Bài tập toán cao cấp Tập 1 part 2 docx

1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 27
T`u
.
d
´othudu
.
o
.
.
c
z
2

=

2(1 + cos ϕ)

cos

−
ϕ
2

+ i sin

−
ϕ
2

v`a do vˆa
.
y
w =

2(1 + cos ϕ)

2(1 + cos ϕ)
×

cos
ϕ
2
+ i sin

ϕ
2

cos

−
ϕ
2

+ i sin

−
ϕ
2

= cos ϕ + i sin ϕ. 
V´ı d u
.
3. 1) T´ınh (
√
3+i)
126
2) T´ınh acgumen cu
’
asˆo
´
ph´u
.
c sau
w = z

4
− z
2
nˆe
´
u argz = ϕ v`a |z| =1.
Gia
’
i. 1) Ta c´o
√
3+i =2

cos
π
6
+ i sin
π
6

.T`u
.
d´o ´a p d u
.
ng cˆong
th ´u
.
c Moivre ta thu du
.
o
.

.
c:
(
√
3+i)
126
=2
126

cos
126π
6
+ i sin
126π
6

=2
126
[cos π + i sin π]=−2
126
.
2) Ta c´o
w = z
4
−z
2
= cos 4ϕ + i sin 4ϕ −[cos 2ϕ −i sin 2ϕ]
= cos 4ϕ − cos 2ϕ + i(sin 4ϕ + sin 2ϕ)
= −2 sin 3ϕ sin ϕ +2i sin 3ϕ cos ϕ
= 2 sin 3ϕ[−sin ϕ + i cos ϕ].

(i) Nˆe
´
u sin 3ϕ>0(t´u
.
c l`a khi
2kπ
3
<ϕ<
(2k +1)π
3
, k ∈ Z )th`ı
w = 2 sin 3ϕ

cos

π
2
+ ϕ

+ i sin

π
2
+ ϕ

.
(ii) Nˆe
´
u sin 3ϕ<0(t´u
.

c l`a khi
(2k − 1)π
3
<ϕ<
2kπ
3
, k ∈ Z )th`ı
w =(−2 sin 3ϕ)[sin ϕ −icos ϕ].
28 Chu
.
o
.
ng 1. Sˆo
´
ph´u
.
c
Ta t`ım da
.
ng lu
.
o
.
.
ng gi´ac cu
’
a v = sin ϕ − i cos ϕ.Hiˆe
’
n nhiˆen |v| =1.
Ta t´ınh argv

argv = arctg

−cos ϕ
sin ϕ

= arctg(−cotgϕ)
= arctg

− tg

π
2
− ϕ

= arctg

tg

ϕ −
π
2

= ϕ −
π
2
·
Nhu
.
vˆa
.

ynˆe
´
u sin 3ϕ<0th`ı
w =(−2 sin 3ϕ)

cos

ϕ −
π
2

+ i sin

ϕ −
π
2

.
(iii) Nˆe
´
u sin 3ϕ =0⇒ ϕ =
kπ
3
⇒ w =0.
Nhu
.
vˆa
.
y
argw =












π
2
+ ϕ nˆe
´
u
2kπ
3
<ϕ<
(2k +1)π
3
,
khˆong x´ac di
.
nh nˆe
´
u ϕ =
kπ
3
,

ϕ −
π
2
nˆe
´
u
(2k − 1)π
3
<ϕ<
2kπ
3
· 
V´ı d u
.
4. Ch´u
.
ng minh r˘a
`
ng
1) cos
π
9
+ cos
3π
9
+ cos
5π
9
+ cos
7π

9
=
1
2
.
2) cos ϕ + cos(ϕ + α)+cos(ϕ +2α)+···+ cos(ϕ + nα)
=
sin
(n +1)α
2
cos

ϕ +
nα
2

sin
α
2
·
Gia
’
i. 1) D˘a
.
t
S = cos
π
9
+ cos
3π

9
+ ···+ cos
7π
9
,
T = sin
π
9
+ sin
3π
9
+ ···+ sin
7π
9
,
z = cos
π
9
+ i sin
π
9
.
1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.

cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 29
Khi d´o
S + iT = z + z
3
+ z
5
+ z
7
=
z(1 − z
8
)
1 −z
2
=
z − z
9
1 −z
2

