Ejercicio 73

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 73
MISCELANEA
Escribir el cociente sin efectuar la división:
  1. x 4 1 1+ x 2 = x 2 1
  2. 8 m 3 + n 6 2m+ n 2 =4 m 2 2m n 2 + n 4
  3. 1 a 5 1a =1+a+ a 2 + a 3 + a 4
  4. x 6 27 y 3 x 2 3y = x 4 +3 x 2 y+9 y 2
  5. x 6 49 y 6 x 3 +7 y 3 = x 3 7 y 3
  6. a 14 b 14 a 2 b 2 = a 12 + a 10 b 2 + a 8 b 4 + a 6 b 6 + a 4 b 8 + a 2 b 10 + b 12
  7. 1+ a 3 1+a =1a+ a 2
  8. 16 x 2 y 4 25 m 6 4x y 2 +5 m 3 =4x y 2 5 m 3
  9. x 27 + y 27 x 3 + y 3 = x 24 x 21 y 3 + x 18 y 6 x 15 y 9 + x 12 y 12 x 9 y 15 + x 6 y 18 x 3 y 21 + y 24
  10. a 27 + y 27 a 9 + y 9 = a 18 a 9 y 9 + y 18
  11. a 4 b 4 64 x 6 a 2 b 2 +8 x 3 = a 2 b 2 8 x 3
  12. 1 a 2 b 4 c 8 1a b 2 c 4 =1+a b 2 c 4
  13. 32 x 5 +243 y 5 2x+3y =16 x 4 24 x 3 y+36 x 2 y 2 54x y 3 +81 y 4
  14. 25 (a+1 ) 2 5+(a+1 ) =4a
  15. 1 x 12 1 x 4 =1+ x 4 + x 8
  16. 64 x 6 343 y 9 4 x 2 7 y 3 =16 x 4 +28 x 2 y 3 +49 y 6
  17. a 18 b 18 a 3 + b 3 = a 15 a 12 b 3 + a 9 b 6 a 6 b 9 + a 3 b 12 b 15
  18. (a+x ) 2 y 2 (a+x ) y =a+x+y
  19. 1+ x 11 x+1 = x 10 x 9 + x 8 x 7 + x 6 x 5 + x 4 x 3 + x 2 x+1
  20. x 40 y 40 x 8 y 8 = x 32 + x 24 y 8 + x 16 y 16 + x 8 y 24 + y 32
  21. 936 x 10 3+6 x 5 =36 x 5
  22. x 8 256 x2 = x 7 +2 x 6 +4 x 5 +8 x 4 +16 x 3 +32 x 2 +64x+128

Ejercicio 72

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 72
Hallar, por simple inspección, el cociente de:
  1. x 6 + y 6 x 2 + y 2 = x 4 x 2 y 2 + y 4
  2. a 8 b 8 a 2 + b 2 = a 6 a 4 b 2 + a 2 b 4 b 6
  3. m 10 n 10 m 2 n 2 = m 8 + m 6 n 2 + m 4 n 4 + m 2 n 6 + n 8
  4. a 12 b 12 a 3 + b 3 = a 9 a 6 b 3 + a 3 b 6 b 9
  5. a 12 x 12 a 3 x 3 = a 9 + a 6 x 3 + a 3 x 6 + x 9
  6. x 15 + y 15 x 3 + y 3 = x 12 x 9 y 3 + x 6 y 6 x 3 y 9 + y 12
  7. m 12 +1 m 4 +1 = m 8 m 4 +1
  8. m 16 n 16 m 4 n 4 = m 12 + m 8 n 4 + m 4 n 8 + n 12
  9. a 18 b 18 a 3 + b 3 = a 15 a 12 b 3 + a 9 b 6 a 6 b 9 + a 3 b 12 b 15
  10. x 20 y 20 x 5 + y 5 = x 15 x 10 y 5 + x 5 y 10 y 15
  11. m 21 + n 21 m 3 + n 3 = m 18 m 15 n 3 + m 12 n 6 m 9 n 9 + m 6 n 12 m 3 n 15 + n 18
  12. x 24 1 x 6 1 = x 18 + x 12 + x 6 +1
  13. a 25 + b 25 a 5 + b 5 = a 20 a 15 b 5 + a 10 b 10 a 5 b 15 + b 20
  14. a 30 m 30 a 6 m 6 = a 24 + a 18 m 6 + a 12 m 12 + a 6 m 18 + m 24

