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Национальный авиационный университет

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Solution. Let u = sinn−1 x

and dv =

sin x

dx . We evaluate

cosm x

du = (n − 1)sinn−2 x cos xdx ,

v = ∫

sin x

= − ∫

d (cos x)

( m ≠ 1 ),

cos

(m − 1)cos

m−1

J n, −m

sinn−1 x

−

n − 1 sinn− 2 x

dx .

(m − 1)cosm−1 x

m − 1 ∫ cosm−2

Thereby

J n, −m =

sinn−1 x

−

n − 1

J n−2, 2−m .

(m − 1)cosm−1 x

m − 1

Т.1

Exercises for class and homework

Find the indefinite integrals.


1.	∫(x − 1)(2x + 3)dx .
4.	∫	x 2 + 2x − 3						dx .
4.	∫		3	x				dx .
				x
7.	∫sin 2			x		dx .
7.	∫sin 2					dx .
				2
10. ∫			1+ 5 cos2 x
									dx .
			1+ cos 2x						dx .
13. ∫			dx .
			x 2 − 4
16. ∫			6 x−1 + 8x					dx .
16. ∫			2		x			dx .
			2
19. ∫			3 − x + 3 + x							dx .
19. ∫				9 − x			2			dx .
				9 − x

2. ∫

x3 − 8

3. ∫ x x dx .

dx .

x − 2

5. ∫ 4 x3 (1− x )dx .

6. ∫

x − 1 dx .

x − x

8. ∫ tg2 xdx .

9. ∫ (2 ctg x − 1)(2 ctg x + 1)dx .

11. ∫

12. ∫

x 2+ 1 + 1 dx .

+ 1

16 − x 2

14. ∫ e x (3 − e− x )dx .

15. ∫

x 4

dx .

+ 1

17. ∫

18. ∫

4 + 9x

9 − 16x 2

20. ∫ (3

x − 1)(2 4 x + 3)dx .

Use appropriate substitutions to find the integrals.

21.

а) ∫sin 3 xd(sin x) ;

б) ∫sin 3 x cos xdx ;

в) ∫cos3 2x sin 2xdx .

22.

а) ∫ ln

xd(ln x) ; б) ∫

ln3 x

dx . 23.

а) ∫ 2

tg x

d(tg x) ; б) ∫

2tg x dx

cos


24.		∫(3x − 4)6 dx .			25. ∫3 2x + 5dx
28.	∫	x 2 dx	.	29.	∫		(2x − 4)dx		.
		3					2
		4 + x					5 + 4x − x
32.	∫	x3dx .		33.	∫ (3x 2 + 1)dx .
		1− x 4					x3 + x + 1
36.	∫ sin xdx .			37.	∫	tg xdx		.

		x					x

. 26. ∫8 3 − xd(3x) . 27. ∫		xdx	.
	5	2
		+ x

30.	∫			xdx		.
		4		4
				+ x
34. ∫		x 2 dx .
		1+ x6
38. ∫	ln xdx				.
			x

∫xdx

31..

		1− x 4
35. ∫	4	sin xdx .
	4	− cos x

39. ∫ x 2 e x3 dx .

sin x cos

xdx

x − 1

∫e

41. ∫

40.

. 42.

44.

∫

45. ∫

xdx .

46.

∫ x tg(x 2

+ 1

1+ x

48.

∫

(2x + 7)dx

49. ∫ (1− 2x)99 xdx .

(3x − 2)

∫	1	dx .		43. ∫	xdx .
	1	+	x		x(x + 1)
− 1)dx .				47. ∫	x3dx .
					x − 1
		50. ∫		dx .
				x x + 1

51.

∫

+ 1)dx

52. ∫

53. ∫

54. ∫

2 dx

x x −

1+ e x

− x 2

x + 1

55.

∫

56. ∫

1+ x

2 dx

57. ∫ x

− x

dx .

x 2 − 4

58.

∫

59. ∫

e2x dx

60. ∫

x 2 − 1

4 1

+ e x

+ x

) arctg x

Use integration by parts to find the integrals.