=
z +1
1 −z
2
=
1
1 − z
=
1

1 −cos
π
9

− i sin
π
9
=

1 −cos
π
9

+ i sin
π
9

1 −cos
π
9


2
+ sin
2
π
9
=
1
2
+
sin
π
9
2

1 −cos
π
9

·
Do d
´o S =
1
2
·
2) Tu
.
o
.
ng tu

.
.
nhu
.
trong 1) ta k´yhiˆe
.
u
S = cos ϕ + cos(ϕ + α)+···+ cos(ϕ + nα),
T = sin ϕ + sin(ϕ + α)+···+ sin(ϕ + nα),
z = cos α + i sin α, c = cos ϕ + i sin ϕ.
Khi d
´o
S + iT = c + cz + ···+ cz
n
=
c(1 − z
n+1
)
1 −z
=
(cos ϕ + i sin ϕ)[1 −cos(n +1)α − i sin(n +1)α]
1 − cos α −i sin α
=
(cos ϕ + i sin ϕ)2 sin
(n +1)α
2

cos
(n +1)α − π
2

+ i sin
(n +1)α − π
2

2 sin
α
2

cos
α − π
2
+ i sin
α −π
2

=
sin
(n +1)α
2
cos

ϕ +
nα
2

sin
α
2
+
sin

(n +1)α
2
sin

ϕ +
nα
2
sin
α
2
i.
T`u
.
d´o so s´anh phˆa
`
n thu
.
.
c v`a phˆa
`
na
’
o ta thu du
.
o
.
.
ckˆe
´
t qua

’
. 
B˘a
`
ng phu
.
o
.
ng ph´ap tu
.
o
.
ng tu
.
.
ta c´o thˆe
’
t´ınh c´ac tˆo
’
ng da
.
ng
a
1
sin b
1
+ a
2
sin b
2

+ ···+ a
n
sin b
n
,
a
1
cos b
1
+ a
2
cos b
2
+ ···+ a
n
cos b
n
30 Chu
.
o
.
ng 1. Sˆo
´
ph´u
.
c
nˆe
´
u c´ac acgumen b
1

,b
2
, ,b
n
lˆa
.
pnˆen cˆa
´
psˆo
´
cˆo
.
ng c`on c´ac hˆe
.
sˆo
´
a
1
,a
2
, ,a
n
lˆa
.
p nˆen cˆa
´
psˆo
´
nhˆan.
V´ı d u

.
5. T´ınh tˆo
’
ng
1) S
n
=1+a cos ϕ + a
2
cos 2ϕ + ···+ a
n
cos nϕ;
2) T
n
= a sin ϕ + a
2
sin 2ϕ + ···+ a
n
sin nϕ.
Gia
’
i. Ta lˆa
.
pbiˆe
’
uth´u
.
c S
n
+ iT
n

v`a thu du
.
o
.
.
c
Σ=S
n
+ iT
n
=1+a(cos ϕ + i sin ϕ)+a
2
(cos 2ϕ + i sin 2ϕ)+
+ a
n
(cos nϕ + i sin nϕ).
D˘a
.
t z = cos ϕ + i sin ϕ v`a ´ap du
.
ng cˆong th´u
.
c Moivre ta c´o:
Σ=1+az + a
2
z
2
+ ···+ a
n
z

n
=
a
n+1
z
n+1
− 1
az − 1
(nhˆan tu
.
’
sˆo
´
v`a mˆa
˜
usˆo
´
v´o
.
i
a
z
−1)
=
a
n+2
z
n
− a
n+1

z
n+1
−
a
2
+1
a
2
− a

z +
1
z

+1
(do z +
1
z
= 2 cos ϕ)
=
a
n+2
(cos nϕ + i sin nϕ) −a
n+1
[cos(n +1)ϕ + i sin(n +1)ϕ]
a
2
− 2a cos ϕ +1
+
−a cos ϕ + ai sin ϕ +1

a
2
− 2a cos ϕ +1
=
a
n+2
cos nϕ − a
n+1
cos(n +1)ϕ − a cos ϕ +1
a
2
− 2a cos ϕ +1
+
+ i
a
n+2
sin nϕ −a
n+1
sin(n +1)ϕ + a sin ϕ
a
2
− 2a cos ϕ +1
·
B˘a
`
ng c´ach so s´anh phˆa
`
n thu
.
.

c v`a phˆa
`
na
’
otathudu
.
o
.
.
c c´ac kˆe
´
t qua
’
cˆa
`
n
d
u
.
o
.
.
c t´ınh.
V´ı d u
.
6. 1) Biˆe
’
udiˆe
˜
n tg5ϕ qua tgϕ.

1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 31
2) Biˆe
’
udiˆe
˜
n tuyˆe
´
n t´ınh sin
5
ϕ qua c´ac h`am sin cu
’

a g´oc bˆo
.
icu
’
a ϕ.
3) Biˆe
’
udiˆe
˜
n cos
4
ϕ v`a sin
4
ϕ·cos
3
ϕ qua h`am cosin cu
’
a c´ac g´oc bˆo
.
i.
Gia
’
i. 1) V`ı tg5ϕ =
sin 5ϕ
cos 5ϕ
nˆen ta cˆa
`
nbiˆe
’
udiˆe

˜
n sin 5ϕ v`a cos 5ϕ
qua sin ϕ v`a cosϕ. Theo cˆong th´u
.
c Moivre ta c´o
cos 5ϕ + i sin 5ϕ = (cos ϕ + i sin ϕ)
5
= sin
5
ϕ +5i cos
4
ϕ sin ϕ
− 10 cos
3
ϕ sin
2
ϕ −10i cos
2
ϕ sin
3
ϕ
+ 5 cos ϕ sin
4
ϕ + i sin
5
ϕ.
T´ach phˆa
`
n thu
.