Ejercicio 71

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 71
Hallar, por simple inspección, el cociente de:
  1. x 4 y 4 xy = x 3 + x 2 y+x y 2 + y 3
  2. m 5 + n 5 m+n = m 4 m 3 n+ m 2 n 2 m n 3 + n 4
  3. a 5 n 5 an = a 4 + a 3 n+ a 2 n 2 +a n 3 + n 4
  4. x 6 y 6 x+y = x 5 x 4 y+ x 3 y 2 x 2 y 3 +x y 4 y 5
  5. a 6 b 6 ab = a 5 + a 4 b+ a 3 b 2 + a 2 b 3 +a b 4 + b 5
  6. x 7 + y 7 x+y = x 6 x 5 y+ x 4 y 2 x 3 y 3 + x 2 y 4 x y 5 + y 6
  7. a 7 m 7 am = a 6 +a m 5 + a 2 m 4 + a 3 m 3 + a 2 m 4 +a m 5 + m 6
  8. a 8 b 8 a+b = a 7 a 6 b+ a 5 b 2 a 4 b 3 + a 3 b 4 a 2 b 5 +a b 6 b 7
  9. x 10 y 10 xy = x 9 + x 8 y+ x 7 y 2 + x 6 y 3 + x 5 y 4 + x 4 y 5 + x 3 y 6 + x 2 y 7 +x y 8 + y 9
  10. m 9 + n 9 m+n = m 8 m 7 n+ m 6 n 2 m 5 n 3 + m 4 n 4 m 3 n 5 + m 2 n 6 m n 7 + n 8
  11. m 9 n 9 mn = m 8 + m 7 n+ m 6 n 2 + m 5 n 3 + m 4 n 4 + m 3 n 5 + m 2 n 6 +m n 7 + n 8
  12. a 10 x 10 a+x = a 9 a 8 x+ a 7 x 2 a 6 x 3 + a 5 x 4 a 4 x 5 + a 3 x 6 a 2 x 7 +a x 8 x 9
  13. 1 n 5 1n =1+n+ n 2 + n 3 + n 4
  14. 1 a 6 1a =1+a+ a 2 + a 3 + a 5
  15. 1+ a 7 1+a =1a+ a 2 a 3 + a 4 a 5 + a 6
  16. 1 m 8 1+m =1m+ m 2 m 3 + m 4 m 5 + m 6 m 7
  17. x 4 16 x2 = x 3 +2 x 2 +4x+8
  18. x 6 64 x+2 = x 5 2 x 4 +4 x 3 8 x 2 +16x32
  19. x 7 128 x2 = x 6 +2 x 5 +4 x 4 +8 x 3 +16 x 2 +32x+64
  20. a 5 +243 a+3 = a 4 3 a 3 +9 a 2 27a+81
  21. x 6 729 x3 = x 5 +3 x 4 +9 x 3 +27 x 2 +81x+243
  22. 625 x 4 x+5 =12525x+5 x 2 x 3
  23. m 8 256 m2 = m 7 +2 m 6 +4 m 5 +8 m 4 +16 m 3 +32 m 2 +64m+128
  24. x 10 1 x1 = x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 +x+1
  25. x 5 +243 y 5 x+3y = x 4 3 x 3 y+9 x 2 y 2 27x y 3 +81 y 4
  26. 16 a 4 81 b 4 2a3b =8 a 3 +12 a 2 b+18a b 2 +27 b 3
  27. 64 m 6 729 n 6 2m+3n =32 m 5 48 m 4 n+72 m 3 n 2 108 m 2 n 3 +162m n 4 243 n 5
  28. 1024 x 10 1 2x1 =512 x 9 +256 x 8 +128 x 7 +64 x 6 +32 x 5 +16 x 4 +8 x 3 +4 x 2 +2x+1
  29. 512 a 9 + b 9 2a+b =256 a 8 128 a 7 b+64 a 6 b 2 32 a 5 b 3 +16 a 4 b 4 8 a 3 b 5 +4 a 2 b 6 2a b 7 + b 8
  30. a 6 729 a3 = a 5 +3 a 4 +9 a 3 +27 a 2 +81a+243

Ejercicio 70

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 70
Hallar, por simple inspección, el cociente de:
  1. 1+ a 3 1+a =1a+ a 2
  2. 1 a 3 1a =1+a+ a 2
  3. x 3 + y 3 x+y = x 2 xy+ y 2
  4. 8 a 3 1 2a1 =4 a 2 +2a+1
  5. 8 x 3 +27 y 3 2x+3y =4 x 2 6xy+9 y 2
  6. 27 m 3 125 n 3 3m5n =9 m 2 +15m+25 n 2
  7. 64 a 3 +343 4a+7 =16 a 2 28a+49
  8. 216125 y 3 65y =36+30y+25 y 2
  9. 1+ a 3 b 3 1+ab =1ab+ a 2 b 2
  10. 729512 b 3 98b =81+72b+64 b 2
  11. a 3 x 3 + b 3 ax+b = a 2 x 2 axb+ b 2
  12. n 3 m 3 x 3 nmx = n 2 +nmx+ m 2 x 2
  13. x 6 27 y 3 x 2 3y = x 4 +3 x 2 y+9 y 2
  14. 8 a 9 + y 9 2 a 3 + y 3 =4 a 6 2 a 3 y 3 + y 6
  15. 1 x 12 1 x 4 =1+ x 4 + x 8
  16. 27 x 6 +1 3 x 2 +1 =9 x 4 3 x 2 +1
  17. 64 a 3 + b 9 4a+ b 3 =16 a 2 4a b 3 + b 6
  18. a 6 b 6 a 2 b 2 = a 4 + a 2 b 2 + b 4
  19. 125343 x 15 57 x 5 =25+35 x 5 +49 x 10
  20. n 6 +1 n 2 +1 = n 4 n 2 +1

