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Рязанский Государственный Университет им. С.А. Есенина

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Задание 57. Образуют ли числа полную систему вычетов по какому-либо мо- дулю? Если нет, то изменить несколько элементов так, чтобы получившаяся система образовывала полную систему вычетов. Составить из получившейся системы приведенную систему вычетов.

1.–84, –38, –725, 388, 73, –166, 155, –105, 426, –195, –127, 116.

2.311, 503, 251, –744, –266, –2558, 452, –291.

3.–19, 18, 310, –300, 875, 273, –2162, 298, –814.

4.30066, –90, –160, –680, –143, 205, –260, 254, 373, 402, –366.

5.93, 240, –452, 685, –307, 495, –344.

6.70, –501, –44, –77, –292, –816, –119, –215, 523, 327.

7.–1032, 125, –455, –17, 2243, 10, 228, 19.

8.407, 510, –35, –166, 105, –356, 91, –249, –436, 565, 161, –254.

9.–142, 373, –260, –366, 254, 492, –90, –680, –160, 300, 205. 10. –1806, –2, –119, –211, –515, –78, –94, 503, 3900, 777.

11. 528, 119, 476, 648, 683, 62, 92, –201, –624, –603, –71. 12. 455, 17, –2246, –10, 1032, –228, –19,–125.

13. –73, 385, 155, 426, –166, –127, –38, 116, –105, –195, –725, –84. 14. –298, –300, –2162, 20, 273, –19, 875, –814, 310.

15. 91, 240, 495, –307, 685, –347, –452.

16. –74, –181, 258, 457, –177, 322, –165.

17. –160, –1715, –6201, –906, –519, –302, –65204, 622, 177, 924. 18. 754, 317, –99, –240, 267, –206, –298, –329, 554.

19. 407, –356, –436, –36, –249, 91, –254, 565, 161, –166, 105, 510. 20. –239, –88, –495, 307, 347, –685, 452.

21. –209, 555, 985, 333, 254, 422, 328, 601, 373, –669, 107. 22. –70, 63, 477, 528, –200, –603, 649, 683, –623, 129, 93. 23. 452, –311, 503, 291, –744, 266, 2558, –251.

24. 554, –298, –206, –240, 754, –102, 317, –3296, 267.

25. –160, –206, 177, –519, –6201, 622, –302, –65204, 923, –1715. 26. 2556, –311, 503, 291, –251, –744, 452, 266.

27. 452, 347, 304, –91, –240, –495, –685.

28. –669, 373, 985, 331, 107, –209, 553, 328, 254, 422, 601. 29. –452, –266, 744, –2558, –291, 311, –503, 251.

30. –555, 392, 713, –801, 900, –705, 1721, –414, –713, 278.

Задание 58. Решить сравнение первой степени.

1.	15x ≡ 21(mod 18).	3.	18x ≡ 12(mod 30).
2.	12x ≡ 16(mod 28).	4.	7x ≡ 15(mod 25).

5.	75x ≡ 54(mod 21).	18.	64x ≡ 51(mod 13).
6.	37x ≡ 16(mod 11).	19.	15x ≡ 21(mod 6).
7.	39x ≡ 5(mod 11).	20.	57x ≡ 15(mod 48).
8.	20x ≡ 35(mod 45).	21.	64x ≡ 5(mod 13).
9.	183x ≡ 93(mod 111).	22.	139x ≡ 7(mod 8).
10.	19x ≡ 4(mod 25).	23.	14x ≡ 9(mod 37).
11.	11x ≡ 15(mod 24).	24.	32x ≡ 13(mod 15).
12.	39x ≡ 19(mod 53).	25.	42x ≡ 105(mod 245).
13.	45x ≡ 21(mod 132).	26.	29x ≡ 35(mod 123).
14.	12x ≡ 15(mod 35).	27.	21x ≡ 15(mod 111).
15.	21x ≡ 10(mod 25).	28.	15x ≡ 120(mod 85).
16.	15x ≡ 7(mod 16).	29.	8x ≡ 15(mod 29).
17.	8x ≡ 17(mod 23).	30.	8x ≡ 17(mod 31).

Задание 59. Решить систему сравнений первой степени.