61.	∫ x cos 4xdx .			62. ∫ (2x − 5) sin 2xdx .		63. ∫ (x + 3)e x dx .
64.	∫ (x3 − 2x 2 + 4)e2x dx .				65. ∫ (x3 + x 2 − 3x − 1) cos xdx .
66.	∫ x 2 2 x dx .			67. ∫ ln(2x − 1)dx .		68. ∫ x ln xdx .
69. ∫ ln(x 2 + 4)dx .				70. ∫ x ln(x − 1)dx .		71.	∫ x ln(x 2 + 1)dx .
	1			73. ∫ ln 2 xdx .
72.	∫ x		ln 2 xdx .			74.	∫ x arctg xdx .
		3
				82

75. ∫ x arcsin xdx .

78. ∫arccos xdx .

81. ∫e x cos 2xdx .

84. ∫ x sin x dx . cos3 x

76.	∫ x tg2 xdx .				77. ∫ x cos2 xdx .
79.	∫	x arctg x		dx .	80. ∫		ln 2 x	dx .
							3
		1+ x2					x
82.	∫sin ln xdx .				83.	∫e px sin qxdx .
85. ∫		x 2 e x	dx .		86.	∫arcsin 2 xdx .
		2
		(x + 2)

Answers

		1.							2		x3 +		x2		− 3x + C . 2.												1				x3 + x2 + 4x + C . 3.																4			x7 / 4 + C . 4.								3	x8 / 3 +						6	x5 / 3 −
								3					2												3																			7												8									5
-	9		x2 / 3 + C . 5.											4		x7 / 4 −				4		x9 / 4										+ C . 6. − x − 2										x + C . 7. (x − sin x) / 2 + C .
-			x2 / 3 + C . 5.											7		x7 / 4 −						x9 / 4										+ C . 6. − x − 2										x + C . 7. (x − sin x) / 2 + C .
2														7			9
																																													tg x +5x
8. tg x −x +C .																	9. −4ctg x −5x +C .																								10.				tg x +5x										+C .				11. ln			x + 1+ x2											+
8. tg x −x +C .																	9. −4ctg x −5x +C .																								10.														+C .				11. ln			x + 1+ x2											+
																																																	2
+arctg x +C .																12. arcsin									x						+ C .									13. ln x +												x2 − 4 + C .								14. 3ex − x + C .
+arctg x +C .																12. arcsin									4						+ C .									13. ln x +												x2 − 4 + C .								14. 3ex − x + C .
																									4
15.					1		x3 − x +arctg x +C .														16.											3x		+	4x			+ C .				17.				1			arctg					3x	+C .				18.		1		arcsin							4x		+ C .
15.		3					x3 − x +arctg x +C .														16.									6ln 3				+	ln 4			+ C .				17.							arctg						+C .				18.				arcsin						3			+ C .
		3																												6ln 3					ln 4									6									2								4								3
19.		2						3 + x − 2						3 − x + C . 20.																		x(−3 +				9	3 x −					8	4 x +									24		12	x7 ) +C				. 21.			а)					sin4 x					+ C;
																																		4								5									19																		4
б)		sin4 x								+ C;			в) −				1		cos4		2x + C.													22. а)					ln4 x			+ C ;								б)				ln4 x			+ C .			23. а)								2tg x				+ C ;
б)		4								+ C;			в) −				8		cos4		2x + C.													22. а)						4		+ C ;								б)			4				+ C .			23. а)								ln 2				+ C ;
		4															8																							4													4															ln 2
б)		2tg x												24.				(3x − 4)7																25.						3	3										4						26. −			9		3									4
									+ C .																			+ C .														(2x +							5)				+ C .												(3		− x)					+ C .
			ln 2						+ C .								21											+ C .												8		(2x +							5)				+ C .							4					(3		− x)					+ C .
			ln 2														21																							8																				4
27.					1		ln(x2 + 5) + C .										28.							1	ln(x3 + 4) + C . 29.																		− ln					5 + 4x − x2									+ C . 30.					1			arctg					x2		+ C .
					1																			1																																						1								x2

