath´u
.
cmˆo
.

tbiˆe
´
nv´o
.
ihˆe
.
sˆo
´
thuˆo
.
c tru
.
`o
.
ng sˆo
´
P d
u
.
o
.
.
cbiˆe
’
udiˆe
˜
nd
o
.
n tri

.
du
.
´o
.
ida
.
ng tˆo
’
ng h˜u
.
uha
.
n
Q(x)=a
0
z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
(2.1)
trong d
´o z l`a biˆe
´

n, a
0
,a
1
,...,a
n
l`a c´ac sˆo
´
;v`amˆo
˜
itˆo
’
ng da
.
ng (2.1) dˆe
`
u
l`a d
ath´u
.
c.
K´yhiˆe
.
u: Q(z) ∈P[z].
Nˆe
´
u a
0
,a
1

,...,a
n
∈ C th`ı ngu
.
`o
.
i ta n´oi r˘a
`
ng Q(z)l`ad
ath´u
.
c trˆen
tru
.
`o
.
ng sˆo
´
ph´u
.
c: Q(z) ∈ C[z]. Nˆe
´
u a
0
,a
1
,...,a
n
∈ R th`ı Q(z)l`ada
th´u

.
c trˆen tru
.
`o
.
ng sˆo
´
thu
.
.
c: Q(z) ∈ R[z].
2.1. D
-
ath´u
.
c 45
Nˆe
´
u Q(z) =0th`ıbˆa
.
ccu
’
a n´o (k´y hiˆe
.
u degQ(z)) l`a sˆo
´
m˜u cao nhˆa
´
t
cu

’
amo
.
i lu˜y th`u
.
acu
’
a c´ac sˆo
´
ha
.
ng =0cu
’
ad
ath´u
.
cv`ahˆe
.
sˆo
´
cu
’
asˆo
´
ha
.
ng c´o lu˜yth`u
.
a cao nhˆa
´

td
´o g o
.
il`ahˆe
.
sˆo
´
cao nhˆa
´
t.
Nˆe
´
u P (z)v`aQ(z) ∈P[z] l`a c˘a
.
pd
ath´u
.
cv`aQ(z) =0th`ıtˆo
`
nta
.
i
c˘a
.
pd
ath´u
.
c h(z)v`ar(z) ∈P[z] sao cho
1
+

P = Qh + r,
2
+
ho˘a
.
c r(z) = 0, ho˘a
.
c degr<degQ.
D
-
i
.
nhl´yB´ezout. Phˆa
`
ndu
.
cu
’
aph´ep chia d
ath´u
.
c P (z) cho nhi
.
th´u
.
c
z − α l`a h˘a
`
ng P (α) (r = P (α)).
2.1.1 D

-
ath´u
.
c trˆen tru
.
`o
.
ng sˆo
´
ph´u
.
c C
Gia
’
su
.
’
Q(z) ∈ C[z]. Nˆe
´
u thay z bo
.
’
isˆo
´
α th`ı ta thu d
u
.
o
.
.

csˆo
´
ph´u
.
c
Q(α)=a
0
α
n
+ a
1
α
n−1
+ ···+ a
n−1
α + a
n
.
D
-
i
.
nh ngh˜ıa 2.1.1. Nˆe
´
u Q(α) = 0 th`ı sˆo
´
z = α d
u
.
o

.
.
cgo
.
il`anghiˆe
.
m
cu
’
ad
ath´u
.
c Q(z) hay cu
’
aphu
.
o
.
ng tr`ınh d
a
.
isˆo
´
Q(z)=0.
D
-
i
.
nh l´y Descate. D
ath´u

.
c Q(z) chia hˆe
´
t cho nhi
.
th´u
.
c z − α khi v`a
chı
’
khi α l`a nghiˆe
.
mcu
’
ad
ath´u
.
c P (z) (t´u
.
cl`aP (α)=0).
D
-
i
.
nh ngh˜ıa 2.1.2. Sˆo
´
ph´u
.
c α l`a nghiˆe
.