.
c v`a phˆa
`
na
’
o ta thu d
u
.
o
.
.
cbiˆe
’
uth´u
.
cd
ˆo
´
iv´o
.
i sin 5ϕ v`a
cos 5ϕ v`a t`u
.
d´o
tg5ϕ =
5 cos
4
ϕ sin ϕ −10 cos
2
ϕ sin

3
ϕ + sin
5
ϕ
cos
5
ϕ −10 cos
3
ϕ sin
2
ϕ + 5 cos ϕ sin
4
ϕ
(chia tu
.
’
sˆo
´
v`a mˆa
˜
usˆo
´
cho cos
5
ϕ)
=
5tgϕ −10tg
3
ϕ +tg
5

ϕ
1 − 10tg
2
ϕ + 5tg
4
ϕ
·
2) D˘a
.
t z = cos ϕ + i sin ϕ. Khi d´o z
−1
= cos ϕ − i sin ϕ v`a theo
cˆong th´u
.
c Moivre:
z
k
= cos kϕ + i sin kϕ, z
−k
= cos kϕ − i sin kϕ.
Do d´o
cos ϕ =
z + z
−1
2
, sin ϕ =
z − z
−1
2i
z

k
+ z
−k
= 2 cos kϕ, z
k
−z
−k
=2i sin kϕ.
´
Ap du
.
ng c´ac kˆe
´
t qua
’
n`ay ta c´o
sin
5
ϕ =

z − z
−1
2i

5
=
z
5
− 5z
3

+10z − 10z
−1
+5z
−3
−z
−5
32i
=
(z
5
− z
−5
) −5(z
3
− z
−3
)+10(z − z
−1
)
32i
=
2i sin 5ϕ −10i sin 3ϕ +20i sin ϕ
32i
=
sin 5ϕ −5 sin 3ϕ + 10 sin ϕ
16
·
32 Chu
.
o

.
ng 1. Sˆo
´
ph´u
.
c
3) Tu
.
o
.
ng tu
.
.
nhu
.
trong phˆa
`
n 2) ho˘a
.
c gia
’
i theo c´ach sau d
ˆay
1
+
cos
4
ϕ =

e

iϕ
+ e
−iϕ
2

4
=
1
16

e
4iϕ
+4e
2iϕ
+6+4e
−2iϕ
+ e
−4iϕ

=
1
8

e
4ϕi
+ e
−4ϕi
2

+

1
2

e
2ϕi
+ e
−2ϕi
2

+
3
8
=
3
8
+
1
2
cos 2ϕ +
1
8
cos 4ϕ.
2
+
sin
4
ϕ cos
3
ϕ =


e
ϕi
− e
−ϕi
2i

4

e
ϕi
+ e
−ϕi
2

3
=
1
128

e
2ϕi
− e
−2ϕi

3

e
ϕi
−e
−ϕi


=
1
128

e
6ϕi
− 3e
2ϕi
+3e
−2ϕi
− e
−6ϕi

e
ϕi
−e
−ϕi

=
1
128

e
7ϕi
−e
5ϕi
− 3e
3ϕi
+3e

ϕi
+3e
−ϕ
i
− 3e
−3ϕi
− e
−5ϕi
+ e
−7ϕi

=
3
64
cos ϕ −
3
64
cos 3ϕ −
1
64
cos 5ϕ −
1
64
cos 7ϕ. 
V´ı d u
.
7. 1) Gia
’
i c´ac phu
.

o
.
ng tr`ınh
1
+
(x +1)
n
−(x − 1)
n
=0
2
+
(x + i)
n
+(x − i)
n
=0, n>1.
2) Ch´u
.
ng minh r˘a
`
ng mo
.
i nghiˆe
.
mcu
’
aphu
.
o

.
ng tr`ınh

1+ix
1 −ix

n
=
1+ai
1 −ai
,n∈ N,a∈ R
d
ˆe
`
u l`a nghiˆe
.
m thu
.
.
c kh´ac nhau.
Gia
’
i. 1) Gia
’
iphu
.
o
.
ng tr`ınh
1

+
Chia hai vˆe
´
cu
’
aphu
.
o
.
ng tr`ınh cho (x −1)
n
ta du
.
o
.
.
c

x +1
x −1

n
=1⇒
x +1
x − 1
=
n
√
1=cos
2kπ

n
+ i sin
2kπ
n
= ε
k
,
k =0, 1, ,n− 1.
1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 33
T`u
.

d
´o suy r˘a
`
ng
x +1=ε
k
(x −1) ⇒ x(ε
k
− 1)=1+ε
k
.
Khi k =0⇒ ε
0
= 1. Do d´ov´o
.
i k =0phu
.
o
.
ng tr`ınh vˆo nghiˆe
.
m. V´o
.
i
k =
1,n−1 ta c´o
x =
ε
k
+1

