Factorización

Polinomios. Factorizar, descomponer. Descomposición en factores

  • Enviado por: Nestor I. Rivera
  • Idioma: castellano
  • País: España España
  • 11 páginas

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1.) 5a2 + a = a·( 5a + 1 )

2.) m2 + 2mx + x2 = ( m + x )2

3.) a2 + a - ab - b = (a2 + a) - (ab + b) = a·(a+1) - b·(a + 1) = ( a - b ) · ( a + 1 )

4.) x2 - 36 = ( x + 6 ) · ( x - 6 )

5.) 9x2 - 6xy + y2 = (3x)2 - 2·3x·y + y2 = ( 3x - y )2

6.) x2 - 3x - 4 = x2 + ( - 4+1)·x - 4·1 = ( x - 4 ) · ( x + 1 )

7.) 6x2 - x - 2 = 6·{6x2 - x - 2} / 6 = { 6·(6x2 ) - 6x - 6·2 } / 6 =

= { (6x)2 - 6x - 12 } / 6 = { (6x)2 + 6x·(- 4 + 3) + (- 4)·(3) } / 6

= { (6x - 4)·(6x + 3) } / 6 = { (6x - 4)·(6x + 3) } / (2·3) =

= { (6x - 4) / 2 } · {(6x + 3) / 3 } = (3x - 2)·(2x + 1) =

= ( 2x + 1 ) · ( 3x - 2 )

8.) 1 + x3 = 13 + x3 = ( 1 + x ) · ( 1 - x + x2 )

9.) 27a3 - 1 = (3a)3 - 13 = ( 3a - 1 ) · ( 9a2 + 3a + 1 )

10.) x5 + m5 = ( x + m ) · ( x4 - mx3 + m2x2 - m3x + m4 )

11.) a3 - 3a2b + 5ab2 = a ( a2 - 3ab + 5b2 )

12.) 2xy - 6y + xz - 3z = 2y(x - 3) + z(x - 3) = (2y + z)·(x - 3) =

( x - 3 ) · ( 2y + z )

13.) 1 - 4b + 4b2 = 12 - 2·2b + (2b)2 = ( 1 - 2b )2

14.) 4x4 + 3x2y2 + y2 = { (2x2)2 + 3x2y2 + x2y2 + y4 } - x2y2 = { (2x2)2 + 4x2y2 + y4 }

- x2y2 = (2x2 + y2)2 - x2y2 = { (2x2 + y2) + xy } · { (2x2 + y2) - xy } =

( 2x2 + xy + y2 ) · ( 2x2 - xy + y2 )

15.) x8 - 6x4y4 + y8 = x8 - 6x4y4 + 4x4y4 + y8 - 4x4y4 = { (x4)2 - 2x4y4 + (y4)2 } -

4x4y4 = (x4 - y4)2 - (2x2y2)2 = { (x4 - y4) + 2x2y2 } · { (x4 - y4) - 2x2y2 } =

( x4 + 2x2y2 - y4 ) · ( x4 - 2x2y2 - y4 )

16.) a2 - a - 30 = ( a - 6 ) · ( a + 5 )

17.) 15m2 + 11m - 14 = 15 { 15m2 + 11m - 14 } / 15 = { (15m)2 + 11·(15m) -

210 } / 15 =

\\\\\\ 210 2 210 = 2·3·5·7

105 3 210 = 6·35

35 5 210 = 14·15

7 7 210 = 10·21

1

\\\\\\ = { (15m + 21) · (15m - 10) } / (3·5) = { (15m + 21) / 3 } · { (15m -

10) / 5} = (15m / 3 + 21 / 3 ) · ( 15m / 5 - 10m / 5) = (5m + 7)·(3m - 2) =

( 3m - 2 ) · ( 5m + 7 )

18.) a6 + 1 = (a 2)3 + 13 = ( a2 + 1 ) · ( a4 - a2 + 1 )

19.) 8m3 - 27y6 = (2m)3 - (3y2)3 = ( 2m - 3y2 ) · ( 4m2 + 6my2 + 9y4 )