mbˆo
.
i m cu
’
ad
ath´u
.
c Q(z)
nˆe
´
uv`achı
’
nˆe
´
u Q(z) chia hˆe
´
tcho(z − α)
m
nhu
.
ng khˆong chia hˆe
´
tcho
(z − α)
m+1
.Sˆo
´
m du
.
o

.
.
cgo
.
il`abˆo
.
i cu
’
a nghiˆe
.
m α. Khi m = 1, sˆo
´
α go
.
i
l`a nghiˆe
.
md
o
.
n cu
’
a Q(z).
Trong tiˆe
´
t 2.1.1 ta biˆe
´
tr˘a
`
ng tˆa

.
pho
.
.
psˆo
´
ph´u
.
c C d
u
.
o
.
.
clˆa
.
pnˆenb˘a
`
ng
c´ach gh´ep thˆem v`ao cho tˆa
.
pho
.
.
psˆo
´
thu
.
.
c R mˆo

.
t nghiˆe
.
ma
’
o x = i cu
’
a
phu
.
o
.
ng tr`ınh x
2
+1=0v`amˆo
.
t khi d˜a gh´ep i v`ao th`ı mo
.
iphu
.
o
.
ng
tr`ınh d
ath´u
.
cd
ˆe
`
uc´onghiˆe

.
mph´u
.
c thu
.
.
csu
.
.
.Dod
´o khˆong cˆa
`
n pha
’
i
s´ang ta
.
o thˆem c´ac sˆo
´
m´o
.
id
ˆe
’
gia
’
iphu
.
o
.

ng tr`ınh (v`ı thˆe
´
C c`on d
u
.
o
.
.
cgo
.
i
l`a tru
.
`o
.
ng d
´ong da
.
isˆo
´
).
D
-
i
.
nh l´y Gauss (d
i
.
nh l´y co
.

ba
’
ncu
’
ad
a
.
isˆo
´
).
46 Chu
.
o
.
ng 2. D
-
ath´u
.
c v`a h`am h˜u
.
uty
’
Mo
.
idath´u
.
cd
a
.
isˆo

´
bˆa
.
c n (n  1) trˆen tru
.
`o
.
ng sˆo
´
ph´u
.
cd
ˆe
`
u c´o ´ıt
nhˆa
´
tmˆo
.
t nghiˆe
.
mph´u
.
c.
T`u
.
d
i
.
nh l´y Gauss r´ut ra c´ac hˆe

.
qua
’
sau.
1
+
Mo
.
idath´u
.
cbˆa
.
c n (n  1) trˆen tru
.
`o
.
ng sˆo
´
ph´u
.
cd
ˆe
`
uc´od´ung n
nghiˆe
.
mnˆe
´
umˆo
˜

i nghiˆe
.
md
u
.
o
.
.
c t´ınh mˆo
.
tsˆo
´
lˆa
`
nb˘a
`
ng bˆo
.
icu
’
an´o,t´u
.
cl`a
Q(x)=a
0
(z − α
1
)
m
1

(z − α
2
)
m
2
···(z − α
k
)
m
k
, (2.2)
trong d
´o α
i
= α
j
∀ i = j v`a m
1
+ m
2
+ ···+ m
k
= n.
D
ath´u
.
c (2.1) v´o
.
ihˆe
.

sˆo
´
cao nhˆa
´
t a
0
=1du
.
o
.
.
cgo
.
il`ad
ath´u
.
c thu
go
.
n.
2
+
Nˆe
´
u z
0
l`a nghiˆe
.
mbˆo
.

i m cu
’
adath´u
.
c Q(z)th`ısˆo
´
ph´u
.
cliˆen ho
.
.
p
v´o
.
in´o
z
0
l`a nghiˆe
.
mbˆo
.
i m cu
’
adath´u
.
c liˆen ho
.
.
p
Q(z), trong d´o d a

th´u
.
c
Q(z)du
.
o
.
.
c x´ac d
i
.
nh bo
.
’
i
Q(z)
def
= a
0
z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
. (2.3)