ε
k
− 1
=
(ε
k
+ 1)(ε
k
−1)
ε
k
− 1)(ε
k
−1)
=
ε
k
ε
k
+ ε
k
− ε
k
− 1
ε
k
ε
k
− ε
k

− ε
k
− 1
=
−2i sin
2kπ
n
2 −2 cos
2kπ
n
= −i
sin
2kπ
n
1 −cos
2kπ
n
= icotg
kπ
n
,k=1, 2, ,n− 1.
2
+
C˜ung nhu
.
trˆen, t`u
.
phu
.
o

.
ng tr`ınh d˜a cho ta c´o

x + i
x − i

n
= −1 ⇐⇒
x + i
x −i
=
n
√
−1=cos
π +2kπ
n
+ i sin
π +2kπ
n
hay l`a
x + i
x −i
= cos
(2k +1)π
n
+ i sin
(2k +1)π
n
= cos ψ + i sin ψ,ψ=
(2k +1)π

n
·
Ta biˆe
´
ndˆo
’
iphu
.
o
.
ng tr`ınh:
x + i
x −i
− 1=cosψ + i sin ψ − 1
⇔
2i
x −i
=2i sin
ψ
2
cos
ψ
2
−2 sin
2
ψ
2
⇔
1
x −i

= sin
ψ
2

cos
ψ
2
−
1
i
sin
ψ
2

= sin
ψ
2

cos
ψ
2
+ i sin
ψ
2

.
34 Chu
.
o
.

ng 1. Sˆo
´
ph´u
.
c
T`u
.
d
´o suy ra
x −i =
1
sin
ψ
2

cos
ψ
2
+ i sin
ψ
2

=
cos
ψ
2
− i sin
ψ
2
sin

ψ
2
= cotg
ψ
2
− i.
Nhu
.
vˆa
.
y
x −i = cotg
ψ
2
− i ⇒ x = cotg
ψ
2
= cotg
(2k +1)π
2n
,k= 0,n− 1.
2) Ta x´et vˆe
´
pha
’
icu
’
aphu
.
o

.
ng tr`ınh d
˜a cho. Ta c´o



1+ai
1 −ai



=1⇒
1+ai
1 − ai
= cos α + i sin α
v`a t `u
.
d
´o
1+xi
1 − xi
=
n

1+ai
1 −ai
= cos
α +2kπ
n
+ i sin

α +2kπ
n
,k= 0,n− 1.
T`u
.
d
´onˆe
´
ud˘a
.
t ψ =
α +2kπ
n
th`ı
x =
cos ψ − 1+i sin ψ
i[cos ψ +1+i sin ψ]
=tg
ψ
2
=tg
α +2kπ
2n
,k= 0,n− 1.
R˜o r`ang d
´o l`a nh˜u
.
ng nghiˆe
.
m thu

.
.
c kh´ac nhau. 
V´ı d u
.
8. Biˆe
’
udiˆe
˜
n c´ac sˆo
´
ph´u
.
csaudˆay du
.
´o
.
ida
.
ng m˜u:
1) z =
(−
√
3+i)

cos
π
12
−i sin
π

12

1 −i
·
2) z =

√
3+i.
1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 35
Gia
’

i. 1) D˘a
.
t z
1
= −
√
3+i, z
2
= cos
π
12
− i sin
π
12
, z
3
=1− i v`a
biˆe
’
udiˆe
˜
n c´ac sˆo
´
ph´u
.
cd´odu
.
´o
.
ida

.
ng m˜u. Ta c´o
z
1
=2e
5π
6
i
;
z
2
= cos
π
12
− i sin
π
12
= cos

−
π
12

+ i sin

−
π
12

= e

−
π
12
i
;
z
3
=
√
2e
−
π
4
i
.
T`u
.
d´othudu
.
o
.
.
c
z =
2e
5π
6
i
· e
−

π
12
i
√
2e
−
π
4
i
=
√
2e
iπ
.
2) Tru
.
´o
.
chˆe
´
tbiˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
c z
1

=
√
3+i du
.
´o
.
ida
.
ng m˜u. Ta c´o
|z
1
| =2; ϕ = arg(
√
3+i)=
π
6
,
do d
´o
√
3+i =2e
π
6
i
.T`u
.
d
´othudu
.
o

.
.
c
w
k
=
4

√
3+i =
4
√
2e
i
(
π
6
+2kπ)
4
=
4
√
2e
i
(12k+1)π
24
,k= 0, 3. 
V´ı d u
.
9. T´ınh c´ac gi´a tri

.
1) c˘an bˆa
.
c3: w =
3
√
−2+2i
2) c˘an bˆa
.
c4: w =
4
√
−4
3) c˘an bˆa
.
c5: w =
5