20.) 16a2 - 24ab + 9b2 = (4a)2 - 2·4a·3b + (3b)2 = ( 4a - 3b )2

21.) 1 + a7 = ( 1 + a ) · ( 1 - a + a2 - a3 + a4 - a5 + a6 )

22.) 8a3 - 12a2 +6a - 1 = (2a)3 - 3·(2a)2·(1) + 3·2a·12 - 13 = ( 2a - 1 )3

23.) 1 - m2 = 12 - m2 = ( 1 + m ) · ( 1 - m )

24.) x4 + 4x2 - 21 = (x2)2 + (7 - 3)x2 + (7)·(-3) = ( x2 + 7 ) · ( x2 - 3 )

25.) 125a6 + 1 = (5a2)3 + 13 = ( 5a2 + 1 ) · ( 25a4 - 5a2 + 1 )

26.) a2 + 2ab + b2 - m2 = (a + b)2 - m2 = ( a + b + m ) · ( a + b - m )

27.) 8a2b + 16a3b - 24a2b2 = 16a3b + 8a2b - 24a2b2 = 8a2b ( 2a - 3b + 1 )

28.) x5 - x4 + x - 1 = x4(x - 1) + 1(x - 1) = ( x4 - 1 ) · ( x - 1 )

29.) 6x2 + 19x - 20 = 6 · { 6x2 + 19x - 20 } / 6 = { (6x)2 + 19·6x - 120 } / 6 =

\\\\\\ 120 2 120 = 2·2·2·3·5 120 = 24·5

60 2 120 = 4·30

30 2 120 = 8·15

15 3 120 = 12·10

5 5 120 = 20·6

\\\\\\ = { (6x + 24)·(6x - 5) } / 6 = { (6x + 24) / 6 } · { 6x - 5 } =

(x + 4)·(6x - 5) = ( 6x - 5 ) · ( x + 4 )

30.) 25x4 - 81y2 = (5x2)2 - (9y)2 = ( 5x2 + 9y ) · ( 5y2 - 9y )

31.) 1 - m3 = ( 1 - m ) · ( 1 + m + m2 )

32.) x2 - a2 + 2xy + y2 + 2ab - b2 = - a2 + 2ab - b2 +x2 +2xy +y2 =

- (a2 - 2ab + b2) + (x2 + 2xy + y2) = - (a - b)2 + (x + y)2 = (x + y)2 - (a - b)2 =

{ x + y + a - b } · { x + y - (a - b) } = ( x + y + a - b ) · ( x + y - a + b )

33.) 21m5n - 7m4n2 + 7m3n3 - 7m2n = 7m2n ( 3m3 - m2n + mn2 - 1 )

34.) a ( x + 1 ) - b ( x + 1 ) + c ( x + 1 ) = ( x + 1 ) · ( a - b + c )

35.) 4 + 4 ( x - y ) + ( x - y )2 = 22 + 2·2·(x - y) + (x - y)2 = ( 2 + x - y )2

36.) 1 - a2b4 = 12 - (ab2)2 = ( 1 + ab2 ) · ( 1 - ab2 )

37.) b2 + 12ab + 36a2 = 36a2 +12ab + b2 = (6a)2 + 2·6a·b + b2 = ( 6a + b )2

38.) x6 + 4x3 - 77 = (x3)2 + (-7 + 11)x3 + (-7)·(11) = ( x3 - 7 ) · ( x3 + 11 )

39.) 15x4 - 17x2 - 4 = 15 · { 15x4 - 17x2 - 4 } / 15 = { (15x2)2 - 17·(15x2) - 60 } / 15 = \\\\\\

60 2 60 = 2·2·3·5

30 2 60 = 4·15

15 3 60 = 6·10

5 5 60 = 2·30

1 60 = 3·20

\\\\\\ = { (15x2 - 20)·(15x2 + 3) } /15 = { (15x2 - 20)·(15x2 + 3) } / (5·3) =

{ (15x2 - 20)/5 } · { (15x2 + 3)/3 } = (3x2 - 4)·(5x2 + 1) = ( 5x2 + 1 ) · ( 3x2 - 4 )