1.x ≡ 10(mod 21),

x≡ 6(mod 15),

x≡ 19(mod 24).

2.x ≡ 7(mod 15),

x≡ 23(mod 35),

x≡ 13(mod 20).

3.3x ≡ 5(mod 7),

2x ≡ 3(mod 5),

3x ≡ 3(mod 9).

4.x ≡ 2(mod 12),

x≡ 12(mod 15),

x≡ 3(mod 33).

5.x ≡ 32(mod 40),

x≡ 23(mod 72),

x≡ 13(mod 24).

6.x ≡ 6(mod 15),

x≡ 18(mod 21),

x≡ 3(mod 12).

7.x ≡ 13(mod 14),

x≡ 6(mod 35),

x≡ 26(mod 45).

8.x ≡ 19(mod 56),

x≡ 3(mod 24),

x≡ 7(mod 20).

9.x ≡ 19(mod 22),

x≡ 8(mod 33),

x ≡ 14(mod 21).

10.4x ≡ 9(mod 7),

2x ≡ 15(mod 9),

5x ≡ 12(mod 13).

11.3x ≡ 2(mod 13),

5x ≡ 11(mod 16),

5x ≡ 2(mod 9).

12.3x ≡ 5(mod 13),

2x ≡ 17(mod 21),

5x ≡ 31(mod 32).

13.x ≡ 1(mod 4), x ≡ 3(mod 5),

x ≡ 2(mod 7).

14.x ≡ 2(mod 3), x ≡ 2(mod 7),

x ≡ –2(mod 11).

15.3x ≡ 5(mod 14),

5x ≡ 1(mod 9),

7x ≡ 2(mod 25).

16.7x ≡ 4(mod 15),

3x ≡ 23(mod 28),

5x ≡ 8(mod 11).

17.2x ≡ 5(mod 21),

5x ≡ 22(mod 31),

4x ≡ 5(mod 29).

18.3x ≡ 5(mod 12),

7x ≡ 3(mod 25),

3x ≡ 2(mod 17).

19.x ≡ 8(mod 15), x ≡ 9(mod 13),

x ≡ 5(mod 14).

20.x ≡ 10(mod 11),

x≡ 9(mod 16),

x≡ 5(mod 7).

21.x ≡ 5(mod 3),

x≡ 7(mod 10),

x≡2 (mod 7).

22.x ≡ 14(mod 19),

x≡ 5(mod 7),

x ≡ 9(mod 10).

23.x ≡ 8(mod 13),

x≡ 9(mod 17),

x≡ 5(mod 11).

24.x ≡ 5(mod 9),

x≡ 3(mod 5),

x≡ 1(mod 7).

25.x ≡ 12(mod 13),

x≡ 10(mod 11),

x≡ 5(mod 12).

26.5x ≡ 2(mod 12),

7x ≡ 2(mod 8),

3x ≡ 1(mod 5).

27.3x ≡ 8(mod 20),

5x ≡ 8(mod 9),

4x ≡ 1(mod 21),

28.2x ≡ 9(mod 15).

5x ≡ 4(mod 7),

7x ≡ 3(mod 9).

29.8x ≡ 1(mod 13),

5x ≡ 7(mod 18),

2x ≡ 1(mod 9).

30.3x ≡ 1(mod 25),

6x ≡ 3(mod 33),

4x ≡ 5(mod 9).

Задание 60. Решить в целых числах уравнение.
1.	77x–16y = 7.	16.	1414x+406y = 42.
2.	24x–37y = 35.	17.	143x+169y = 5.
3.	27x+31y = 29.	18.	237x+44y = 1.
4.	37x–29y = 53.	19.	275x+145y = 10.
5.	31x+55y = 77.	20.	23x+49y = 53.
6.	45x–37y = 25.	21.	43x+37y = 21.
7.	38x+114y = 209.	22.	258x–172y = 56.
8.	53x+47y = 11.	23.	122x+129y = 2.
9.	119x–68y = 34.	24.	45x–37y = 25.
10.	91x+117y = 156.	25.	81x–48y = 33.
11.	673x+103y = 2.	26.	53x+47y = 100.
12.	258x–172y = 112.	27.	42x+31y = 67.
13.	531x+118y = 112.	28.	43x+35y = 21.
14.	253x+1001y = 22.	29.	17x–25y = 117.
15.	311x–273y = 446.	30.	70x+33y = 2.