√
3 −i
8+8i
.
Gia
’
i. Phu
.
o
.
ng ph´ap tˆo
´

t nhˆa
´
td
ˆe
’
t´ınh gi´a tri
.
c´ac c˘an th´u
.
cl`abiˆe
’
u
diˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
idˆa
´
u c˘an du
.
´o
.
ida
.

ng lu
.
o
.
.
ng gi´ac (ho˘a
.
cda
.
ng m˜u) rˆo
`
i
´ap du
.
ng c´ac cˆong th´u
.
ctu
.
o
.
ng ´u
.
ng.
1) Biˆe
’
udiˆe
˜
n z = −2+2i du
.
´o

.
ida
.
ng lu
.
o
.
.
ng gi´ac. Ta c´o
r = |z| =
√
8=2
√
2; ϕ = arg(−2+2i)=
3π
4
·
36 Chu
.
o
.
ng 1. Sˆo
´
ph´u
.
c
Do d´o
w
k
=

3

√
8

cos
3π
4
+2kπ
3
+ i sin
3π
4
+2kπ
3

,k=
0, 2.
T`u
.
d´o
w
0
=
√
2

cos
π
4

+ i sin
π
4

=1+i,
w
1
=
√
2

cos
11π
12
+ i sin
11π
12

,
w
2
=
√
2

cos
19π
12
+ i sin
19π

12

.
2) Ta c´o
−4 = 4[cos π + i sin π]
v`a do d
´o
w
k
=
4
√
4

cos
π +2kπ
4
+ i sin
π +2kπ
4

,k= 0, 3.
T`u
.
d
´o
w
0
=
√

2

cos
π
4
+ i sin
π
4

=1+i,
w
1
=
√
2

cos
3π
4
+ i sin
3π
4

= −1+i,
w
2
=
√
2


cos
5π
4
+ i sin
5π
4

= −1 − i,
w
3
=
√
2

cos
7π
4
+ i sin
7π
4

=1−i.
3) D
˘a
.
t
z =
√
3 −i
8+8i

·
Khi d
´o |z| =
√
3+1
√
64 + 64
=
1
4
√
2
. Ta t´ınh argz.Tac´o
argz = arg(
√
3 −i) − arg(8 + 8i)=−
π
6
−
π
4
= −
5π
12
·
1.4. Biˆe
’
udiˆe
˜
nsˆo

´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 37
Do vˆa
.
y
w
k
=
5


1
4
√
2



cos
−
5π
12
+2kπ
5
+ i sin
−
5π
12
+2kπ
5

=
1
√
2

cos

−
π
12
+
2kπ
5

+ i sin

−

π
12
+
2kπ
5

,k= 0, 4. 
V´ı d u
.
10. 1) T´ınh tˆo
’
ng mo
.
i c˘an bˆa
.
c n cu
’
a1.
2) T´ınh tˆo
’
ng1+2ε +3ε
2
+ ···+ nε
n−1
, trong d´o ε l`a c˘an bˆa
.
c n
cu
’
ado

.
nvi
.
.
3) T´ınh tˆo
’
ng c´ac lu˜y th`u
.
abˆa
.
c k cu
’
amo
.
i c˘an bˆa
.
c n cu
’
asˆo
´
ph´u
.
c α.
Gia
’
i. 1) D
ˆa
`
u tiˆen ta viˆe
´

t c´ac c˘an bˆa
.
c n cu
’
a 1. Ta c´o
ε
k
=
n
√
1=cos
2kπ
n
+ i sin
2kπ
n
,k= 0,n− 1.
T`u
.
d
´o
ε
0
=1,ε
1
= ε = cos
2π
n
+ i sin
2π

n
,
ε
k
= cos
2kπ
n
+ i sin
2kπ
n
=

cos
2π
n
+ i sin
2π
n

k
= ε
k
,k=1, 2, ,n− 1.
Nhu
.
vˆa
.
ymo
.
i nghiˆe

.
mcu
’
a c˘an bˆa
.
c n cu
’
a1c´othˆe
’
viˆe
´
tdu
.
´o
.
ida
.
ng
1,ε,ε
2
, ,ε
n−1
.
Bˆay gi`o
.
ta t´ınh
S =1+ε + ε
2
+ ···+ ε
n−1

=
1 −ε
n
1 − ε
·
Nˆe
´
u n>1th`ıε
n
= 1 v`a do d´o
S =
1 − ε
n
1 −ε
=0.
38 Chu
.
o
.
ng 1. Sˆo
´
ph´u
.
c
2) Ta k´yhiˆe
.
utˆo
’
ng cˆa
`

n t´ınh l`a S.Tax´et biˆe
’
uth´u
.
c
(1 −ε)S = S − εS =1+2ε +3ε
2
+ ···+ nε
n−1
− ε − 2ε
2
−···−(n −1)ε
n−1
− nε
n
=1+ε + ε
2
+ ···+ ε
n−1
  