40.) 1 + ( a - 3b )3 = 13 + (a - 3b)3 = {1 + a - 3b}·{1 - (a - 3b) + (a - 3b)2} =

{ 1 + a - 3b } · { 1 - a + 3b + a2 - 6ab + 9b2 }

41.) x4 + x2 + 25 = x4 + x2 + 9x2 + 25 - 9x2 = x4 + 10x2 + 25 - 9x2 =

(x2)2 + 2·5x2 + 52 - 9x2 = (x2 + 5) - (3x)2 =(x2 + 5 + 3x)·(x2 + 5 - 3x) =

( x2 + 3x + 5 ) · ( x2 - 3x +5 )

42.) a8 - 28a4 + 36 = (a4)2 + 28a4 + 16a4 + 36 - 16a4 = (a4)2 - 2·6a4 + 62 - (4a2)2 = (a4 - 6)2 - (4a2)2 = (a4 - 6 + 4a2)·(a4 - 6 - 4a2) =

( a4 + 4a2 - 6 ) · ( a4 - 4a2 - 6 )

43.) 343 + 8a3 = 73 + (2a)3 = ( 7 + 2a ) · ( 49 - 14a +4a2 )

44.) 12a2bx - 15a2by = 3a2b ( 4x - 5y )

45.) x2 + 2xy - 15y2 = x2 + 2xy - 15y2 + 16y2 - 16y2 = x2 + 2xy + y2 - (4y)2 =

(x + y)2 - (4y)2 = (x + y + 4y)·(x +y - 4y) = (x + 5y)·(x - 3y) =

( x - 3y ) · ( x + 5y )

46.) 6am - 4an - 2n + 3m = 6am + 3m - 4an - 2n = 3m(2a + 1) -2n(2a + 1 ) =

( 3m - 2n ) · ( 2a + 1 )

47.) 81a6 - 4b2c8 = (9a3)2 - (2bc4)2 = ( 9a3 + 2bc4 ) · ( 9a3 - 2bc4 )

48.) 16 - (2a + b)2 = 42 - (2a + b)2 = {4 + (2a + b)}·{4 - (2a + b)} =

( 4 + 2a + b ) · ( 4 - 2a - b )

49.) 20 - x - x2 = -1( - 20 + x + x2) = -1(x2 + x - 20) = -1(x2 + (- 4 + 5)x + (- 4)·(5) = -1{ (x + 5)·(x - 4) } = ( 5 + x ) · ( x - 4 )

50.) n2 + n - 42 = n2 + (7 - 6)n - (7)·(-6) = ( n + 7 ) · ( n - 6 )

51.) a2 - d2 + n2 - c2 - 2an - 2cd = a2 - 2an + n2 - d2 - 2cd - c2 =

(a2 - 2an + n2) - 1(d2 + 2cd + c2) = (a - n)2 - (d + c)2 = {a - n + (d + c) } · {

a - n - (d + c)} = ( a - n + d + c ) · ( a - n - d - c )

52.) 1 + 216x9 = 13 + (6x3)3 = ( 1 + 6x3 ) · ( 1 - 6x3 + 36x6 )

53.) x3 - 64 = x3 - 43 = ( x - 4 ) · ( x2 + 4x + 16 )

54.) x3 - 64x4 = x3 ( 1 - 64x )

55.) 18ax5y3 - 36x4y3 - 54x2y8 = 18x2y3 ( ax3 - 2x2 - 3y5 )

56.) 49a2b2 - 14ab + 1 = (7ab)2 - 2·7ab·1 + 12 = ( 7ab - 1 )2

57.) ( x + 1 )2 - 81 = (x + 1)2 - 92 = (x + 1 + 9)·(x + 1 - 9) = ( x + 10 ) · ( x - 8 )

58.) a2 - ( b + c )2 = {a + (b + c)}·{a - (b + c)} = ( a + b + c ) · ( a - b - c )

59.) ( m + n )2 - 6 ( m + n ) + 9 = ( m + n - 3 )2

60.) 7x2 + 31x - 20 = 7·{7x2 + 31x - 20} / 7 = {(7x)2 + 31(7x) - 140} / 7 = \\\\\\