2.1.2 D
-
ath´u
.
c trˆen tru
.
`o
.
ng sˆo
´
thu
.
.
c R
Gia
’
su
.
’
Q(z)=z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
(2.4)

l`a d
ath´u
.
c quy go
.
nv´o
.
ihˆe
.
sˆo
´
thu
.
.
c a
1
,a
2
,...,a
n
.
D
ath´u
.
c n`ay c´o t´ınh chˆa
´
td
˘a
.
cbiˆe

.
t sau dˆa y .
D
-
i
.
nh l´y 2.1.1. Nˆe
´
usˆo
´
ph´u
.
c α l`a nghiˆe
.
mbˆo
.
i m cu
’
ad
ath´u
.
c (2.4) v´o
.
i
hˆe
.
sˆo
´
thu
.

.
c th`ı sˆo
´
ph´u
.
c liˆen ho
.
.
pv´o
.
in´o
α c˜ung l`a nghiˆe
.
mbˆo
.
i m cu
’
a
d
ath´u
.
cd
´o.
Su
.
’
du
.
ng d
i

.
nh l´y trˆen dˆay ta c´o thˆe
’
t`ım khai triˆe
’
ndath´u
.
cv´o
.
ihˆe
.
sˆo
´
thu
.
.
c Q(z) th`anh t´ıch c´ac th`u
.
asˆo
´
.Vˆe
`
sau ta thu
.
`o
.
ng chı
’
x´et d
a

th´u
.
cv´o
.
ihˆe
.
sˆo
´
thu
.
.
cv´o
.
ibiˆe
´
nchı
’
nhˆa
.
n gi´a tri
.
thu
.
.
cnˆen biˆe
´
nd
´o t a k ´y
hiˆe
.

ul`ax thay cho z.
2.1. D
-
ath´u
.
c 47
D
-
i
.
nh l´y 2.1.2. Gia
’
su
.
’
d
ath´u
.
c Q(x) c´o c´ac nghiˆe
.
m thu
.
.
c b
1
,b
2
,...,b
m
v´o

.
ibˆo
.
itu
.
o
.
ng ´u
.
ng β
1
,β
2
,...,β
m
v`a c´ac c˘a
.
p nghiˆe
.
mph´u
.
cliˆen ho
.
.
p a
1
v`a a
1
, a
2

v`a a
2
,...,a
n
v`a a
n
v´o
.
ibˆo
.
itu
.
o
.
ng ´u
.
ng λ
1
,λ
2
,...,λ
n
. Khi d´o
Q(x)=(x− b
1
)
β
1
(x− b
2

)
β
2
···(x − b
m
)
β
m
(x
2
+ p
1
x + q
1
)
λ
1
×
× (x
2
+ p
2
x + q
2
)
λ
2
···(x
2
+ p

n
x + q
b
)
λ
n
. (2.5)
D
-
i
.
nh l´y 2.1.3. Nˆe
´
ud
ath´u
.
c Q(x)=x
n
+ a
1
x
n−1
+ ···+ a
n−1
x + a
n
v´o
.
ihˆe
.

sˆo
´
nguyˆen v`a v´o
.
ihˆe
.
sˆo
´
cao nhˆa
´
tb˘a
`
ng 1 c´o nghiˆe
.
mh˜u
.
uty
’
th`ı
nghiˆe
.
md
´o l`a sˆo
´
nguyˆen.
D
ˆo
´
iv´o
.

id
ath´u
.
cv´o
.
ihˆe
.
sˆo
´
h˜u
.
uty
’
ta c´o
D
-
i
.
nh l´y 2.1.4. Nˆe
´
u phˆan sˆo
´
tˆo
´
i gia
’
n

m
(, m ∈ Z,m>0) l`a nghiˆe

.
m
h˜u
.
uty
’
cu
’
a phu
.
o
.
ng tr`ınh v´o
.
ihˆe
.
sˆo
´
h˜u
.
uty
’
a
0
x
n
+a
1
x
n−1

