Ejercicio 106

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CAPITULO X

Descomposición Factorial
Ejercicio 106
MISCELANIA SOBRE LOS 10 CASOS DE DESCOMPOSICION EN FACTORES
Descomponer en factores:
  1. 5 a 2 +a=a(5a+1 )
  2. m 2 +2mx+ x 2 = (m+x ) 2
  3. a 2 +aabb =a(a+1 ) b(a+1 ) =(a+1 ) (ab )
  4. x 2 36=(x6 ) (x+6 )
  5. 9 x 2 6xy+ y 2 = (3xy ) 2
  6. x 2 3x4 = x 2 +x4x4 . 4 1 | 4 =x(x+1 ) 4(x+1 ) =(x+1 ) (x4 )
  7. 6 x 2 x2 =6 x 2 +3x4x2 . 12 6 3 1 | 2 2 3 =3x(2x+1 ) 2(2x+1 ) =(2x+1 ) (3x2 )
  8. 1+ x 3 =(1+x ) (1x+ x 2 )
  9. 27 a 3 1 =(3a1 ) [ (3a ) 2 +3a+1 ] =(3a1 ) (9 a 2 +3a+1 )
  10. x 5 + m 5 =(x+m ) ( x 4 x 3 m+ x 2 m 2 x m 3 + m 4 )
  11. a 3 3 a 2 b+5a b 2 =a( a 2 3ab+5 b 2 )
  12. 2xy6y+xz3z =2y(x3 ) +z(x3 ) =(x3 ) (2y+z )
  13. 14b+4 b 2 = (12b ) 2
  14. 4 x 4 +3 x 2 y 2 + y 4 =4 x 4 +3 x 2 y 2 + y 4 + x 2 y 2 x 2 y 2 =4 x 4 +4 x 2 y 2 + y 4 x 2 y 2 = (2 x 2 + y 2 ) 2 x 2 y 2 =[(2 x 2 + y 2 ) +xy ] [(2 x 2 + y 2 ) xy ] =(2 x 2 +xy+ y 2 ) (2 x 2 xy+ y 2 )
  15. x 8 6 x 4 y 4 + y 8 = x 8 6 x 4 y 4 + y 8 +4 x 4 y 4 4 x 4 y 4 = x 8 2 x 4 y 4 + y 8 4 x 4 y 4 = ( x 4 y 4 ) 2 4 x 4 y 4 =[( x 4 y 4 ) 2 x 2 y 2 ] [( x 4 y 4 ) +2 x 2 y 2 ] =( x 4 2 x 2 y 2 y 4 ) ( x 4 +2 x 2 y 2 y 4 )
  16. a 2 a30 = a 2 6a+5a30 . 30 15 5 1 | 2 3 5 =a(a6 ) +5(a6 ) =(a6 ) (a+5 )
  17. 15 m 2 +11m14 =15 m 2 10m+21m14 . 210 105 35 7 1 | 2 3 5 7 =5m(3m2 ) +7(3m2 ) =(3m2 ) (5m+7 )
  18. a 6 +1 =( a 2 +1 ) [ ( a 2 ) 2 a 2 +1 ] =( a 2 +1 ) ( a 4 a 2 +1 )
  19. 8 m 3 27 y 6 =(2m3 y 2 ) [ (2m ) 2 +(2m ) (3 y 2 ) + (3 y 2 ) 2 ] =(2m3 y 2 ) (4 m 2 +6m y 2 +9 y 4 )
  20. 16 a 2 24ab+9 b 2 = (4a3b ) 2
  21. 1+ a 7 =(1+a ) (1a+ a 2 a 3 + a 4 a 5 + a 6 )
  22. 8 a 3 12 a 2 +6a1 =(8 a 3 1 ) (12 a 2 6a ) =(2a1 ) [ (2a ) 2 +2a+1 ] 6a(2a1 ) =(2a1 ) (4 a 2 +2a+1 ) 6a(2a1 ) =(2a1 ) [(4 a 2 +2a+1 ) 6a ] =(2a1 ) [4 a 2 +2a+16a ] =(2a1 ) (4 a 2 4a+1 ) =(2a1 ) (2a1 ) 2 = (2a1 ) 3
  23. 1 m 2 =(1m ) (1+m )
  24. x 4 +4 x 2 21 = x 4 3 x 2 +7 x 2 21 . 21 7 1 | 3 7 = x 2 ( x 2 3 ) +7( x 2 3 ) =( x 2 3 ) ( x 2 +7 )