140 2 140 = 2·2·5·7

70 2 140 = 4·35

35 5

7 7

1

\\\\\\ = { (7x + 35)·(7x - 4) } / 7 = { (7x + 35) / 7 } · { (7x - 4) } =

{ x + 5 } · { 7x - 4 } = ( x + 5 ) · ( 7x - 4 )

61.) 9a3 + 63a - 45a2 = 9a3 - 45a2 + 63a = 9a ( a2 - 5a + 7 )

62.) ax + a - x - 1 = a(x + 1) - 1(x + 1) = ( a - 1 ) · ( x + 1 )

63.) 81x4 + 25y2 - 90x2y = 81x4 - 90x2y + 25y2 = (9x2)2 - 2·(9x2)·(5y) + (5y)2 =

( 9x2 - 5y )2

64.) 1 - 27b2 + b4 = 1 - 27b2 + 25b2 + b4 - 25b2 = 1 - 2b2 + b4 - 25b2 =

(1 - b2) - (5b)2 = (1 - b2 + 5b)·(1 - b2 + 5b) = ( 1 + 5b - b2 ) · ( 1 - 5b - b2 )

65.) m4 + m2n2 + n4 = m4 + m2n2 + n4 + m2n2 - m2n2 = { (m2)2 + 2m2n2 + (n2)2 } - m2n2 = (m2 + n2)2 - m2n2 = (m2 + n2 + mn)·(m2 + n2 - mn) =

( m2 + mn + n2 ) · ( m2 - mn + n2 )

66.) c4 - 4d4 = (c2)2 - (2d2)2 = ( c2 + 2d2 ) · ( c2 - 2d2 )

67.) 15x4 - 15x3 + 20x2 = 5x2 ( 3x2 - 3x + 4 )

68.) a2 - x2 - a - x = (a + x)·(a - x) - a - x = (a + x)·(a - x) - (a + x) =

( a + x ) · ( a - x - 1 )

69.) x4 - 8x2 - 240 = \\\\\\

240 2 240 = 2·2·2·2·3·5

120 2 240 = 4·60

60 2 240 = 8·30

30 2 240 = 16·15

15 3 240 = 20·12

5 5

1

\\\\\\ = { (x2)2 + (- 20 + 12)x2 + (-20)·(12) } = ( x2 + 12 ) · ( x2 - 20 )

70.) 6m4 + 7m2 - 20 = 6·{6m4 + 7m2 - 20 } / 6 = { (6m2)2 + 7·6m - 120 } / 6 =

= \\\\\\ 120 2 120 = 2·2·2·3·5

60 2 120 = 4·30 120 = 8·15

30 2 120 = 24·5

15 3 120 = 12·10

5 5 120 = 20·6

\\\\\\ = { (6m2 + 15)·(6m2 - 8) } / (3·2) = { (6m2 + 15) / 3 }·{ (6m2 - 8) / 2 }

= ( 2m2 + 5 ) · ( 3m2 - 4 )

71.) 9n2 + 4a2 - 12an = 9n2 - 12an + 4a2 = (3n)2 - 2·3n·2a + (2a)2 =

= ( 3n - 2a )2

72.) 2x2 + 2 = 2 ( x2 + 1 )

73.) 7a ( x + y - 1 ) - 3b ( x + y - 1 ) = ( 7a - 3b ) · ( x + y - 1 )

74.) x2 + 3x - 18 = x2 + (6 - 3)x - (6)·(-3) = ( x + 6 ) · ( x - 3 )

75.) ( a + m )2 - ( b + n )2 = { a + m + b + n } · { a + m - (b + n) } =

( a + m + b + n ) · ( a + m - b - n )

76.) x3 + 6x2y + 12xy2 + 8y3 = x3 + 3·x2·(2y) + 3·x·(2y)2 + (2y)3 = ( x + 2y )3

77.) 8a2 - 22 a - 21 = 8·{8a2 - 22a - 21} / 8 = { (8a)2 - 22·(8a) - 168 } / 8 =\\\\\\