+···+a
n−1
x+
a
n
=0th`ı  l`a u
.
´o
.
ccu
’
asˆo
´
ha
.
ng tu
.
.
do a
n
v`a m l`a u
.
´o
.
ccu
’
ahˆe
.
sˆo
´

cao
nhˆa
´
t a
0
.
C
´
AC V
´
IDU
.
V´ı d u
.
1. Gia
’
su
.
’
P (z)=a
0
z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a

n
.Ch´u
.
ng
minh r˘a
`
ng:
1
+
Nˆe
´
u P (z) ∈ C[z]th`ıP (z)=P (z).
2
+
Nˆe
´
u P (z) ∈ R[z]th`ıP (z)=P (z).
Gia
’
i. 1
+
´
Ap du
.
ng c´ac t´ınh chˆa
´
tcu
’
a ph´ep to´an lˆa
´

y liˆen ho
.
.
p ta thu
d
u
.
o
.
.
c
p(Z)=a
0
z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
= a
0
z
n
+ a
1
z

n−1
+ ···+ a
n−1
z + a
n
= a
0
(z)
n
+ a
1
(z)
n−1
+ ···+ a
n−1
z + a
n
= P (z).
48 Chu
.
o
.
ng 2. D
-
ath´u
.
c v`a h`am h˜u
.
uty
’

2
+
Gia
’
su
.
’
P (z) ∈ R[z]. Khi d
´o
P (z)=a
0
z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
= a
0
z
n
+ a
1
z
n−1
+ ···+ a

n−1
z + a
n
= a
0
(z)
n
+ a
1
(z)
n−1
+ ···+ a
n−1
z + a
n
= a
0
(z)
n
+ a
1
(z)
n−1
+ ···+ a
n−1
z + a
n
= P (z).
T`u
.

d
´oc˜ung thu du
.
o
.
.
c P (z)=
P (z)v`ı P(z)=P(z). 
V´ı d u
.
2. Ch´u
.
ng minh r˘a
`
ng nˆe
´
u a l`a nghiˆe
.
mbˆo
.
i m cu
’
ad
ath´u
.
c
P (z)=a
0
z
n

+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
,a
0
=0
th`ı sˆo
´
ph´u
.
c liˆen ho
.
.
p
a l`a nghiˆe
.
mbˆo
.
i m cu
’
adath´u
.
c
P (z)=a
0

z
n
+ a
1
z
n−1
+ ···+ a
n−1
z + a
n
(go
.
il`adath´u
.
c liˆen ho
.
.
pph´u
.
cv´o
.
id
ath´u
.
c P (z)).
Gia
’
i. T`u
.
v´ıdu

.
1 ta c´o
P (z)=P (z). (2.6)
V`ı a l`a nghiˆe
.
mbˆo
.
i m cu
’
a P (z)nˆen
P (z)=(z − a)
m
Q(z),Q(a) = 0 (2.7)
trong d
´o Q(z)l`adath´u
.
cbˆa
.
c n − m.T`u
.
(2.6) v`a (2.7) suy ra
P (z)=P (z)=(z − a)
m
Q(z)=(z − a)
m
Q(z). (2.8)
Ta c`on cˆa
`
nch´u
.

ng minh r˘a
`
ng
Q(a) = 0. Thˆa
.
tvˆa
.
y, nˆe
´
u Q(a)=0th`ı
b˘a
`
ng c´ach lˆa
´
y liˆen ho
.
.
pph´u
.
cmˆo
.
tlˆa
`
nn˜u
.
a ta c´o
Q(a)=Q(a)=0 ⇒ Q(a)=0.
D
iˆe
`

u n`ay vˆo l´y. B˘a
`
ng c´ach d˘a
.
t t = z,t`u
.
(2.8) thu d
u
.
o
.
.
c
P (t)=(t − a)
m
Q(t), Q(a) =0.
2.1. D
-
ath´u
.
c 49
D˘a
’
ng th´u
.
c n`ay ch´u
.
ng to
’
r˘a

`
ng t =
a l`a nghiˆe
.
mbˆo
.
i m cu
’
adath´u
.
c
P (t). 
V´ı d u
.
3. Ch´u
.
ng minh r˘a
`
ng nˆe
´
u a l`a nghiˆe
.
mbˆo
.
i m cu
’
ad
ath´u
.
cv´o