		2																			3																																						1 ln x3			4							2
		2																			3																																									4							2
31.				1		arcsin x2							+ C .				32.				−		1		1							− x4		+ C .				33.			2 x3 + x + 1 + C .															34.						+ 1 + x3										+ C .
		2																			2																																						3
35.			−8ln(4 −										cos x ) − 2								cos x + C .																	36. −2cos												x + C .									37. − 2 ln cos												x + C .
38.		2										3 / 2					39.								1							x3						40.						sin					x										41.	2				x			− 1 3
						(ln x)							+ C .																e				+ C .									e									+ C .																x					+ C .
		3				(ln x)							+ C .												3				e				+ C .									e									+ C .									3												+ C .
		3																							3																																			3
42.			2 x − 2ln(1 +													x ) + C . 43.										2arctg								x + C .44.								x − ln(ex + 1) + C														45.			2 x −2 arctg												x +C .

46.	−	1	ln	cos(x2 − 1)			+ C .			47.			2	(x

		2										7

48.	−	2			−	25			+ C .			49.		−
		9(3x − 2)				18(3x − 2)2
51.	2	x − 2 +			2 arctg			x − 2	+ C .			52. 3(
								2
53.	ln		ex + 1 − 1			+ C .			54.	1	arcsin x
			ex + 1 + 1							2

− 1)7 +	6	(x − 1)5 + 2 (x − 1)3 + 2 x − 1 + C .
	5

(1− 2x)100

+ C .

50. ln

x + 1 − 1

+ C .

200

x + 1 + 1

− 2t + ln

) + C ,

де

t = 1 + 3 x + 1 .

1 − x2

+ C .

55.

+ C .

−

arccos

56.		−			(1 + x2 )3							+ C .								57.				81	arcsin			x	−					x			9 − x2				(9 − 2x2 ) + C .																	58.				x2 − 1						+ C .
56.		−					3x3					+ C .								57.				8	arcsin				−				8				9 − x2				(9 − 2x2 ) + C .																	58.						x				+ C .
							3x3																	8			3						8																															x
59.	4		4		(1 + ex )7 −							4		4			(1 + ex )3				+ C . 60.								ln		arctg x							+ C .					61.							x		sin 4x +							1		cos 4x + C .
59.	4		4		(1 + ex )7 −							4		4			(1 + ex )3				+ C . 60.								ln		arctg x							+ C .					61.							x		sin 4x +							1		cos 4x + C .
	7											3																																	3					4								16
	7											3																																						4								16
		5 − 2x											1																		x																			7				2			7										2 x
		5 − 2x											1																		x											x								7				2			7					9					2 x
62.								cos 2x +								sin 2x + C .								63. (x + 2)e									+ C .							64.								−					x		+			x +					e					+C .
	2												2																																					4							4
																																										2								4							4					8
65.		(3x2 + 2x − 9)cos x + + (x3 + x2 − 9x − 3)sin x + C .																																							66.								x2 ln2 2 − 2x ln 2 + 2																	2x + C .
65.		(3x2 + 2x − 9)cos x + + (x3 + x2 − 9x − 3)sin x + C .																																							66.																									2x + C .
																																																							ln3 2
67.		xln(2x − 1) − x −																1	ln(2x − 1) + C .																	68.				2 x		x ln x −								4 x					x + C .
67.		xln(2x − 1) − x −															2		ln(2x − 1) + C .																	68.				2 x		x ln x −								4 x					x + C .
																	2																							3										9
69.		xln(x2 + 4) − 2x + 4arctg																				x		+ C .			70.			x2					ln(x −1) −							1	(x +1)2 −												1	ln		x −1			+C .
69.		xln(x2 + 4) − 2x + 4arctg																				x		+ C .			70.			x2					ln(x −1) −							1	(x +1)2 −												1	ln		x −1			+C .
																					2										2										4														2
																					2										2										4														2
71.		1		[(x2 + 1) ln(x2 + 1) − x2 − 1] + C .																							72.				3				3 x4				(8ln2 x − 12ln x + 9) + C .
71.	2			[(x2 + 1) ln(x2 + 1) − x2 − 1] + C .																							72.								3 x4				(8ln2 x − 12ln x + 9) + C .
	2																														32
73.		x(ln2 x − 2ln x + 2) + C .																									74.					1			(x2 +1)arctg x −																x		+C .
																															2																			2
75.		2x			2 −		1			ar c sin x +							1		x	1 − x2				+ C .			76.					x tg x −								x2	+ln					cos x						+C .
		2x			2 −		1										1																							x2
		x2			4		x										4																							2
					4										1		4																							2
77.					+				sin 2x +						1	cos 2x + C .									78.		xarccos x −												1− x2 + C .															79.				1+ x2 arctg x −
	4					4							8
−ln(x + 1 + x2 ) +C .																				80. −				2ln2 x + 2ln x + 1											+ C .						81.			1			ex			(2sin 2x + cos 2x) + C .
																										4x2																	5
82.				x	(sin ln x − cosln x) + C .																		83.			e px	( p sin qx − q cos qx)														+ C.							84.									x			−		1			tg x			+ С .
	2																												p2		+ q2																							2cos2 x								2
85.		x − 2				e		x			+ C .	86.						x ar c sin			2		x	+ 2arcsin x 1 − x												2	− 2x + C .
85.		x + 2				e					+ C .	86.						x ar c sin					x	+ 2arcsin x 1 − x													− 2x + C .
		x + 2