168 2 168 = 2·2·2·3·7

84 2 168 = 8·21

42 2 168 = 4·42

21 3 168 = 12·14

7 7 168 = 28·6

1

\\\\\\ = { (8a)2 + (6 - 28)·(8a) + (-28)·(8) } / 8 = { (8a + 6)·(8a - 28) } / 8 =

{ (8a + 6)·(8a - 28) } / (2·4) = { (8a + 6) / 2 } · { (8a - 28) / 4 } =

( 4a + 3 ) · ( 2a - 7 )

78.) 1 + 18ab + 81a2b2 = 12 + 2·9ab + (9ab)2 = ( 1 + 9ab )2

79.) 4a6 - 1 = (2a3)2 - 12 = ( 2a3 + 1 ) · ( 2a3 - 1 )

80.) x6 - 4x3 - 480 = \\\\\\

480 2 480 = 2·2·2·2·2·3·5

240 2 480 = 32·15

120 2 480 = 16·30

60 2 480 = 8·60

30 2 480 = 12·40

15 3 480 = 24·20

5 5

1

\\\\\\ = (x3)2 + (20 - 24)x3 + (20)·(-24) = ( x3 + 20 ) · ( x3 - 24 )

81.) ax - bx + b - a - by + ay = (ax + ay - a) - bx -by + b = (ax + ay - a) - (bx + by - b) = a(x + y - 1) - b(x + y - 1) = ( a - b ) · ( x + y - 1 )

82.) 6am - 3m - 2a + 1 = (6am - 3m) - (2a - 1) = 3m(2a - 1) - (2a - 1) =

( 3m - 1 ) · ( 2a - 1 )

83.) 15 + 14x - 8x2 = - 8x2 + 14x + 15 = -(8x2 - 14x - 15) =

8·{ - (8x2 - 14x - 15) } / 8 = - { (8x)2 - 14·(8x) - 120 } / 8 = \\\\\\

120 2 120 = 2·2·2·3·5

60 2 120 = 4·30

30 2 120 = 24·5

15 3 120 = 12·10

5 5 120 = 8·15

1 120 = 20·6

\\\\\\ = - { (8x)2 + (6 - 20)·(8x) + (6)·(- 20) } / 8 =

- [ { (8x + 6)·(8x - 20) } / (2·4) ] = - [ { (8x + 6) / 2 } · { (8x - 20) / 4 } ] =

- [ { 4x + 3 } · {2x - 5} ] = - (4x + 3)·(2x - 5) = ( 3 + 4x ) · ( 2x - 5 )

84.) a10 - a8 + a6 + a4 = a4 ( a6 - a4 + a2 + 1 )

85.) 2x ( a + 1 ) - a - 1 = 2x(a + 1) - (a + 1) = ( 2x - 1 ) · ( a + 1 )

86.) ( m + n ) · ( m - n ) + 3n ( m - n ) = ( m + 4n ) · ( m - n )

87.) a2 - b3 + 2b3x2 - 2a2x2 = a2 - 2a2x2 - b3 + 2b3x2 = (a2 - 2a2x2) - (b3 - 2b3x2) =

a2(1 - 2x2) - b3(1 - 2x2) = ( a2 - b3 ) · ( 1 - 2x2 )

88.) 2am - 3b - c - cm - 3bm + 2a = (2am - 3bm - cm) + 2a - 3b - c =

m(2a - 3b - c) + 1(2a - 3b - c) = ( m + 1 ) · ( 2a - 3b - c )

89.) x2 - 2x/3 + 1/9 = x2 - 2·(1/3)·x + (1/3)2 = ( x - 1/3 )2

90.) 4a2n - b4n = (2an)2 - (b2n)2 = ( 2an + b2n ) · ( 2an - b2n )

91.) 81x2 - ( a + x )2 = (9x)2 - (a + x)2 = {9x + a + x}·{9x - (a + x) } =

(10x + a)·(9x - a - x) = ( 10x + a ) · ( 8x - a )

92.) a2 + 9 - 6a - 16x2 = (a2 - 6a + 9) - (4x)2 = (a - 3)2 - (4x)2 =

( a - 3 + 4x ) · ( a - 3 - 4x )