Задание 61. Представить число в виде цепной дроби. Найти δ3 и погрешность приближения числа с помощью δ3.


1.		105					.						11.		30			.								22.		22				.
	38													37													81

2.		245								.			12.		127							.				23.		269							.
														52													106
	83
3.		235								.			13.		24				.							24.	–			31			.
																													99
	69													35
4.	46					.							14.	1,23.												25.	43				.
		19											15.	–		71				.								53
	375																												177
5.								.						41												26.	–									.
		824											16.		157								.							67
	990													225															241

6.										.					999										.	27.	–										.
		577											17.																	195
7.	875								.					2195												28.			352										.

		576											18.	0,459.														1513
8.	–		55									.	19.	–			251							.		29.	–			182								.
			117																											225
																	764
9.	71				.								20.	0,907.												30.	–		64					.
		41											21.	163							.									53
10. –				187							.
															59
	64

Задание 62. Найти рациональное число, которое обращается в данную цепную дробь.

1.	[0; 1, 2, 3, 4, 5].	16.	[2; 1, 3, 4, 2].
2.	[0; 2, 5, 14, 14].	17.	[2; 1, 19, 1, 3].
3.	[0; 2, 5, 1, 1, 2, 16].	18.	[3; 2, 2, 6, 2].
4.	[–1; 1, 2, 22, 1, 4, 2].	19.	[2; 2, 2, 1, 2].
5.	[0; 1, 9, 1, 3, 23].	20.	[0; 2, 5, 14, 1, 4].
6.	[2; 1, 3, 4, 1, 2].	21.	[1; 1, 2, 1, 1, 13, 6].
7.	[0; 3, 1, 2, 7].	22.	[1; 1, 1, 12, 1, 1, 2, 4].
8.	[2; 1, 1, 6, 8].	23.	[–1; 1, 1, 7, 1, 6].
9.	[–1; 1, 2, 5, 6].	24.	[1; 1, 2, 1, 2, 1, 2].
10.	[0; 1, 4, 3, 2, 1].	25.	[–2; 1, 30, 2].
11.	[–3; 2, 1, 3, 1, 4].	26.	[0; 1, 4, 3, 2].
12.	[–2; 1, 3, 4, 5, 2].	27.	[2; 2, 3, 1, 5].
13.	[0; 4, 3, 2, 1, 5, 6].	28.	[0; 1, 2, 5, 2].
14.	[–1; 5, 4, 3, 2, 1].	29.	[1; 4, 2, 1, 7].
15.	[–2; 1, 3, 1, 4, 2].	30.	[–2; 3, 1, 2, 1, 2].

Задание 63. Представить число в виде периодической цепной дроби.

1.	22.	4.	29.	7.	19.
2.	21.	5.	20.	8.	3.
3.	18.	6.	17.	9.	7.

10.												3			.										− 3		.
10.			8.					21.	1+			3								26.		77			− 3
11.								21.	1+	4										26.
11.	11.									4											1 +		2
12.											15														35
12.		12.						22.	3+		15					.				27.					35		.
13.										3													2
13.		13.								3													2
14.									7 +					77							2 +					7
14.		14.						23.	7 +					77				.		28.	2 +					7
15.								23.	14									.		28.	4
15.	15 .								14												4
16.	1–						.		1+				21							29.	9 +			21
						3		24.										.			9 +			21
						3															6
17.	2–2									2
				2.
																					9 +
																							101
									22 +											30.								.
																	15
18.	1+			6 .				25.											.

										7													5
19.			23		.					7
20.
20.			31.

Задание 64. Найти действительное число, которое имеет данное представление в виде периодической цепной дроби.