Т.1

Individual test problems

1.1. Find the integrals.

1.1.1. ∫

x2 − 3x + 2

dx .

1.1.2. ∫

(2 −

x )dx .

1.1.3. ∫ 5 x3 (4 + 3

x )dx .

1.1.4. ∫ 3 x2 (1− 4 x )dx .

1.1.5. ∫ 5 x4 (1+ 3 x )dx .

1.1.6. ∫ 6 x5 (3 − 3 x )dx .

1.1.7. ∫

x3 + 1

dx .

1.1.8. ∫

x − x + 1

dx .

7 x2

3 x4

1.1.9. ∫

x +

x − 2

dx .

1.1.10. ∫ 3 x7 (5 − 5 x )dx .

5 x4

1.1.11. ∫ (5 x4

+ 1) 3 x4 )dx .

1.1.12. ∫ 4 x5 (2 − 6 x )dx .

1.1.13. ∫ 3 x8 (1+ 3 x )dx .

1.1.14. ∫ (4 x9 + 1) 3 x2 )dx .

1.1.15. ∫ 7 x5 (1− 7 x )dx .

1.1.16. ∫

x + 1

dx .

3 x2

1.1.17. ∫

3 x − x2 + 2

dx .

1.1.18. ∫

x4 +

x − 3

dx .

3 x4

3 x5

1.1.19. ∫ 3 x7 (3 + 4 x )dx .

1.1.20. ∫ 3 x4 (1+ 5 x )dx .

1.1.21. ∫ 5 x3 (5 −

x )dx .

1.1.22. ∫

x + 4

dx .

3 x4

1.1.23. ∫

3 x − 2x3 + 1

dx .

1.1.24. ∫

x2 +

x − 1

dx .

3 x2

3 x7

1.1.25. ∫ 4 x9 (1−

x )dx .

1.1.26. ∫ (5 x + 2) 3 x2 )dx .

1.1.27. ∫ 4 x3 (3 − 3 x )dx .

1.1.28. ∫

x + 1

dx .

1.1.29. ∫

3 x − 3x2 + 1

dx .

1.1.30. ∫

x4 +

x − 2

dx .

3 x5

4 x3

1.2. Find the integrals.

1.2.1. ∫

−

dx .

1.2.2. ∫

3 − 5x

dx .

1.2.3. ∫

8 − 13x

dx .