93.) 9a2 - x2 - 4 + 4x = (3a2) - (x2 - 4x + 4) = (3a)2 - (x2 - 2·2·x + 22) =

(3a)2 - (x - 2)2 = {3a + (x - 2) } · { 3a - (x - 2) } = ( 3a + x - 2 ) · ( 3a - x + 2 )

94.) 9x2 - y2 + 3x - y = (3x)2 - y2 - (3x + y) = (3x + y)·(3x - y) +(3x - y) =

( 3x - y ) · ( 3x + y + 1 )

95.) x2 - x - 72 = \\\\\\

72 2 72 = 2·2·2·3·3

36 2 72 = 8·9

18 2

9 3

3 3

1

\\\\\\ = x2 + (8 - 9)x + ((8)·(-9) = ( x + 8 ) · ( x - 9 )

96.) 36a4 - 120a2b2 + 49b4 = (6a2)2 - 120a2b2 + 36a2b2 + (7b2)2 - 36a2b2 =

(6a2)2 - 84a2b2 + (7b2)2 - 36a2b2 = (6a2)2 - 2·(6a2)·(7b2) + (7b2)2 - (6ab)2 =

(6a2 - 7b2)2 - (6ab)2 = ( 6a2 - 7b2 + 6ab ) · ( 6a2 - 7b2 + 6ab )

97.) a2 - m2 - 9n2 - 6mn + 4ab + 4b2 = a2 + 4ab + 4b2 - m2 - 6mn - 9n2 =

a2 + 2·a·2b + (2b)2 - { m2 - 2·m·3n - (3n)2 } = (a + 2b)2 - (m - 3n)2 =

{ a + 2b + (m - 3n) }·{ a + 2b - (m - 3n) } =

( a + 2b + m - 3n ) · ( a + 2b - m + 3n )

98.) 1 - 4a8/9 = 12 - (2a4/3)2 = ( 1 + 2a4/3 ) · ( 1 - 2a4/3 )

99.) 81a8 + 64b12 = (9a4)2 + (8b6)2 = (9a4)2 + 144a4b6 + (8b6) - 144a4b6 =

(9a4)2 + 2·(9a4)·(8b6) + (8b6) - (12a2b3)2 = (9a4 + 8b6)2 - (12a2b3)2 =

( 9a4 + 12a2b3 + 8b6 ) · ( 9a4 - 12a2b3 + 8b6 )

100.) 49x2 - 77x + 30 = (7x)2 - 11·7x + 30 = ( 7x - 5 ) · ( 7x - 6 )

101.) x2 - 2abx - 35a2b2 = x2 - 2abx - 35a2b2 + 36a2b2 - 36a2b2 = x2 - 2abx + a2b2 - 36a2b2 = (x - ab)2 - (6ab)2 = (x - ab + 6ab)·(x - ab - 6ab) =

( x + 5ab ) · ( x - 7ab )

102.) 125x3 - 225x2 + 135x - 27 = (5x)3 - 3·(5x)2·3 + 3·5x·32 - 33 = ( 5x - 3 )3

103.) ( a - 2 )2 - ( a + 3 )2 = { a - 2 + (a + 3) }·{ a - 2 - (a + 3) } =

(a - 2 + a + 3)·(a - 2 - a - 3) = (2a + 1)·(-5) = - 5 ( 2a + 1 )

104.) 4a2m + 12a2n - 5bm - 15bn = (4a2m + 12a2n) - (5bm + 15bn) =

4a2(m + 3n) - 5b(m + 3n) = ( 4a2 - 5b ) · ( m + 3n )

105.) 1 + 6x3 + 9x6 = 12 + 2·3x3 + (3x3)2 = ( 1 + 3x3 )2

106.) a4 + 3a2b - 40b2 = a4 + (8b - 5b)a2 + (8b)·(-5b) = ( a2 + 8b ) · ( a2 - 5b )

107.) m3 + 8a3x3 = m3 + (2ax)3 = ( m + 2ax ) · ( m2 - 2amx + 4a2x2 )