1.	[3; 3, 1, (2)].	11.	[3; (2, 6)].	21.	[3; (2, 5)].
2.	[2; 1, 5, 2, 1,(2, 1)].	12.	[3; (1, 1, 1, 1, 6)].	22.	[(3; 2, 5)].
3.	[4; (1, 3, 1, 8)].	13.	[3; (1, 6)].	23.	[(3; 2, 1, 4)].
4.	[2; (1, 1, 1, 4)].	14.	[4; 3, (2, 4)].	24.	[(3; 1, 4)].
5.	[2; (1, 4)].	15.	[5; (2, 1, 1, 2, 10)].	25.	[(1; 6, 5, 6)].
6.	[3; (3, 6)].	16.	[1; 2, (3, 4)].	26.	[4; (1, 3, 1, 8)].
7.	[3; (1, 2, 1, 6)].	17.	[4; (3, 2)].	27.	[2; (3, 1)].
8.	[–1; 3, (1, 2)].	18.	[(5; 4, 3)].	28.	[4; (2, 8)].
9.	[4; (4, 8)].	19.	[(2; 6, 1)].	29.	[4; (1, 1, 2, 1, 1, 8)].
10.	[–1; 5, (1, 4)].	20.	[3; 1, (2, 3)].	30.	[2; (1, 7)].

Задание 65. Решить сравнение первой степени с помощью цепных дробей.

1.	32x ≡ 63(mod 119).	16.	–73x ≡ 60(mod 311).
2.	69x ≡ 192(mod 201).	17.	–53x ≡ 84(mod 219).
3.	37x ≡ 5(mod 217).	18.	143x ≡ 75(mod 119).
4.	271x ≡ 25(mod 119).	19.	13x ≡ 178(mod 153).
5.	91x ≡ 143(mod 222).	20.	243x ≡ 271(mod 317).
6.	143x ≡ 41(mod 221).	21.	221x ≡ 111(mod 360).
7.	82x ≡ 14(mod 202).	22.	141x ≡ 73(mod 320).
8.	354x ≡ 567(mod 639).	23.	139x ≡ 118(mod 239).
9.	67x ≡ 64(mod 183).	24.	327x ≡ 78(mod 379).
10.	89x ≡ 86(mod 241).	25.	239x ≡ 302(mod 471).
11.	213x ≡ 137(mod 516).	26.	23x ≡ 667(mod 693).
12.	111x ≡ 81(mod 447).	27.	57x ≡ 15(mod 482).
13.	186x ≡ 374(mod 422).	28.	21x ≡ 15(mod 111).
14.	129x ≡ 321(mod 471).	29.	15x ≡ 120(mod 851).
15.	–50x ≡ 67(mod 177).	30.	2560x ≡ 45(mod 3605).

Задание 66. Решить сравнение.
1.	6x10–12x+1 ≡ 0(mod 5).	16.	x12–2x7+x3+1 ≡ 0(mod 5).
2.	x5–2x3+x2–2 ≡ 0(mod 3).	17.	13x23–8x22–2x13+1 ≡ 0(mod 11).
3.	x5–7x4+9x2–x+13 ≡ 0(mod 3).	18.	10x91+x11+9x2–x+6≡ 0(mod 11).
4.	x7–x6+5x2–3 ≡ 0(mod 5).	19.	10x42–5x30+2x2+4 ≡ 0(mod 17).
5.	x5+x4+x3–x2–2 ≡ 0(mod 5).	20.	28x9+x8–28x7+x4+23 ≡0(mod 3).
6.	x7–6 ≡ 0(mod 5).	21.	34x10+x7+3x4+x+37 ≡ 0(mod 5).
7.	x8+2x7+x5–x4–x+3≡ 0(mod 5).	22.	x12–53x11–24x6–x+1 ≡ 0(mod 7).
8.	6x4+17x2–16≡ 0(mod 3).	23.	x10–2x+1 ≡ 0(mod 5).
9.	4x7–2x3+8 ≡ 0(mod 5).	24.	x8+2x7–x4+3 ≡ 0(mod 5).
10.	x7–2x6+2x2+13 ≡ 0(mod 5).	25.	x16+3x8–x4+x–2 ≡ 0(mod 7).
11.	x12+2x11–2x–1 ≡ 0(mod 11).	26.	x10+x8–x4–x2+5x–3 ≡ 0(mod 7).
12.	x10+x8+x7–x4–x2+4x–3≡ 0(mod 7).	27.	x101+6x5+9x2–5 ≡ 0(mod 11).
13. x29+3x15+x11–3x5+9x2+10x–5≡0(mod11).		28.	x14–x13–x2+2x+1 ≡ 0(mod 13).
14.	x12–2x7+x3+1 ≡ 0(mod 5).	29.	323x90+164x67–1 ≡ 0(mod 23).
15.	x14–4x13–x+6 ≡ 0(mod 13).	30.	2x35–9x15+13x8+x+5≡0(mod11).