+ 2

− x2

x2 − 1

1.2.4. ∫

6x + 1

dx .

1.2.5. ∫

x − 2

dx .

1.2.6. ∫

3 − 7x

dx .

− x2

− 4x2

− 1

1.2.7. ∫

5 − 3x

dx .

1.2.8. ∫

1 + x

dx .

1.2.9. ∫

3x + 2

dx .

2x2 + 1

2 − x2

+ 1

1.2.10. ∫

1 − 5x

dx .

1.2.11. ∫

4x − 3

dx .

1.2.12. ∫

5x + 1

dx .

25x

−

x2 − 6

1.2.13. ∫

x − 3

dx .

1.2.14. ∫

5 − 3x

dx .

1.2.15. ∫

4 − 2x

dx .

+ 7

1 − 4x2

4 − 3x2

1.2.16. ∫

5 − x

dx .

1.2.17. ∫

1 + 3x

dx .

1.2.18. ∫

5 − 4x

dx .

1 − x2

4x2 + 1

1.2.19. ∫

5x − 1

dx .

1.2.20. ∫

1 − 3x

dx .

1.2.21. ∫

x − 5

dx .

−

3 −

x2 − 3

1.2.22. ∫

x + 4

dx .

1.2.23. ∫

2x − 7

dx .

1.2.24. ∫

7x − 2

dx .

− 5

9 − x2

x2 − 1

1.2.25. ∫

1 + 3x

dx .

1.2.26. ∫

x − 5

dx .

1.2.27. ∫

3 − 7x

dx .

+ 7

x2 + 1

1.2.28. ∫

8 −

dx .

1.2.29. ∫

3x + 7 dx .

1.2.30. ∫

2x − 1

dx .

+ 1

x2 + 4

3x2 − 4

1.3. Find the integrals.

1.3.1. ∫sin 2 (1 − x)dx .

1.3.2. ∫sin3 (1 − 2x)dx .

1.3.3. ∫ (1− 2 sin x)2 dx .

1.3.4. ∫ cos3 (5x − 1)dx .

1.3.5. ∫cos3 (1 + 3x)dx .

1.3.6. ∫(3 − sin 2x)2 dx .

1.3.7. ∫sin

2 3x

dx .

1.3.8. ∫(cos x +

dx .

1.3.9. ∫cos

(2x + 3)dx .

1.3.10. ∫sin

3 4x

dx .

1.3.11. ∫(1 − cos 2x)2 dx .

1.3.12. ∫sin 2 (2x − 1)dx .

1.3.13. ∫sin 3 6xdx .

1.3.15. ∫sin 2 (4x − 3)dx .

1.3.17. ∫ (1+ 2 cos x)2 dx .

1.3.19. ∫ sin2 (2x − 1)dx .

1.3.21. ∫ (1− 3cos x)2 dx .

1.3.23. ∫sin3 (5x − 1)dx .

1.3.25. ∫ cos2 (2x + 1)dx .

1.3.27. ∫cos2 7xdx .

1.3.29. ∫sin3 4xdx .

1.4. Find the integrals. 1.4.1. ∫(4 − 3x) cos 2xdx .

1.4.3. ∫(x2 + 4)e− x dx .

1.4.5. ∫ x ln(4x − 1)dx .

1.4.7. ∫ln(x2 + 2x + 2)dx .

1.4.9. ∫(2 + x2 )cos xdx .

1.4.11. ∫(x + 2)3−2x+1 dx .

1.4.13. ∫ x ln(x + 3)dx .

1.4.15. ∫(3x − 1)ctg2 (3x − 1)dx .

1.4.17. ∫(2x − 1) cos2 xdx .

1.3.25. ∫ cos2 (2x + 1)dx .

1.3.27. ∫cos2 7xdx .

1.3.29. ∫sin 3 4xdx .

1.3.14.∫sin 2 2x dx .

1.3.16.∫ cos2 (1− 2x)dx .

1.3.18.∫cos2 3xdx .

1.3.20.∫ sin2 (1− x)dx .