0(ε=1)
−nε
n
= −n
v`ı ε
n
=1.
Nhu
.
vˆa

.
y
(1 − ε)S = −n → S =
−n
1 −ε
nˆe
´
u ε =1.
Nˆe
´
u ε =1th`ı
S =1+2+···+ n =
n(n +1)
2
·
3) Gia
’
su
.
’
β
0
l`a mˆo
.
t trong c´ac gi´a tri
.
c˘an cu
’
a α. Khi d´o ( v ´o
.

i
α = 0) mo
.
i c˘an bˆa
.
c n cu
’
a α c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.
´o
.
ida
.
ng t´ıch β
0
ε
k
,
k =1, 2, ,n − 1, trong d
´o ε
k
= cos
2kπ
n

+ i sin
2kπ
n
l`a c˘an bˆa
.
c n
cu
’
a1.
T`u
.
d´otˆo
’
ng cˆa
`
nt`ımS b˘a
`
ng
S = β
k
0
+(β
0
ε
1
)
k
+(β
0
ε

2
)
k
+ ···+(β
0
ε
n−1
)
k
= β
k
0
(1 + ε
k
1
+ ε
k
2
+ ···+ ε
k
n−1
)

ε
k
m
=

cos
2mπ

n
+ i sin
2mπ
n

k
=

cos
2π
n
+ i sin
2π
n

mk

= β
k
0

1+ε
k
1
+ ε
2k
1
+ ···+ ε
(n−1)k
1


.
Biˆe
’
uth´u
.
c trong dˆa
´
u ngo˘a
.
c vuˆong l`a cˆa
´
psˆo
´
nhˆan. Nˆe
´
u ε
k
1
=1,t´u
.
cl`a
k khˆong chia hˆe
´
tchon th`ı
S = β
k
0
1 −ε
nk

1
1 −ε
k
1
= β
k
0
1 −1
1 −ε
k
1
=0 (v`ıε
n
1
=1).
1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu
.
´o
.
ida
.
ng lu

.
o
.
.
ng gi´ac 39
Nˆe
´
u ε
k
1
=1t´u
.
cl`ak chia hˆe
´
tchon, k = nq th`ı
S = β
nq
0
[1+1+···+1]=β
nq
0
n = nα
q
(v`ı β
n
0
= α).
Nhu
.
vˆa

.
y
S =



0nˆe
´
u k chia hˆe
´
tchon;
nα
q
nˆe
´
u k = nq, q ∈ Z. 
B
`
AI T
ˆ
A
.
P
1. Biˆe
’
udiˆe
˜
n c´ac sˆo
´
ph´u

.
csaudˆay du
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac
1) −1+i
√
3(DS. 2

cos
2π
3
+ i sin
2π
3

)
2)
√
3 −i (DS. 2

cos

11π
6
+ i sin
11π
6

)
3) −
√
3 −i (DS. 2

cos
7π
6
+ i sin
7π
6

)
4)
√
3
2
+
i
2
(DS. cos
π
6
+ i sin

π
6
)
5)
−
√
3
2
+
1
2
i (DS. cos
5π
6
+ i sin
5π
6
)
6)
1
2
− i
√
3
2
(DS. cos
5π
3
+ i sin
5π

3
)
7) −
1
2
−i
√
3
2
(DS. cos
4π
3
+ i sin
4π
3
)
8) 2 +
√
3 − i (DS. 2

2+
√
3

cos
23π
12
+ i sin
23π
12


)
9) 2 −
√
3 −i (DS. 2

2 −
√
3

cos
19π
12
+ i sin
19π
12

)
2. Biˆe
’
udiˆe
˜
n c´ac sˆo
´
ph´u
.
csaud
ˆay du
.
´o

.
ida
.
ng lu
.
o
.
.
ng gi´ac
1) −cos ϕ + i sin ϕ (D
S. cos(π − ϕ)+i sin(π −ϕ))
2) −sin ϕ + i cos ϕ (D
S. cos

π
2
+ ϕ

+ i sin

π
2
+ ϕ))
40 Chu
.
o
.
ng 1. Sˆo
´
ph´u

.
c
3) cos ϕ −i sin ϕ (DS. cos(−ϕ)+i sin(−ϕ))
4) −cos ϕ −i sin ϕ (D
S. cos(π + ϕ)+i sin(π + ϕ))
B˘a
`
ng c´ach d˘a
.
t α = θ +2kπ, trong d´o0 θ<2π, ta c´o:
5) 1+cos α+i sin α (DS. 2 cos
θ
2

cos
θ
2
+i sin
θ
2

v´o
.
i0 θ<π;
−2 cos
θ
2

cos
θ +2π

2
+ i sin
θ +2π
2

v´o
.
i π  θ<2π)
6) 1 −cos α + i sin α (D
S. 2 sin
θ
2

cos
π − θ
2
+ i sin
π − θ
2

)
7) sin α + i(1 + cos α)
(D
S. 2 cos
θ
2

cos
π − θ
2

+ i sin
π − θ
2

v´o
.
i0 θ<π;
−2 cos
θ
2

cos
3π − θ
2
+ i sin
3π − θ
2

v´o
.
i π  θ<2π)
8) −sin α + i(1 + cos α)
(DS. 2 cos
θ
2

cos
π + θ
2
+ i sin

π + θ
2

v´o
.
i0 θ<π;
−2 cos
θ
2

cos
3π + θ
2
+ i sin
3π + θ
2

v´o
.
i π  θ<2π)
3. T´ınh:
1)

cos
π
6
− i sin
π
6


100
(DS. −
1
2
− i
√
3
2
)
2)