  25. 125 a 6 +1 =(5 a 2 +1 ) [ (5 a 2 ) 2 +5 a 2 +1 ] =(5 a 2 +1 ) (25 a 4 +5 a 2 +1 )
  26. a 2 +2ab+ b 2 m 2 = (a+b ) 2 m 2 =[(a+b ) m ] [(a+b ) +m ] =(a+bm ) (a+b+m )
  27. 8 a 2 b+16 a 3 b24 a 2 b 2 =8 a 2 b(1+2a3b )
  28. x 5 x 4 +x1 = x 4 (x1 ) +(x1 ) =(x1 ) ( x 4 +1 )
  29. 6 x 2 +19x20 =6 x 2 +24x5x20 . 120 60 30 15 5 1 | 2 2 2 3 5 =6x(x+4 ) 5(x+4 ) =(x+4 ) (6x5 )
  30. 25 x 4 81 y 2 =(5 x 2 +9y ) (5 x 2 9y )
  31. 1 m 3 =(1m ) (1+m+ m 2 )
  32. x 2 a 2 +2xy+ y 2 +2ab b 2 =( x 2 +2xy+ y 2 ) ( a 2 2ab+ b 2 ) = (x+y ) 2 (ab ) 2 =[(x+y ) +(ab ) ] [(x+y ) (ab ) ] =(x+y+ab ) (x+ya+b )
  33. 21 m 5 n7 m 4 n 2 +7 m 3 n 3 7 m 2 n=7 m 2 n(3 m 3 m 2 n+m n 2 1 )
  34. a(x+1 ) b(x+1 ) +c(x+1 ) =(x+1 ) (ab+c )
  35. 4+4(xy ) + (xy ) 2 = [2+(xy ) ] 2 = (xy+2 ) 2
  36. 1 a 2 b 4 =(1+a b 2 ) (1a b 2 )
  37. b 2 +12ab+36 a 2 = (b+6a ) 2
  38. x 6 +4 x 3 77 = x 6 7 x 3 +11 x 3 77 . 77 11 1 | 7 11 = x 3 ( x 3 7 ) +11( x 3 7 ) =( x 3 7 ) ( x 3 +11 )
  39. 15 x 4 17 x 2 4 =15 x 4 20 x 2 +3 x 2 4 . 60 30 15 5 1 | 2 2 3 5 =5 x 2 (3 x 2 4 ) +(3 x 2 4 ) =(3 x 2 4 ) (5 x 2 +1 )
  40. 1+ (a3b ) 3 =[1+(a3b ) ] [1(a3b ) + (a3b ) 2 ] =(a3b+1 ) (1a+3b+ a 2 6ab+9 b 2 )
  41. x 4 + x 2 +25 = x 4 + x 2 +25+9 x 2 9 x 2 = x 4 +10 x 2 +259 x 2 = ( x 2 +5 ) 2 9 x 2 =[( x 2 +5 ) 3x ] [( x 2 +5 ) +3x ] =( x 2 3x+5 ) ( x 2 +3x+5 )
  42. a 8 28 a 4 +36 = a 8 28 a 4 +36+16 a 4 16 a 4 = a 8 12 a 4 +3616 a 4 = ( a 4 6 ) 2 16 a 4 =[( a 4 6 ) +4 a 2 ] [( a 4 6 ) 4 a 2 ] =( a 4 +4 a 2 6 ) ( a 4 4 a 2 6 )
  43. 343+8 a 3 =(7+2a ) [ 7 2 7(2a ) + (2a ) 2 ] =(7+2a ) (4914a+4 a 2 )
  44. 12 a 2 bx15 a 2 by=3 a 2 b(4x5y )
  45. x 2 +2xy15 y 2 = x 2 3xy+5xy15 y 2 . 15 5 1 | 3 5 =x(x3y ) +5y(x3y ) =(x3y ) (x+5y )
  46. 6am4an2n+3m =6am+3m4an2n =3m(2a+1 ) 2n(2a+1 ) =(2a+1 ) (3m2n )
  47. 81 a 6 4 b 2 c 8 =(9 a 3 2b c 4 ) (9 a 3 +2b c 4 )
  48. 16 (2a+b ) 2 =[4(2a+b ) ] [4+(2a+b ) ] =(42a+b ) (4+2a+b )
  49. 20x x 2 =20+4x5x x 2 . 20 10 5 1 | 2 2 5 =4(5+x ) 5x(5+x ) =(5+x ) (45x )