1.3.22.∫cos2 25x dx .

1.3.24.∫ cos2 (3 − x)dx .

1.3.26.∫cos3 4xdx .

1.3.28.∫(sin x − 5)2 dx .

1.3.30.∫sin 2 34x dx .

1.4.2.∫(x2 − 1)sin 3xdx .

1.4.4.∫(x2 − 3)e2x dx .

1.4.6.	∫ 2x − 1 ln (2x − 1)dx .
1.4.8.	∫ln(x2 − 2x + 5)dx .

1.4.10.∫(3x − 7)2− x dx .

1.4.12.∫arctg(3x)dx .

1.4.14. ∫ 4 x ln xdx .

1.4.16. ∫ x tg2 (2x)dx .

1.4.18.∫(3 − x)sin2 2xdx .

1.3.26.∫cos3 4xdx .

1.3.28.∫(sin x − 5)2 dx .

1.3.30. ∫sin 2 34x dx .

1.4. Find the integrals. 1.4.1. ∫(4 − 3x)cos 2xdx .

1.4.3. ∫(x2 + 4)e− x dx .

1.4.5. ∫ x ln(4x − 1)dx .

1.4.7. ∫ln(x2 + 2x + 2)dx .

1.4.9. ∫(2 + x2 )cos xdx .

1.4.11. ∫(x + 2)3−2x+1 dx .

1.4.13. ∫ x ln(x + 3)dx .

1.4.15. ∫(3x − 1)ctg2 (3x − 1)dx .

1.4.17. ∫(2x − 1)cos2 xdx .

1.4.19. ∫ x arcsin x2dx .

1.4.21. ∫ x ln(4 − x)dx .

1.4.23. ∫ln(x2 + 4x + 5)dx .

1.4.25. ∫(1+ 2x2 )cos 4xdx .

1.4.27. ∫ x arccos3xdx .

1.4.29. ∫ x ln(x2 + 3)dx .

1.4.2.∫(x2 − 1)sin 3xdx .

1.4.4.∫(x2 − 3)e2x dx .

1.4.6.	∫ 2x − 1 ln (2x − 1)dx .
1.4.8.	∫ln(x2 − 2x + 5)dx .

1.4.10.∫(3x − 7)2− x dx .

1.4.12.∫arctg(3x)dx .

1.4.14. ∫ 4 x ln xdx .

1.4.16. ∫ x tg2 (2x)dx .

1.4.18.∫(3 − x)sin2 2xdx .

1.4.20.∫ x arcsin x4dx .

1.4.22.∫ 3 4 − 3x ln (4 − 3x)dx .

1.4.24.∫ln(x2 − 6x + 13)dx .

1.4.26.∫(x2 − 6)3x dx .

1.4.28.∫ x arctg(2x2 )dx .

1.4.30.∫ x arccos x2dx .

Topic 2. Polynomials. The rational functions

Values of polynomial functions. Fundamental theorem of algebra. Factor theorem. The factored form of a function. Proper and improper rational fractions. Long division to transform to mixed-number form — polynomial plus proper function. Linear factors of a polynomial function.

Literature: [1, ch. 4], [3, ch. 7, § 1], [4, section 7, § 22], [6, section 7], [7, ch.10, § 7—8], [8, 1 section, ch. 7, § 31].

Т.2	Main concepts

2.1. Polynomial functions. Linear factors of a polynomial function

A polynomial function is a function that can be expressed in the form

Pn (x) = a0 xn +a1 xn−1 +...+an−1 x +an ,

Theorem 3

Where the degree n is a nonnegative integer, the coefficients a0, a1, …, an are real numbers, and an ≠ 0.

If Pn (x0 ) = 0 , the number x0 is called a root of the polynomial function Pn(x).