Задание 67. Найти порядок элемента а по модулю m.

1.a = 6, m = 23.

2.a = 3, m = 25.

3.a = 7, m = 29.

4.a = 27, m = 47.

5.a = 2, m = 17.

6.a = 5, m = 17.

7.a = 10, m = 13.

8.a = 10, m = 31.

9.a = 13, m = 53. 10. a = 18, m = 29. 11. a = 5, m = 61. 12. a = 10, m = 47. 13. a = 32, m = 61. 14. a = 32, m = 71. 15. a = 2, m = 35.

16.a = 3, m = 55.

17.a = 10, m = 17.

18.a = 97, m = 17.

19.a = 18, m = 11.

20.a = 23, m = 41.

21.a = 6, m = 59.

22.a = 14, m = 59.

23.a = 3, m = 17.

24.a = 3, m = 19.

25.a = 5, m = 18.

26.a = 5, m = 23.

27.a = 18, m = 41.

28.a = 8, m = 19.

29.a = 15, m = 23.

30.a = 18, m = 23.

Задание 68. Решить сравнение с помощью индексов.


1.	2x ≡ 7(mod 67).	6.	20x ≡		21(mod 41).
2.	13x ≡ 12(mod 47).	7.	37x15 ≡ 62(mod 73).
3.	x27 ≡ 39(mod 43).	8.	5x4	≡ 3(mod 11).
4.	8x26 ≡ 37(mod 41).	9.	2x8	≡ 5(mod 13).
5.	25x7 ≡ –7(mod 31).	10.	2x3	≡ 17(mod 41).

11.	27x5 ≡ 25(mod 31).		21.	x8 ≡ 23(mod 41).
12.	11x3	≡ 6(mod 79).	22.	x5 ≡ 74(mod 71).
13.	23x3	≡ 15(mod 73).	23.	16x ≡ 11(mod 53).
14.	52x ≡ 38(mod 61).		24. x8 ≡ 29(mod 13).
15.	37x8	≡ 59(mod 61).	25.	12x ≡ 17(mod 31).
16.	18x8	≡ 6(mod 13).	26.	13x8 ≡ –36(mod 61).
17.	x12 ≡ 37(mod 41).		27.	7x13 ≡ –23(mod 47).
18.	x55 ≡ 17(mod 97).		28.	19x5 ≡ –13(mod 53).
19.	x35 ≡ 17(mod 67).		29.	23x ≡ 37(mod 41).
20.	x30 ≡ 46(mod 73).		30.	5x11 ≡ –19(mod 29).

Задание 69. Представить многочлен f(х) по степени двучлена х–α.

1.f (x) = x3–8x2+23x–24, α = 2.

2.f (x) = x4–7x3+3x2+63x–108, α = 2.

3.f (x) = x5–4x3+6x2–8x+10, α = 2.

4.f (x) = x4–2x2+3, α = –1.

5.f (x) = 2x3+13x2+25x+14, α = –2.

6.f (x)=3x4+24x3+70x2+87x+38, α =–2.

7.f (x) = x4–8x3+24x2–50x+90, α = 2.

8.f (x) = x5, α = 1.

9.f (x) = x3–9x2+27x–27, α = 3. 10. f (x) = x3–5x2+3x+9, α = 3. 11. f (x) = 2x3–18x2+108, α = 3. 12. f (x) = 3x3–81x+162, α = 3.

13. f (x) = x4–12x3+54x2–108x+81, α = 3. 14. f (x) = x4–6x3+18x2–54x+81, α = 3. 15. f (x) = x4+3x2+1, α = –1.

16. f (x) = 2x5–x3–2x2–6x+10, α = 2. 17. f (x) = 4x3–2x2+5x–1, α = 2. 18. f (x) = x4–4x3+5x–2, α = 3.