4
√
3+i

12
(DS. 2
12
)
3)
(
√
3+i)
6
(−1+i)
8
− (1 + i)
4
(DS. −3, 2)
4)

(−i −
√
3)
15
(1 −i)
20
+
(−i +
√
3)
15
(1 + i)
20
(DS. −64i)
5)
(1 + i)
100
(1 −i)
96
+(1+i)
96
(DS. −2)
6)
(1 + icotgϕ)
5
1 −icotgϕ)
5
(DS. cos(π − 10ϕ)+i sin(π −10ϕ))
7)
(1 −i

√
3)(cos ϕ + i sin ϕ)
2(1 − i)(cos ϕ − i sin ϕ)
(DS.
√
2
2

cos

6ϕ −
π
12

+ i sin

6ϕ −
π
12

)
1.4. Biˆe
’
udiˆe
˜
nsˆo
´
ph´u
.
cdu

.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 41
8)
(1 + i
√
3)
3n
(1 + i)
4n
(DS. 2)
4. Ch´u
.
ng minh r˘a
`
ng z +
1
z
= 2 cos ϕ ⇒ z
n
+
1

z
n
= 2 cos nϕ.
5. H˜ay biˆe
’
udiˆe
˜
n c´ac h`am sau d
ˆay qua sin ϕ v`a cos ϕ
1) sin 3ϕ (D
S. 3 cos
2
ϕ sin ϕ −sin
3
ϕ)
2) cos 3ϕ (D
S. cos
3
ϕ −3 cos ϕ sin
2
ϕ)
3) sin 4ϕ (D
S. 4 cos
3
ϕ sin ϕ −4 cos ϕ sin
3
ϕ)
4) cos 4ϕ (D
S. cos
4

ϕ −6 cos
2
ϕ sin
2
ϕ + sin
4
ϕ)
6. H˜ay biˆe
’
udiˆe
˜
n c´ac h`am sau qua tgx
1) tg4ϕ (D
S.
4tgϕ −4tg
3
ϕ
1 −6tg
2
ϕ +tg
4
ϕ
)
2) tg6ϕ (D
S.
6tgϕ − 20tg
3
ϕ + 6tg
5
ϕ

1 −15tg
2
ϕ + 15tg
4
ϕ −tg
6
ϕ
)
7. Ch´u
.
ng minh r˘a
`
ng
1 −C
2
n
+ C
4
n
− C
6
n
+ =2
n
2
cos
nπ
4
·
C

1
n
− C
3
n
+ C
5
n
− C
7
n
+ =2
n
2
sin
nπ
4
·
Chı
’
dˆa
˜
n. T´ınh (1 + i)
n
b˘a
`
ng c´ach su
.
’
du

.
ng cˆong th´u
.
c Moivre v`a
su
.
’
du
.
ng cˆong th´u
.
c nhi
.
th ´u
.
c Newton rˆo
`
i so s´anh phˆa
`
n thu
.
.
c v`a phˆa
`
n
a
’
o c´ac sˆo
´
thu d

u
.
o
.
.
c.
8. Ch´u
.
ng minh r˘a
`
ng
1) cos
π
5
+ cos
3π
5
=
1
2
2) cos
π
7
+ cos
3π
7
+ cos
5π
7
=

1
2
3) cos
2π
5
+ cos
4π
5
= −
1
2
4) cos
2π
7
+ cos
4π
7
+ cos
6π
7
= −
1
2
5) cos
2π
9
+ cos
4π
9
+ cos

6π
9
+ cos
8π
9
= −
1
2
42 Chu
.
o
.
ng 1. Sˆo
´
ph´u
.
c
9. Gia
’
iphu
.
o
.
ng tr`ınh

i −x
i + x

n
=

cotgα + i
cotgα − i
,n∈ N,α∈ R.
(D
S. x =tg
α + kπ
n
, k = 0,n−1)
10. Ch´u
.
ng minh r˘a
`
ng nˆe
´
u A l`a sˆo
´
ph´u
.
cc´omod
un = 1 th`ı mo
.
i nghiˆe
.
m
cu
’
aphu
.
o
.

