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4
◦
.
x
4
− 4x
3
+ 4x
2
+ 4x − 5.
5
◦
. |z| < 3, w = (2 + 3i)z + i −6.
1) z
1
+z
2
= 1+7i, z
1
−z
2
= −3−i, z
1
z
2
= −14+2i, z
1
/z
2
= (1+i)/2
z
1
+ z
2
= 2 + 3i, z
1
− z
2
= 4 − 7i, z
1
z
2
= 7 + 17i, z
1
/z
2
= −(1 + i)/2,
z
1
+ z
2
= 1 + 3i, z
1
− z
2
= −1 + i, z
1
z
2
= −2 + 2i, z
1
/z
2
= 1 + i,
z
1
+ z
2
= 2 + 4i, z
1
− z
2
= 2 − 6i, z
1
z
2
= 5 + 10i, z
1
/z
2
= −(1 + 2i)/5.
1) 5 − 12i, 2) 2i − 2, 3) − i, 4) 1.
1) x
1,2
= ±5i, 2) x
1,2
= ±7i, 3) x
1,2
= 1 ± 2i, 4) x
1,2
= −2 ± 3i,
5) x
1,2
= 2 ± i
√
3, 6) x
1,2
= 1 ± i.
w
1
= −2i, w
2
= 12e
π i
4
, w
3
= 10e
5πi
12
, w
4
= 3e
−
2π i
3
.
z
1
= e
2
(cos 3 + i sin 3), z
2
= e
−1
(cos 3 + i sin 3), z
3
= cos 4 + i sin 4,
z
4
= cos 5 − i sin 5.
z
1
=
√
2e
−
πi
4
, z
2
=
√
2e
πi
4
, z
3
= 2e
5π i
6
, z
4
= 2e
−
πi
6
, z
5
= 5e
−
πi
2
,
z
6
= 6e
πi
2
, z
7
= 64e
πi
, z
8
= 4.
1) − 4, 2) − 8i, 3) 256e
2π i
3
, 4) 32e
−
5π i
6
.
1) z
1,2
= ±
√
6 e
πi/4
, 2) z
k
= 2 e
(−π/12+kπ/3)i
, k = 0, 5,
3) z
k
=
6
√
2 e
(−π/12+2kπ/3)i
, k = 0, 2, 4) z
k
=
4
√
2 e
(−π/24+kπ/2)i
, k = 0, 3,
5) z
1,2
= ±e
−πi/4
, 6) z
k
= e
2kπi/3
, k = 0, 2, 7) z
k
= 2 e
(π/15+2kπ/5)i
, k = 0, 4,
8) z
k
= 2 e
(π/3+2kπ/3)i
, k = 0, 2.
1) z
k
= e
2kπi/8
, k = 0, 7, 2) z
k
= e
(π/3+2kπ/3)i
, k = 0, 2,
3) z
k
=
10
√
2 e
(3π/20+2kπ/5)i
, k = 0, 4, 4) z
k
= e
(−π/6+2kπ/3)i
, k = 0, 2.
1)
|
w
|
<
3
,
2)
|
w
|
<
4
,
3)
|
w
+ 1
|
<
4
,
4)
|
w
−
2
|
<
1
,
5) |w − 1 + 2i| < 2, 6) |w − 3 + i| < 2.
|w − b| < |a|r.
1) z
k
= ln 5 + i[(2k + 1)π − arctg
4
3
], 2) z
k
=
1
2
ln 2 + iπ(
1
4
+ 2k),
4◦ . Íàéòè êîðíè ìíîãî÷ëåíà è ðàçëîæèòü åãî íà ìíîæèòåëè ñ âåùå-
ñòâåííûìè êîýôôèöèåíòàìè: x4 − 4x3 + 4x2 + 4x − 5.
5◦ . Íàéòè îáðàç êðóãà |z| < 3, åñëè w = (2 + 3i)z + i − 6.
ÎÒÂÅÒÛ
2 1) z1 +z2 = 1+7i, z1 −z2 = −3−i, z1 z2 = −14+2i, z1 /z2 = (1+i)/2
2) z1 + z2 = 2 + 3i, z1 − z2 = 4 − 7i, z1 z2 = 7 + 17i, z1 /z2 = −(1 + i)/2,
3) z1 + z2 = 1 + 3i, z1 − z2 = −1 + i, z1 z2 = −2 + 2i, z1 /z2 = 1 + i,
4) z1 + z2 = 2 + 4i, z1 − z2 = 2 − 6i, z1 z2 = 5 + 10i, z1 /z2 = −(1 + 2i)/5.
3 1) 5 − 12i, 2) 2i − 2, 3) − i, 4) 1.
4 1) x1,2 = ±5i, 2) x1,2 = ±7i, 3) x1,2 = 1 ± 2i, 4) x1,2 = −2 ± 3i,
√
5) x1,2 = 2 ± i 3, 6) x1,2 = 1 ± i.
πi 5πi 2πi
6 w1 = −2i, w2 = 12e 4 , w3 = 10e 12 , w4 = 3e− 3 .
7 z1 = e2 (cos 3 + i sin 3), z2 = e−1 (cos 3 + i sin 3), z3 = cos 4 + i sin 4,
z4 = cos 5 − i sin 5.
√ πi √ πi 5πi πi πi
8 z1 = 2e− 4 , z2 = 2e 4 , z3 = 2e 6 , z4 = 2e− 6 , z5 = 5e− 2 ,
πi
z6 = 6e 2 , z7 = 64eπi , z8 = 4.
10 1) Îêðóæíîñòü, 2) ýëëèïñ, 3) îêðóæíîñòü, 4) îêðóæíîñòü,
5) äóãà îêðóæíîñòè, 6) îòðåçîê ïðÿìîé, 7) ïàðàáîëà, 8) äóãà îêðóæíîñòè.
2πi 5πi
12 1) − 4, 2) − 8i, 3) 256e 3 , 4) 32e− 6 .
√
13 1) z1,2 = ± 6 eπi/4 , 2) zk = 2 e(−π/12+kπ/3)i , k = 0, 5,
√ √
3) zk = 6
2 e(−π/12+2kπ/3)i , k = 0, 2, 4) zk = 4
2 e(−π/24+kπ/2)i , k = 0, 3,
5) z1,2 = ± e−πi/4 , 6) zk = e2kπi/3 , k = 0, 2, 7) zk = 2 e(π/15+2kπ/5)i , k = 0, 4,
8) zk = 2 e(π/3+2kπ/3)i , k = 0, 2.
14 1) zk = e2kπi/8 , k = 0, 7, 2) zk = e(π/3+2kπ/3)i , k = 0, 2,
√
3) zk = 10
2 e(3π/20+2kπ/5)i , k = 0, 4, 4) zk = e(−π/6+2kπ/3)i , k = 0, 2.
16 1) |w| < 3, 2) |w| < 4, 3) |w + 1| < 4, 4) |w − 2| < 1,
5) |w − 1 + 2i| < 2, 6) |w − 3 + i| < 2.
17 |w − b| < |a|r.
18 1) zk = ln 5 + i[(2k + 1)π − arctg 43 ], 2) zk = 12 ln 2 + iπ( 14 + 2k),
14