19. f (x) = x4+7x3+4x2–25x+1, α = –5. 20. f (x) = x4–8x3+24x2–50x+22, α = 2. 21. f (x) = x4+3x3–4x2+6x–5, α = –2.

22. f (x) = 2x5–3x3+6x2–8x–4, α = 3. 23. f (x) = 4x3–2x2+5x+11, α = 2. 24. f (x) = 3x4+8x3–2x2+6x–5, α = –3. 25. f (x) = x4+2x3–3x2–4x+1, α = –1.

26. f (x) = x5–5x4+7x3–2x2+4x–8, α = 2.

27. f (x) = x5+9x4+32x3+57x2+51x+18, α = –2.

28. f (x) = x6–12x5+60x4–161x3+246x2–204x+72, α = 2.

29.f (x) = x5–11x4+49x3–111x2+129x–63, α = 2.

30.f (x) = x6+12x5+59x4+152x3+215x2+156x+45, α = –2.

Задание 70. Выделить кратные множители многочлена.

1.	x5–8x4+25x3–38x2+28x–8.	16.	4x5+15x4+20x3+10x2–1.
2.	2x5+3x4+4x3+x2–1.	17.	8x5–28x4+38x3–25x2+8x–1.
3.	2x6+6x5+6x4+x3–3x2–3x–1.	18.	x6+3x5+3x4–x3–6x2–6x–2.
4.	x6+6x5+3x4+6x3+9x2+12x+3.	19.	3x6+12x5+9x4+6x3+3x2+6x+1.
5.	x6–6x4–4x3+9x2+12x+4.	20.	4x6+12x5+9x4–4x3–6x2+1.
6.	x5–10x3–20x2–15x–4.	21.	4x5+15x4+20x3+10x2–1.
7.	x6–15x4+8x3+51x2–72x+27.	22.	8x5–20x4+24x3–16x2+6x–1.
8.	x5–6x4+16x3–24x2+20x–8.	23.	4x6+4x5+5x4–2x3–x2–2x+1.
9.	x6–2x5–x4–2x3+5x2+4x+4.	24.	2x5–5x4+x3–6x2+12x+8.
10.	x7–3x6+5x5–7x4+7x3–5x2+3x–1.	25.	8x5+12x4–6x3+x2–5x+2.
11.	8x5–20x4–10x3+25x2+20x+4.	26.	16x5–16x4–72x3–64x2–23x–3.
12.	x5–15x3–10x2+60x+72.	27.	4x5+20x4+25x3–10x2–20x+8.
13.	x6+2x5+x4–x2–2x–1.	28.	3x5+23x4+64x3+72x2+16x–16.
14.	x9+x8–2x5–2x4+x+1.	29.	2x5+11x4+18x3+8x2–4x–3.
15.	x5–x3–4x2–3x–2.	30.	3x5+4x4–8x3–18x2–11x–2.

Задание 71. Составить многочлен степени ≤ 3, график которого проходит через заданные точки.

1.	(0; 3), (1; 5), (–1; –1), (2; 17).	16.	(0; –1), (1; 2), (–1; –2), (–2; –7).
2.	(0; 5), (1; 6), (–1; 4), (–2; –3).	17.	(0; 1), (1; 4), (–1; 0), (–2; –5).
3.	(0; 3), (1; –3), (2; –7), (–1; 11).	18.	(0; 2), (1; 6), (–1; 4), (–2; 6).
4.	(0; 2), (1; 5), (–1; –3), (–2; –16).	19.	(0; 4), (1; 6), (–1; 10), (2; 4).
5.	(0; 2), (1; 7), (–1; –1), (–2; –8).	20.	(0; –5), (1; 2), (–1; 0), (–2; 17).
6.	(0; –8), (1; –5), (–1; –11), (2; 10).	21.	(0; 2), (1; 0), (–1; 4), (2; 4).
7.	(0; 1), (1; –3), (–1; 9), (2; –3).	22.	(0; 1), (1; –1), (–1; –5), (2; 1).
8.	(0; 3), (1; 8), (–1; 10), (2; 19).	23.	(0; 2), (1; 2), (–1; –2), (2; 10).
9.	(0; –4), (1; –4), (–1; –8), (2; –2).	24.	(0; 4), (1; 2), (–1; 10), (2; –2).
10.	(0; 3), (1; 4), (–1; 2), (–3; –24).	25.	(0; 0), (1; 5), (–1; –5), (2; 22).
11.	(0; 1), (1; –1), (–1; 3), (2; 9).	26.	(0; –2), (1; –3), (–1; 9), (2; 6).
12.	(0; 2), (1; 5), (–1; –1), (–2; –22).	27.	(0; 1), (1; 0), (–1; 2), (2; 5).
13.	(0; –1), (1; 3), (–1; –5), (2; 7).	28.	(0; 12), (1; 15), (–1; 9), (–2; –12).
14.	(0; 1), (1; 0), (–1; 12), (2; 9).	29.	(0; –4), (1; –4), (–1; 0), (–2; 2).
15.	(0; 3), (1; 7), (–1; –1), (2; –1).	30.	(0; 1), (1; 2), (–1; 0), (–2; –7).