.
i
hˆe
.
sˆo
´
thu
.
.
c P (z)=a
0
z
n
+ a
1
z
n−1
+ ···+ a
n
(a
0
= 0) th`ı sˆo
´
ph´u
.
c liˆen
ho
.
.
p

a c˜ung l`a nghiˆe
.
mbˆo
.
i m cu
’
ach´ınh dath´u
.
cd
´o.
Gia
’
i. T`u
.
v´ıdu
.
1, 2
+
ta c´o
P (z)=P (z) (2.9)
v`a do a l`a nghiˆe
.
mbˆo
.
i m cu
’
a n´o nˆen
P (z)=(z − a)
m
Q(z) (2.10)

trong d
´o Q(z)l`adath´u
.
cbˆa
.
c n − m v`a Q(a) =0.
Ta cˆa
`
nch´u
.
ng minh r˘a
`
ng
P (z)=(z −
a)
m
Q(z),Q(a) =0. (2.11)
Thˆa
.
tvˆa
.
yt`u
.
(2.9) v`a (2.10) ta c´o
P (z)=
(z − a)
m
Q(z)=(z − a)
m
· Q(z)

=

(z − a)

m
Q(z)=(z − a)
m
Q(z)
v`ı theo (2.9)
Q(
z)=Q(z) ⇒ Q(z)=Q(z).
Ta c`on cˆa
`
nch´u
.
ng minh Q(
a) = 0. Thˆa
.
tvˆa
.
yv`ı Q(a) =0nˆen
Q(a) =0v`adod´o Q(a) =0v`ıdˆo
´
iv´o
.
id
ath´u
.
cv´o
.

ihˆe
.
sˆo
´
thu
.
.
cth`ı
Q(t)=Q(t). 
V´ı du
.
4. Gia
’
iphu
.
o
.
ng tr`ınh z
3
− 4z
2
+4z − 3=0.
Gia
’
i. T`u
.
d
i
.
nh l´y 4 suy r˘a

`
ng c´ac nghiˆe
.
m nguyˆen cu
’
aphu
.
o
.
ng tr`ınh
v´o
.
ihˆe
.
sˆo
´
nguyˆen d
ˆe
`
ul`au
.
´o
.
ccu
’
asˆo
´
ha
.
ng tu

.
.
do a = −3. Sˆo
´
ha
.
ng tu
.
.
do
50 Chu
.
o
.
ng 2. D
-
ath´u
.
c v`a h`am h˜u
.
uty
’
a = −3 c´o c´ac u
.
´o
.
cl`a±1, ±3. B˘a
`
ng c´ach kiˆe
’

m tra ta thu d
u
.
o
.
.
c z
0
=3
l`a nghiˆe
.
m nguyˆen. T`u
.
d
´o
z
3
− 4z
2
+4z − 3=(z − 3)(z
2
− z +1)
=(z − 3)(z −
1
2
+ i
√
3
2


z −
1
2
− i
√
3
2

hay l`a phu
.
o
.
ng tr`ınh d
˜a cho c´o ba nghiˆe
.
ml`a
z
0
=3,z
1
=
1
2
− i
√
3
2
; z
2
=

1
2
+ i
√
3
2
· 
V´ı d u
.
5. Biˆe
’
udiˆe
˜
nd
ath´u
.
c P
6
(z)=z
6
− 3z
4
+4z
2
− 12 du
.
´o
.
ida
.

ng:
1
+
t´ıch c´ac th`u
.
asˆo
´
tuyˆe
´
n t´ınh;
2
+
t´ıch c´ac th`u
.
asˆo
´
tuyˆe
´
n t´ınh v´o
.
i tam th´u
.
cbˆa
.
c hai v´o
.
ihˆe
.
sˆo
´

thu
.
.
c.
Gia
’
i. Tat`ımmo
.
i nghiˆe
.
mcu
’
ad
ath´u
.
c P (z). V`ı
z
6
− 3z
4
+4z
2
− 12 = (z
2
− 3)(z
4
+4)
nˆen r˜o r`ang l`a
z
1