ng tr`ınh

1+ix
1 −ix

n
= A
d
ˆe
`
u l`a nghiˆe
.
m thu
.
.
c v`a kh´ac nhau.
11. Gia
’
iphu
.
o
.
ng tr`ınh
x
n
−nax
n−1
− C
2
n

a
2
x
n−2
−···−a
n
=0.
(DS. x
k
=
a
ε
k
√
2 −1
, k = 0,n− 1)
Chı
’
dˆa
˜
n. D`ung cˆong th´u
.
c nhi
.
th ´u
.
c Newton dˆe
’
du
.

aphu
.
o
.
ng tr`ınh
vˆe
`
da
.
ng x
n
=(x + a)
n
− x
n
.
12. Gia
’
iphu
.
o
.
ng tr`ınh
x
5
+ x
4
+ x
3
+ x

2
+ x +1=0.
(D
S. x
k
= cos
kπ
3
+ i sin
kπ
3
, k =1, 2, 3, 4, 5)
13. Gia
’
iphu
.
o
.
ng tr`ınh
x
5
+ αx
4
+ α
2
x
3
+ α
3
x

2
+ α
4
x + α
5
=0,α∈ C,α=0.
(D
S. x
k
= α

cos
kπ
3
+ i sin
kπ
3

, k =1, 2, 3, 4, 5)
Chı
’
dˆa
˜
n. Vˆe
´
tr´ai l`a tˆo
’
ng cˆa
´
psˆo

´
nhˆan v´o
.
i cˆong bˆo
.
ib˘a
`
ng
α
x
.
14. Gia
’
su
.
’
n ∈ N, n>1, c =0,c ∈ R. Gia
’
i c´ac phu
.
o
.
ng tr`ınh sau
d
ˆa y
1.4. Biˆe
’
udiˆe
˜
nsˆo

´
ph´u
.
cdu
.
´o
.
ida
.
ng lu
.
o
.
.
ng gi´ac 43
1) (x + c)
n
−(x − c)
n
=0 (DS. x = −ccotg
kπ
n
, k =
1,n−1)
2) (x + ci)
n
−(x − ci)
n
=0 (DS. x = −cicotg
kπ

n
, k = 1,n− 1)
3) (x + ci)
n
+ i(x − ci)
n
=0
(DS. x = −cicotg
(3+4k)π
4n
, k = 0,n−1)
4) (x + ci)
n
− (cos α + i sin α)(x −ci)
n
=0,α=2kπ.
(D
S. x = −cicotg
α +2kπ
2n
,k= 0,n− 1)
15. T´ınh
D
n
(x)=
1
2π

1
2

+ cos x + cos 2x + ···+ cos nx

.
(DS. D
n
(x)=
1
2π
sin
2n +1
2
x
2 sin
x
2
)
16. 1) Biˆe
’
udiˆe
˜
n cos 5x v`a sin5x qua cos x v`a sinx.
2) T´ınh cos
2π
5
v`a sin
2π
5
.
(D
S. 1) cos 5x = cos

5
x −10 cos
3
x sin
2
x + 5 cos x sin
4
x,
sin 5x = 5 cos
4
x sin x −10 cos
2
x sin
3
x + sin
5
x.
2) sin
2π
5
=

10 + 2
√
5
4
, cos
2π
5
=

√
5 −1
4
)
Chı
’
dˆa
˜
n. Dˆe
’
t´ınh sin
2π
5
cˆa
`
nsu
.
’
du
.
ng biˆe
’
uth´u
.
ccu
’
a sin 5x.
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 du
.
o
.
.
cbiˆe
’
udiˆe
˜
ndo
.
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 dath´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`adath´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 dath´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
’

adath´u
.
c Q(z) hay cu
’
aphu
.
o
.
ng tr`ınh da
.
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
’
adath´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
.
mdo
.
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 dath´u
.
cdˆe
`
uc´onghiˆe
.
mph´u
.
c thu
.
.
csu
.
.
.Dod´o khˆong cˆa
`
n pha
’
i
s´ang ta

.
o thˆem c´ac sˆo
´
m´o
.
idˆe
’
gia
’
iphu
.
o
.
ng tr`ınh (v`ı thˆe
´
C c`on du
.
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
’
ada
.
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
.
mdu
.
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`adath´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´oz
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 di
.
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 dath´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
’
adath´u
.
c (2.4) v´o
.
i
hˆe
.
sˆo

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

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 da
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
´
udath´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
.
idath´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

du
.
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
’
adath´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
.
idath´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 du
.
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
’
adath´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 d u
.
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 dˆe
`
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
˜
ndath´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 du
.
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 d u
.
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`adath´u
.
cbˆa
.
c n −1v´o
.
ihˆe
.
sˆo
´
cao nhˆa
´
tb˘a
`
ng 2n.Dˆo
´
i
v´o
.
idath´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 dath´u

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

.
ntrˆenc`ung tru
.
`o
.
ng P d´o. Nh˜u
.
ng dath´u
.
cn`aydu
.
o
.
.
cgo
.
il`ada
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)