Задание 72. Найти рациональные корни многочлена. Разложить многочлен на неприводимые над Q множители.

1.	4x5–4x4+5x3–14x2–21x–6.	16.	6x5+21x4+14x3–5x2+4x–4.
2.	2x5–x4–10x3–10x2–12x–9.	17.	9x5+12x4+10x3+10x2+x–2.
3.	4x5–14x4+14x3–13x2+12x+12.	18.	12x5+12x4–13x3+14x2–14x+4.
4.	6x5–3x4+26x3–13x2+8x–4.	19.	4x5–8x4+13x3–26x2+3x–6.
5.	8x5+32x4–50x3–38x2+57x–15.	20.	15x5–57x4+38x3+50x2–32x–8.
6.	18x5+48x4+71x3+124x2+65x+10.	21.	10x5+65x4+124x3+71x2+48x+18.
7.	3x5–13x4+2x3+17x2+33x+18.	22.	18x5+33x4+17x3+2x2–13x+3.
8.	x5+5x4–6x3–30x2+8x+40.	23.	40x5+8x4–30x3–6x2+5x+1.
9.	6x5–25x4–34x3+44x2+7x–10.	24.	10x5–7x4–44x3+34x2+25x–6.
10.	x5–3x4–x3–x2+8x+12.	25.	12x5+8x4–x3–x2–3x+1.
11.	9x5+x4–108x3–12x2+243x+27.	26.	27x5+243x4–12x3–108x2+x+9.
12.	8x5+32x4–6x3–24x2+x+4.	27.	4x5+x4–24x3–6x2+32x+8.
13.	7x5+29x4–17x3–73x2+46x+8.	28.	8x5+46x4–73x3–17x2+29x+7.
14.	3x5+8x4–12x3–42x2–15x+10.	29.	10x5–15x4–42x3–12x2+8x+3.
15.	10x5–5x4–42x3+21x2+8x–4.	30.	4x5–8x4–21x3+42x2+5x–10.
Задание 73. Разложить многочлен на неприводимые над Q множители, исполь-
зуя метод Кронекера.
1.	2x4–x3+2x2+1.	16.	x4–6x3+7x2+6x+1.
2.	x4+4x3+2x2–4x+1.	17.	x4+2x3–3x2–4x–1.
3.	x4+2x3+x2–1.	18.	x4–4x3+5x2–4x+1.
4.	x4+2x3–x2–2x+1.	19.	x4–2x3–3x2–4x–1.
5.	x4+2x3+3x2+2x+1.	20.	x4–4x3+3x2–2x–1.
6.	x4–2x3+x2–1.	21.	x4–4x3+x2+4x+1.
7.	x4–x2+2x–1.	22.	x4+2x3–5x2–2x+1.
8.	x4–4x3+2x2+4x+1.	23.	x4+4x3+x2–4x+1.
9.	x4–x3–4x2+x+1.	24.	x4–2x3–5x2+2x+1.
10.	x4+3x–2.	25.	x4+6x3+11x2+6x+1.
11.	x4–2x3–x2+2x+1.	26.	x4+2x2+x+2.
12.	x4–2x3+3x2–2x+1.	27.	x4–x3+2x2–x+1.
13.	x4+x3+3x2+2x+2.	28.	x4+2x2–x+2.
14.	x4–x3–x2+2x–2.	29.	x4–2x2–3x–2.
15.	x4–3x–2.	30.	x4+2x3+2x2+x–2.