= −
√
3,z
2
=
√
3,z
3
=1+i,
z
4
=1− i, z
5
= −1+i, z
6
= −1 − i.
T`u
.
d
´o
1
+
P
6
(z)=(z−
√
3)(z +
√
3)(z−1−i)(z−1+i)(z +1−i)(z +1+i)
2

+
B˘a
`
ng c´ach nhˆan c´ac c˘a
.
p nhi
.
th´u
.
c tuyˆe
´
n t´ınh tu
.
o
.
ng ´u
.
ng v´o
.
i c´ac
nghiˆe
.
mph´u
.
c liˆen ho
.
.
pv´o
.
i nhau ta thu d

u
.
o
.
.
c
P
6
(z)=(z −
√
3)(z +
√
3)(z
2
− 2z + 2)(z
2
+2z +2). 
V´ı d u
.
6. T`ım d
ath´u
.
chˆe
.
sˆo
´
thu
.
.
cc´olu˜yth`u

.
a thˆa
´
p nhˆa
´
t sao cho c´ac
sˆo
´
z
1
=3,z
2
=2− i l`a nghiˆe
.
mcu
’
a n´o.
2.1. D
-
ath´u
.
c 51
Gia
’
i. V`ıdath´u
.
cchı
’
c´o hˆe
.

sˆo
´
thu
.
.
cnˆen c´ac nghiˆe
.
mph´u
.
c xuˆa
´
thiˆe
.
n
t`u
.
ng c˘a
.
p liˆen ho
.
.
pph´u
.
c, ngh˜ıa l`a nˆe
´
u z
2
=2− i l`a nghiˆe
.
mcu

’
an´oth`ı
z
2
=2+i c˜ung l`a nghiˆe
.
mcu
’
a n´o. Do d´o
P (z)=(z − 3)(z − 2+i)(z − 2 − i)=z
3
− 7z
2
+17z − 15. 
V´ı du
.
7. Phˆan t´ıch d
ath´u
.
c
(x +1)
n
− (x − 1)
n
th`anh c´ac th`u
.
asˆo
´
tuyˆe
´

n t´ınh.
Gia
’
i. Ta c´o
P (x)=(x +1)
n
− (x − 1)
n
=[x
n
+ nx
n−1
+ ...]− [x
n
− nx
n−1
+ ...]=2nx
n−1
+ ...
Nhu
.
vˆa
.
y P (x)l`ad
ath´u
.
cbˆa
.
c n − 1v´o
.

ihˆe
.
sˆo
´
cao nhˆa
´
tb˘a
`
ng 2n.D
ˆo
´
i
v´o
.
id
ath´u
.
c n`ay ta d
˜abiˆe
´
t(§1) nghiˆe
.
mcu
’
a n´o:
x
k
= icotg
kπ
n

,k=1, 2,...,n− 1.
Do d
´o
(x +1)
n
− (x − 1)
n
=2n

x − icotg
π
n

x − icotg
2π
n

···

x − icotg
(n − 1)π
n

.
Khi phˆan t´ıch d
ath´u
.
c trˆen tru
.
`o

.
ng P th`anh th`u
.
asˆo
´
ta thu
.
`o
.
ng
g˘a
.
pnh˜u
.
ng d
ath´u
.
c khˆong thˆe
’
phˆan t´ıch th`anh t´ıch hai d
ath´u
.
c c´o bˆa
.
c
thˆa
´
pho
.
ntrˆenc`ung tru

.
`o
.
ng P d
´o. Nh˜u
.
ng d
ath´u
.
cn`ayd
u
.
o
.
.
cgo
.
il`ad
a
th´u
.
cbˆa
´
t kha
’
quy.
Ch˘a
’
ng ha
.

n: d
ath´u
.
c x
2
− 2l`akha
’
quy trˆen tru
.
`o
.
ng sˆo
´
thu
.
.
cv`ı:
x
2
− 2=(x −
√
2)(x +
√
2)