  50. n 2 +n42 = n 2 6n+7n42 . 42 21 7 1 | 2 3 7 =n(n6 ) +7(n6 ) =(n6 ) (n+7 )
  51. a 2 d 2 + n 2 c 2 2an2cd =( a 2 2an+ n 2 ) ( d 2 +2cd+ c 2 ) = (an ) 2 (d+c ) 2 =[(an ) +(d+c ) ] [(an ) (d+c ) ] =(an+d+c ) (andc )
  52. 1+216 x 9 =(1+6 x 3 ) [16 x 3 + (6 x 3 ) 2 ] =(1+6 x 3 ) (16 x 3 +36 x 6 )
  53. x 3 64=(x4 ) ( x 2 +4x+16 )
  54. x 3 64 x 4 = x 3 (164x )
  55. 18a x 5 y 3 36 x 4 y 3 54 x 2 y 8 =18 x 2 y 3 (a x 3 2 x 2 3 y 5 )
  56. 49 a 2 b 2 14ab+1= (7ab1 ) 2
  57. (x+1 ) 2 81 =[(x+1 ) 9 ] [(x+1 ) +9 ] =(x+19 ) (x+1+9 ) =(x8 ) (x+10 )
  58. a 2 (b+c ) 2 =[a+(b+c ) ] [a(b+c ) ] =(a+b+c ) (abc )
  59. (m+n ) 2 6(m+n ) +9 = [(m+n ) 3 ] 2 = (m+n3 ) 2
  60. 7 x 2 +31x20 =7 x 2 +35x4x20 . 140 70 35 1 | 2 2 35 =7x(x+5 ) 4(x+5 ) =(x+5 ) (7x4 )
  61. 9 a 3 +63a45 a 2 =9 a 3 45 a 2 +63a =9a( a 2 5a+7 )
  62. ax+ax1 =a(x+1 ) (x+1 ) =(x+1 ) (a1 )
  63. 81 x 4 +25 y 2 90 x 2 y =81 x 4 90 x 2 y+25 y 2 = (9 x 2 5y ) 2
  64. 127 b 2 + b 4 =127 b 2 + b 4 +25 b 2 25 b 2 =12 b 2 + b 4 25 b 2 = (1 b 2 ) 2 25 b 2 =[(1 b 2 ) +5b ] [(1 b 2 ) 5b ] =(1+5b b 2 ) (15b b 2 )
  65. m 4 + m 2 n 2 + n 4 = m 4 + m 2 n 2 + n 4 + m 2 n 2 m 2 n 2 = m 4 +2 m 2 n 2 + n 4 m 2 n 2 = ( m 2 + n 2 ) 2 m 2 n 2 =[( m 2 + n 2 ) mn ] [( m 2 + n 2 ) +mn ] =( m 2 mn+ n 2 ) ( m 2 +mn+ n 2 )
  66. c 4 4 d 4 =( c 2 2 d 2 ) ( c 2 +2 d 2 )
  67. 15 x 4 15 x 3 +20 x 2 =5 x 2 (3 x 2 3x+4 )
  68. a 2 x 2 ax =(a+x ) (ax ) ax =(a+x ) (ax ) (a+x ) =(a+x ) [(ax ) 1 ] =(a+x ) (ax1 )
  69. x 4 8 x 2 240 = x 4 20 x 2 +12 x 2 240 . 240 120 60 30 15 5 1 | 2 2 2 2 3 5 = x 2 ( x 2 20 ) +12( x 2 20 ) =( x 2 20 ) ( x 2 +12 )
  70. 6 m 4 +7 m 2 20 =6 m 4 8 m 2 +15 m 2 20 . 120 60 30 15 1 | 2 2 2 15 =2 m 2 (3 m 2 4 ) +5(3 m 2 4 ) =(3 m 2 4 ) (2 m 2 +5 )
  71. 9 n 2 +4 a 2 12an =9 n 2 12an+4 a 2 = (3n2a ) 2
  72. 2 x 2 +2=2( x 2 +1 )
  73. 7a(x+y1 ) 3b(x+y1 ) =(x+y1 ) (7a3b )
  74. x 2 +3x18 = x 2 +6x3x18 . 18 9 3 1 | 2 3 3 =x(x+6 ) 3(x+6 ) =(x+6 ) (x3 )
  75. (a+m ) 2 (b+n ) 2 =[(a+m ) +(b+n ) ] [(a+m ) (b+n ) ] =(a+b+m+n ) (ab+mn )