Задание 74. Разложить на неприводимые над C и над R множители.

1.	x5+x3+x.	7.	(x2+x)2+4(x2+x)–12.
2.	x4–x3+2x2+x–3.	8.	(x2+x)2–14(x2+x)+24.
3.	x4+x3+3x2+x+2.	9.	(x2+4x+8)2+3x(x2+4x+8)+2x2.
4.	x5–x4+5x3–5x2+9x–9.	10.	(x2+x+1)(x2+x+2)–12.
5.	27x4–9x2+14x–4.	11.	(x2+x+4)2+8x(x2+x+4)+15x2.
6.	x6+27.	12.	4x4–24x3+29x2+42x–63.
		69

13.	(x+1)(x+2)(x+3)(x+4)–24.	22.	12x4–4x3–9x2+1.
14.	(x+1)(x+3)(x+5)(x+7)+15.	23. x5–6x4+16x3–32x2+48x–32.
15.	4(x+5)(x+6)(x+10)(x+12)–3x2.	24. x4–x3+2x2+x–3.
16.	2x4+7x3–2x2–13x+6.	25.	2x4+7x3–2x2–13x+6.
17.	2x4–x3–9x2+13x–5.	26. x4–4x3+8x2–16x+16.
18.	x4–2x3–3x2+4x+4.	27.	6x4+5x3–74x2+11x+12.
19.	x4–2x3–11x2+12x+36.	28.	10x4+21x3–55x2–72x+36.
20.	x4+2x3–16x2–2x+15.	29. x7–x6–x5+x4–x3+x2+x–1.
21.	x4+2x3–x2+2x+1.	30.	2(x2+6x+1)2+5(x2+6x+1)(x2+1)+2(x2+1)2.

Задание 75. Известно, что α является корнем многочлена f(x). Найти остальные корни многочлена и разложить его на неприводимые множители над C, над R

инад Q.

1.f (x) = x4–2x3+3x2–2x+2, α = 1+i.

2.f (x) = x4–4x3+3x2+8x–10, α = 2–i.

3.f (x) = x4+3x3+5x2+4x+2, α = –1+i.

4.f (x) = x4–2x3+7x2–4x+10, α = 1–2i.

5.f (x) = x4–5x3+3x2+16x–10, α = 3–i.

6.f (x) = x4+7x3+14x2+3x–15, α = –2+i.

7.f (x) = x4+4x3+6x2+4x–15, α = –1–2i.

8.f (x) = x4–2x3+x2–8x–12, α = 2i.

9.f (x) = x4–4x3+5x2+12x–24, α = 2+2i. 10. f (x) = x4–2x3+12x2–4x+20, α = 1–3i. 11. f (x) = x4+4x3–4x2–32x–20, α = –3+i.

12. f (x) = x4+5x3+6x2–19x–13, α = –3+2i. 13. f (x) = x4–4x3+10x2+12x–39, α =2–3i. 14. f (x) = x4+5x3+14x2+26x–20, α=–1+3i. 15. f (x) = x4+x3+6x2+9x–27, α = –3i.

16. f (x) = x4–7x3+14x2+17x–65, α = 3–2i. 17. f (x) = x4+2x3+2x2–38x–39, α = –2+3i.

18.f (x) = x4–x3–2x2–3x–1, α = − 12 + 23 i.

19.f (x) = x4–x2+2x–1, α = 12 – 23 i.

20.f (x) = x4+6x3+14x2–12x–36, α = – 3+3i.

21.f (x) = x4+4x3+16x2+12x+39, α = –2+3i.

22.f (x) = x4–x3–12x2–9x–27, α = 3i.

23.f (x) = x4+2x3–5x2–36x–40, α = –2– 2i.

24.f (x) = x4+4x3–3x2–50x–52, α = –3–2i.

25.f (x) = x4+x3+15x2+43x+68, α = 1–4i.

26.f (x) = x4+9x3+27x2+33x+34, α = –4+i.

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