108.) 1 - 9x2 + 24xy - 16 y2 = 12 - (3x)2 + 2·3x·4y - (4y)2 = 12 - { (3x - 2·3x·4y + (4y)2 } = 12 - (3x - 4y)2 = { 1 + (3x - 4y) }·{ 1 - (3x - 4y) } =

( 1 + 3x - 4y ) · ( 1 - 3x + 4y )

109.) 1 + 11x + 24x2 = 24x2 + 11x + 1 = 24·{ 24x2 + 11x + 1} / 24 =

{ (24x)2 + 11·(24x) + 24 } / 24 = { (24x)2 + (8 + 3)·(24x) + (8·3) } / 24 =

{ (24x + 3)·(24x + 8) } / (3·8) = { (24x + 3) / 3 } · { (24x + 8 ) / 8 } =

( 8x + 1 ) · ( 3x + 1 )

110.) 9x2y3 - 27x3y3 - 9x5y3 = 9x2y3 ( 1 - 3x - x3 )

111.) ( a2 + b2 - c2 )2 - 9x2y2 = (a2 + b2 - c2)2 - (3xy)2 =

( a2 + b2 - c2 + 3xy ) · ( a2 + b2 - c2 + 3xy )

112.) 8(a + 1)3 - 1 = { 2(a + 1) }3 - 13 = (2a + 2)3 - 13 =

{ (2a + 2) - 1 }·{ (2a + 2)2 + (2a + 2)·1 + 12 } =

(2a + 1)·( 4a2 + 8a + 4 + 2a + 2 + 1) = ( 2a + 1 ) · ( 4a2 + 10a + 7 )

113.) 100x4y6 - 121m4 = (10x2y3)2 - (11m2)2 =

( 10x2y3 + 11m2 ) · ( 10x2y3 - 11m2 )

114.) ( a2 + 1 )2 + 5( a2 + 1 ) - 24 = (a2 + 1)2 + (8 - 3)·(a2 + 1) + (8)·(-3) =

{ (a2 + 1) + 8}·{ (a2 + 1) - 3 } = (a2 + 1 + 8)·(a2 + 1 - 3) =

( a2 + 9 ) · ( a2 - 2 )

115.) 1 + 1000x6 = 13 + (10x2)3 = ( 1 + 10x2 ) · ( 1 - 10x2 + 100x4 )

116.) 49a2 - x2 - 9y2 + 6xy = 49a2 - (x2 - 6xy + 9y2) = 49a2 - { x2 - 2·x·3y + (3y)2}

= (7a)2 - (x - 3y)2 = { 7a + (x - 3y) }·{ 7a - (x - 3y) } =

( 7a + x - 3y ) · ( 7a - x + 3y )

117.) x4 - y2 + 4x2 + 4 - 4yz - 4z2 = x4 + 4x2 + 4 - y2 - 4yz - 4z2 =

(x4 + 4x2 + 4) - (y2 + 4yz + 4z2) = { (x2)2 + 2·x·2 + 22 }·{y2 + 2·y·2z + (2z)2 }

= (x2 + 2)2 - (y + 2z)2 = { (x2 + 2) + (y + 2z) }·{ (x2 + 2) - (y + 2z) } =

( x2 + 2 + y + 2z ) · ( x2 + 2 - y - 2z )

118.) a3 - 64 = a3 - 43 = ( a - 4 ) · ( a2 + 4a + 16 )

119.) a5 + x5 = ( a + x ) · ( a4 - a3x + a2x2 - ax3 + x4 )

120.) a6 - 3a3b - 54b2 = (a3)2 + (6b - 9b)·a3 - (6b)·(-9b) = ( a3 + 6b ) · ( a3 - 9b )

121.) 165 + 4x - x2 = -x2 + 4x + 165 = -(x2 - 4x - 165) = \\\\\\

165 3 165 = 3·5·11

55 5 165 = 33·5

11 11 165 = 15·11

1

\\\\\\ = - { x2 + (11 - 15)x - (11)·(-15)} = - { (x + 11)·(x - 15) } =

( 11 + x ) · ( x - 15 )