  76. x 3 +6 x 2 y+12x y 2 +8 y 3 = (x+2y ) 3
  77. 8 a 2 22a21 =8 a 2 +6a28a21 . 168 84 42 21 7 1 | 2 2 2 3 7 =2a(4a+3 ) 7(4a+3 ) =(4a+3 ) (2a7 )
  78. 1+18ab+81 a 2 b 2 = (1+9ab ) 2
  79. 4 a 6 1=(2 a 3 1 ) (2 a 3 +1 )
  80. x 6 4 x 3 480 = x 6 24 x 3 +20 x 3 480 . 480 240 120 60 30 15 5 1 | 2 2 2 2 2 3 5 = x 3 ( x 3 24 ) +20( x 3 24 ) =( x 3 24 ) ( x 3 +20 )
  81. axbx+baby+ay =ax+ayabxby+b =a(x+y1 ) b(x+y1 ) =(x+y1 ) (ab )
  82. 6am3m2a+1 =3m(2a1 ) (2a1 ) =(2a1 ) (3m1 )
  83. 15+14x8 x 2 =15+20x6x8 x 2 . 120 60 30 15 5 1 | 2 2 2 3 5 =5(3+4x ) 2x(3+4x ) =(3+4x ) (52x )
  84. a 10 a 8 + a 6 + a 4 = a 4 ( a 6 a 4 + a 2 +1 )
  85. 2x(a1 ) a+1 =2x(a1 ) (a1 ) =(a1 ) (2x1 )
  86. (m+n ) (mn ) +3n(mn ) =(mn ) [(m+n ) +3n ] =(mn ) (m+n+3n ) =(mn ) (m+4n )
  87. a 2 b 3 +2 b 3 x 2 2 a 2 x 2 = a 2 b 3 2 a 2 x 2 +2 b 3 x 2 = a 2 b 3 2 x 2 ( a 2 b 3 ) =( a 2 b 3 ) 2 x 2 ( a 2 b 3 ) =( a 2 b 3 ) (12 x 2 )
  88. 2am3bccm3bm+2a =2am3bmcm+2a3bc =m(2a3bc ) +2a3bc =m(2a3bc ) +(2a3bc ) =(2a3bc ) (m+1 )
  89. x 2 2 3 x+ 1 9 = (x 1 3 ) 2
  90. 4 a 2n b 4n =(2 a n + b 2n ) (2 a n b 2n )
  91. 81 x 2 (a+x ) 2 =[9x+(a+x ) ] [9x(a+x ) ] =[9x+a+x ] [9xax ] =(10x+a ) (8xa )
  92. a 2 +96a16 x 2 = a 2 6a+916 x 2 = (a3 ) 2 16 x 2 =[(a3 ) +4x ] [(a3 ) 4x ] =(a+4x3 ) (a4x3 )
  93. 9 a 2 x 2 4+4x =9 a 2 x 2 +4x4 =9 a 2 ( x 2 4x+4 ) =9 a 2 (x2 ) 2 =[3a+(x2 ) ] [3a(x2 ) ] =(3a+x2 ) (3ax+2 )
  94. 9 x 2 y 2 +3xy =(3xy ) (3x+y ) +(3xy ) =(3xy ) [(3x+y ) +1 ] =(3xy ) (3x+y+1 )
  95. x 2 x72 = x 2 9x+8x72 . 72 36 18 9 1 | 2 2 2 9 =x(x9 ) +8(x9 ) =(x9 ) (x+8 )
  96. 36 a 4 120 a 2 b 2 +49 b 4 =36 a 4 120 a 2 b 2 +49 b 4 +36 a 2 b 2 36 a 2 b 2 =36 a 4 84 a 2 b 2 +49 b 4 36 a 2 b 2 = (6 a 2 7 b 2 ) 2 36 a 2 b 2 =[(6 a 2 7 b 2 ) +6ab ] [(6 a 2 7 b 2 ) 6ab ] =(6 a 2 +6ab7 b 2 ) (6 a 2 6ab7 b 2 )
  97. a 2 m 2 9 n 2 6mn+4ab+4 b 2 = a 2 +4ab+4 b 2 ( m 2 +6mn+9 n 2 ) = (a+2b ) 2 (m+3n ) 2 =[(a+2b ) +(m+3n ) ] [(a+2b ) (m+3n ) ] =(a+2b+m+3n ) (a+2bm3n )
  98. 1 4 9 a 8 =(1 2 3 a 4 ) (1+ 2 3 a 4 )