		(Factor		theorem). x −x0 is a factor of polyno-
	Theorem 1	(Factor		theorem). x −x0 is a factor of polyno-
		mial Pn (x)		if and only if the polynomial equals 0 when x0
		mial Pn (x)		if and only if the polynomial equals 0 when x0
is substituted for x. On the other hand,
					,
			Pn (x) = (x −x0 )Qn−1 (x)		,
where Qn−1 (x) is a polynomial of n–1 degree.
		(Bezoo’s remainder theorem). If polynomial Pn(x) is
	Theorem 2	(Bezoo’s remainder theorem). If polynomial Pn(x) is
		divided by (x - λ), then the remainder R is Pn(λ). That is, if

then		Pn(x) = (x - λ) Pn − 1(x) + R
then
				Pn(λ) = R.

This is not difficult to show to be true, for if

Pn(x) = (x - λ) Pn − 1(x) + R

then

Pn(λ) = (λ - λ) Pn − 1(λ) + R = R.

(Fundamental theorem of algebra). If Pn(x) is a

polynomial function of degree n > 0, then Pn(x) has at least one complex zero.

Of course, a direct result of this is that the polynomial equation Pn(x) = 0 has at least one complex root.

Now, suppose that we have a polynomial function Pn(x) and that x1 is zero of P. By the theorem 1, this implies that (x − x1) is a factor of P and that we can write P as

Pn(x) = (x − x1) Qn − 1(x)

where Qn − 1(x) is a polynomial function of degree n − 1. Now, by the theorem 3, Qn − 1(x) in tern has a zero x2, and Qn − 1(x) can be written as

Qn − 1(x) = (x − x2) Qn − 2(x)

where Qn − 2(x) is a polynomial function of degree n − 2. This implies that

Pn(x) = (x − x1) (x − x2) Qn − 2(x). If we continue this process, eventually arrive at

Pn(x) = (x − x1) (x − x2) … (x − xn)Q0(x)

where Q0(x) is a polynomial function of degree 0, namely, a constant function. We have the rational function it to you as an exercise to show that Q0(x) = a0, the leading coefficient of P.

Theorem 6

Theorem 4		If Pn(x) is polynomial function of degree n > 0 and
		leading coefficient a0, then
		leading coefficient a0, then

		Pn(x) = a0 (x − x1) (x − x2) … (x − xn)

where x1, x2, x3, …, xn are the complex zeros of Pn(x). Particularly,

ax2 +bx +c = a(x −x )(x −x )

where x1, x2, x3, …, xn are the complex zeros of ax2 +bx +c = 0 .

If P (x)

is divided a without remainder by (x −x )k , but is not divided by

(x −x )k+1 , then x

is called the repeated k times roots of

P (x) .

In this case

P (x) = (x −x )k Q

(x) , Q

(x ) ≠ 0 .

n−k

Whether

a polynomial Pn(x) has the roots

x1 , x2 ,

, xm (m ≤n) ,

with

repeated accordingly k1 , k2 ,…, km , then

P (x) = a (x −x )k1 (x −x )k2

...(x −x

)km

(*)

(Identity equations). An identity is an equation that is

Theorem 5

true for all values of the variable.

(Conditional equations). A conditional equation is one that is true for some value(s) of the variable and not true

for other values of the variable

			a0 =b0 , a1 = b1 , …, an = bn .
			(Conjugate root theorem). If Pn(x) is a polynomial
	Theorem 7		(Conjugate root theorem). If Pn(x) is a polynomial
	Theorem 7		function of degree n > 0 with real coefficients, and if a +bi
			is a zero of P, then a −bi		is also a zero of P. Moreover, the
number of the conjugate roots a +bi				and a −bi is the same.
Let us consider the product
(x −(a +bi))(x −(a −bi)) = ((x −a)−bi)((x −a) +bi) = (x−a)2 +b2 =
			= x2 −2ax +a2 +b2 = x2 + px +q ,
where p =−2a,		q = a2 +b2 .
	Therefore, if	a	polynomial Pn(x) has a pair of conjugate complex roots
a ±bi , we can substitute (see (*))				product	(x −(a +bi))(x −(a −bi)) for a
quadratic trinomial			x2 + px +q	with real	coefficients and a negative
discriminant.
				90

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