122.) a4 + a2 + 1 = a4 + a2 + a2 + 1 - a2 = {(a2)2 + 2·(a2)·(1) + 12 } - a2 =

(a2 + 1)2 - a2 = (a2 + 1 + a)·(a2 + 1 - a) =( a2 + a +1 ) · ( a2 - a + 1 )

123.) x2 / 4 - y6 / 81 = (x / 2)2 - (y3 / 9)2 = ( x / 2 + y3 / 9 ) · ( x / 2 - y3 / 9 )

124.) 16x2 + 8xy / 5 + y2 / 25 = (4x )2 + 2·(4x)·(y/5) + (y / 5)2 =

( 4x + y / 5 )2

125.) a4b4 + 4a2b2 - 96 = \\\\\\

96 2 96 = 2·2·2·2·2·3

48 2 96 = 32·3

24 2 96 = 16·6

12 2 96 = 8·12

6 2 96 = 4·24

3 3 96 = 2·48

1

\\\\\\ = (a2b2)2 + (12 - 8)·a2b2 + (12)·(-8) = ( a2b2 + 12 ) · ( a2b2 - 8 )

126.) 8a2x + 7y + 21by - 7ay - 8a3x + 24a2bx =

8a2x - 8a3x + 24a2bx + 7y - 7ay + 21by =

8a2x (1 - a + 3b) + 7y(1 - a + 3b) = ( 8a2x + 7y ) · ( 1 - a + 3b )

127.) x4 + 11x2 - 390 = \\\\\\

390 2 390 = 2·3·5·13

195 3 390 = 6·65

65 5 390 = 10·39

13 13 390 = 26·15

1

\\\\\\ = (x2)2 + (26 - 15)x2 + (26)·(-15) = ( x2 + 26 ) · ( x2 - 15 )

128.) 7 + 33m - 10m2 = - 10m2 + 33m + 7 = - { 10m2 - 33m - 7 } =

- [ 10·{ 10m2 - 33m - 7 } / 10 ] = - [ { (10m)2 - 33·(10m) - 70 } / 10 ] = \\\\\\

70 2 70 = 2·5·7

35 5 70 = 10·7

7 7 70 = 35·2

1

\\\\\\ = - [ { (10m)2 + (2 - 35)·(10m) + (2)·(-35) } / 10 ] =

- [ { (10m + 2)·(10m - 35) } / (2·5) ] =

- [ { (10m + 2) / 2} · { (10m - 35) / 5 } ] =

- [ { 5m + 1}·{2m - 7} ] = - (5m + 1)·(2m - 7) = (5m + 1)·(7 - 2m)

= ( 1 + 5m ) · ( 7 - 2m )

129.) 4 ( a + b )2 - 9 ( c + d )2 = { 2(a + b) } 2 - { 3(c + d) } 2 =

{ 2(a + b) + 3(c + d) } · { 2(a + b) - 3(c + d) } =

( 2a + 2b + 3c + 3d ) · ( 2a +2b - 3a - 3d )

130.) 729 - 125x3y12 = 93 - (5xy4)3 = { 9 - 5xy4 } · { 92 + 9·5xy4 + (5xy4)2 } =

( 9 - 5xy4 ) · ( 81 + 45xy4 + 25x2y8 )

131.) (x + y)2 + x + y = (x + y)·(x + y) + 1·(x + y) = ( x + y ) · ( x + y + 1 )

132.) 4 - ( a2 + b2 ) + 2ab = 4 - a2 - b2 + 2ab = 4 - a2 + 2ab - b2 =

4 - ( a2 - 2ab + b2 ) = 22 - (a - b)2 = {2 + (a - b) } · {2 - (a - b) } =

( 2 + a - b ) · ( 2 - a + b )

133.) x3 - y3 + x - y = (x - y)·(x2 + xy + y2) + 1·(x - y) =( x - y ) · ( x2 + xy + y2 + 1 )

134.) a2 - b2 + a3 - b3 = (a + b)·(a - b) + (a - b)·(a2 + ab + b2) =

( a - b ) · ( a2 + ab + b2 + a + b )

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