  99. 81 a 8 +64 b 12 =81 a 8 +64 b 12 +144 a 4 b 6 144 a 4 b 6 =81 a 8 +144 a 4 b 6 +64 b 12 144 a 4 b 6 = (9 a 4 +8 b 6 ) 2 144 a 4 b 6 =[(9 a 4 +8 b 6 ) 12 a 2 b 3 ] [(9 a 4 +8 b 6 ) +12 a 2 b 3 ] =(9 a 4 12 a 2 b 3 +8 b 6 ) (9 a 4 +12 a 2 b 3 +8 b 6 )
  100. 49 x 2 77x+30 =49 x 2 42x35x+30 . 1470 735 245 49 7 1 | 2 3 5 7 7 =7x(7x6 ) 5(7x6 ) =(7x6 ) (7x5 )
  101. x 2 2abx35 a 2 b 2 = x 2 7abx+5abx35 a 2 b 2 . 35 7 1 | 5 7 =x(x7ab ) +5ab(x7ab ) =(x7ab ) (x+5ab )
  102. 125 x 3 225 x 2 +135x27= (5x3 ) 3
  103. (a2 ) 2 (a+3 ) 2 =[(a2 ) +(a+3 ) ] [(a2 ) (a+3 ) ] =[a2+a+3 ] [ a 2 a 3 ] =5(2a+1 )
  104. 4 a 2 m+12 a 2 n5bm15bn =4 a 2 (m+3n ) 5b(m+3n ) =(m+3n ) (4 a 2 5b )
  105. 1+6 x 3 +9 x 6 = (1+3 x 3 ) 2
  106. a 4 +3 a 2 b40 b 2 = a 4 +8 a 2 b5 a 2 b40 b 2 . 40 20 10 5 1 | 2 2 2 5 = a 2 ( a 2 +8b ) 5b( a 2 +8b ) =( a 2 +8b ) ( a 2 5b )
  107. m 3 +8 a 3 x 3 =(m+2ax ) ( m 2 2amx+4 a 2 x 2 )
  108. 19 x 2 +24xy16 y 2 =1(9 x 2 24xy+16 y 2 ) =1 (3x4y ) 2 =[1(3x4y ) ] [1+(3x4y ) ] =(13x+4y ) (1+3x4y )
  109. 1+11x+24 x 2 =1+3x+8x+24 x 2 . 24 12 6 3 1 | 2 2 2 3 =(1+3x ) +8x(1+3x ) =(1+3x ) (1+8x )
  110. 9 x 2 y 3 27 x 3 y 3 9 x 5 y 3 =9 x 2 y 3 (13x x 2 )
  111. ( a 2 + b 2 c 2 ) 2 9 x 2 y 2 =[( a 2 + b 2 c 2 ) 3xy ] [( a 2 + b 2 c 2 ) +3xy ] =( a 2 + b 2 c 2 3xy ) ( a 2 + b 2 c 2 +3xy )
  112. 8 (a+1 ) 3 1 =[2(a+1 ) 1 ] { [2(a+1 ) ] 2 +2(a+1 ) +1 } =[2a+21 ] {4 (a+1 ) 2 +2a+2+1 } =(2a+1 ) {4( a 2 +2a+1 ) +2a+3 } =(2a+1 ) {4 a 2 +8a+4+2a+3 } =(2a+1 ) (4 a 2 +10a+7 )
  113. 100 x 4 y 6 121 m 4 =(10 x 2 y 3 11 m 2 ) (10 x 2 y 3 +11 m 2 )
  114. ( a 2 +1 ) 2 +5( a 2 +1 ) 24 = ( a 2 +1 ) 2 3( a 2 +1 ) +8( a 2 +1 ) 24 . 24 12 6 3 1 | 2 2 2 3 =( a 2 +1 ) [( a 2 +1 ) 3 ] +8[( a 2 +1 ) 3 ] =[( a 2 +1 ) 3 ] [( a 2 +1 ) +8 ] =[ a 2 +13 ] [ a 2 +1+8 ] =( a 2 2 ) ( a 2 +9 )
  115. 1+1000 x 6 =(1+10 x 2 ) (110 x 2 + [10 x 2 ] 2 ) =(1+10 x 2 ) (110 x 2 +100 x 4 )
  116. 49 a 2 x 2 9 y 2 +6xy =49 a 2 x 2 +6xy9 y 2 =49 a 2 ( x 2 6xy+9 y 2 ) =49 a 2 (x3y ) 2 =[7a(x3y ) ] [7a+(x3y ) ] =(7ax+3y ) (7a+x3y )
  117. x 4 y 2 +4 x 2 +44yz4 z 2 = x 4 +4 x 2 +4 y 2 4yz4 z 2 =( x 4 +4 x 2 +4 ) ( y 2 +4yz+4 z 2 ) = ( x 2 +2 ) 2 (y+2z ) 2 =[( x 2 +2 ) (y+2z ) ] [( x 2 +2 ) +(y+2z ) ] =[ x 2 +2y2z ] [ x 2 +2+y+2z ] =( x 2 y2z+2 ) ( x 2 +y+2z+2 )
  118. a 3 64 =(a4 ) ( a 2 +4a+ 4 2 ) =(a4 ) ( a 2 +4a+16 )
  119. a 5 + x 5 =(a+x ) ( a 4 a 3 x+ a 2 x 2 a x 3 + x 4 )
  120. a 6 3 a 3 b54 b 2 = a 6 +6 a 3 b9 a 3 b54 b 2 . 54 27 9 1 | 2 3 9 = a 3 ( a 3 +6b ) 9b( a 3 +6b ) =( a 3 +6b ) ( a 3 9b )
  121. 165+4x x 2 =16511x+15x x 2 . 165 55 11 1 | 3 5 11 =11(15x ) +x(15x ) =(15x ) (11+x )
  122. a 4 + a 2 +1 = a 4 + a 2 +1+ a 2 a 2 = a 4 +2 a 2 +1 a 2 = ( a 2 +1 ) 2 a 2 =[( a 2 +1 ) a ] [( a 2 +1 ) +a ] =( a 2 a+1 ) ( a 2 +a+1 )
  123. x 2 4 y 6 81 =( x 2 y 3 9 ) ( x 2 + y 3 9 )
  124. 16 x 2 + 8xy 5 + y 2 25 = (4x+ y 5 ) 2
  125. a 4 b 4 +4 a 2 b 2 96 = a 4 b 4 8 a 2 b 2 +12 a 2 b 2 96 . 96 48 24 12 1 | 2 2 2 12 = a 2 b 2 ( a 2 b 2 8 ) +12( a 2 b 2 8 ) =( a 2 b 2 8 ) ( a 2 b 2 +12 )
  126. 8 a 2 x+7y+21by7ay8 a 3 x+24 a 2 bx =8 a 2 x8 a 3 x+24 a 2 bx+7y7ay+21by =8 a 2 x(1a+3b ) +7y(1a+3b ) =(1a+3b ) (8 a 2 x+7y )
  127. x 4 +11 x 2 390 = x 4 15 x 2 +26 x 2 390 . 390 195 65 13 1 | 2 3 5 13 = x 2 ( x 2 15 ) +26( x 2 15 ) =( x 2 15 ) ( x 2 +26 )
  128. 7+33m10 m 2 =72m+35m10 m 2 . 70 35 1 | 2 35 =(72m ) +5m(72m ) =(72m ) (1+5m )
  129. 4 (a+b ) 2 9 (c+d ) 2 =[2(a+b ) 3(c+d ) ] [2(a+b ) +3(c+d ) ] =(2a+2b3c3d ) (2a+2b+3c+3d )
  130. 729125 x 3 y 12 =(95x y 4 ) [ 9 2 +9(5x y 4 ) + (5x y 4 ) 2 ] =(95x y 4 ) (81+45x y 4 +25 x 2 y 8 )
  131. (x+y ) 2 +x+y = (x+y ) 2 +(x+y ) =(x+y ) [(x+y ) +1 ] =(x+y ) (x+y+1 )
  132. 4( a 2 + b 2 ) +2ab =4 a 2 b 2 +2ab =4 a 2 +2ab b 2 =4( a 2 2ab+ b 2 ) =4 (ab ) 2 =[2(ab ) ] [2+(ab ) ] =(2a+b ) (2+ab )
  133. x 3 y 3 +xy =( x 3 y 3 ) +xy =(xy ) ( x 2 +xy+ y 2 ) +(xy ) =(xy ) [( x 2 +xy+ y 2 ) +1 ] =(xy ) ( x 2 +xy+ y 2 +1 )
  134. a 2 b 2 + a 3 b 3 =( a 2 b 2 ) +( a 3 b 3 ) =(ab ) (a+b ) +(ab ) ( a 2 +ab+ b 2 ) =(ab ) [(a+b ) +( a 2 +ab+ b 2 ) ] =(ab ) (a+b+ a 2 +ab+ b 2 )