Ejercicio 69

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 69
Escribir por simple inspección el resultado de:
  1. x 2 1 x+1 =x1
  2. 1 x 2 1x =1+x
  3. x 2 y 2 x+y =xy
  4. y 2 x 2 yx =y+x
  5. x 2 4 x+2 =x2
  6. 9 x 4 3 x 2 =3+ x 2
  7. a 2 4 b 2 a+2b =a2b
  8. 2536 x 4 56 x 2 =5+6 x 2
  9. 4 x 2 9 m 2 n 4 2x+3m n 2 =2x3m n 2
  10. 36 m 2 49 n 2 x 4 6m7n x 2 =6m+7n x 2
  11. 81 a 6 100 b 8 9 a 3 +10 b 4 =9 a 3 10 b 4
  12. a 4 b 6 4 x 8 y 10 a 2 b 3 +2 x 4 y 5 = a 2 b 3 2 x 4 y 5
  13. x 2n y 2n x n + y n = x n y n
  14. a 2x+2 100 a x+1 10 = a x+1 +10
  15. 19 x 2m+4 1+3 x m+2 =1+3 x m+2
  16. (x+y ) 2 z 2 (x+y ) z =(x+y ) +z=x+y+z
  17. 1 (a+b ) 2 1+(a+b ) =1(a+b ) =1ab
  18. 4 (m+n ) 2 2+(m+n ) =2(m+n ) =2mn
  19. x 2 (xy ) 2 x+(x+y ) =x(x+y ) =y
  20. (a+x ) 2 9 (a+x ) +3 =(a+x ) 3=a+x3

Ejercicio 68

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 68
Miscelanea
Escribir, por simple inspección, el resultado de:
  1. (x+2 ) 2 = ( x ) 2 +2( x ) ( 2 ) + ( 2 ) 2 = x 2 +4x+4
  2. (x+2 ) (x+3 ) = ( x ) 2 +(2+3 ) ( x ) +(2 × 3 ) = x 2 +5x+6
  3. (x+1 ) (x1 ) = ( x ) 2 ( 1 ) 2 = x 2 1
  4. (x1 ) 2 = ( x ) 2 2( x ) ( 1 ) + ( 1 ) 2 = x 2 2x+1
  5. (n+3 ) (n+5 ) = ( n ) 2 +(3+5 ) ( n ) +[3 × 5 ] = n 2 +8n+15
  6. (m3 ) (m+3 ) = ( m ) 2 ( 3 ) 2 = m 2 9
  7. (a+b+1 ) (a+b1 ) =[(a+b ) +1 ] [(a+b ) 1 ] = (a+b ) 2 ( 1 ) 2 = a 2 +2ab+ b 2 1
  8. (1+b ) 3 = ( 1 ) 3 +3 ( 1 ) 2 ( b ) +3( 1 ) ( b ) 2 + ( b ) 3 =1+3b+3 b 2 + b 3
  9. ( a 2 +4 ) ( a 2 4 ) = ( a 2 ) 2 ( 4 ) 2 = a 4 16
  10. (3ab5 x 2 ) 2 = (3ab ) 2 2(3ab ) (5 x 2 ) + (5 x 2 ) 2 =9 a 2 b 2 30ab x 2 +25 x 4
  11. (ab+3 ) (3ab ) =(3+ab ) (3ab ) = ( 3 ) 2 (ab ) 2 =9 a 2 b 2
  12. (14ax ) 2 = ( 1 ) 2 +2( 1 ) (4ax ) + (4ax ) 2 =18ax+16 a 2 x 2
  13. ( a 2 +8 ) ( a 2 7 ) = ( a 2 ) 2 +(87 ) ( a 2 ) +[8(7 ) ] = a 4 + a 2 56
  14. (x+y+1 ) (xy1 ) =[x+(y+1 ) ] [x(y+1 ) ] = ( x ) 2 (y+1 ) 2 = x 2 ( y 2 +2y+1 ) = x 2 y 2 2y1
  15. (1a ) (a+1 ) =(1a ) (1+a ) =1 a 2
  16. (m8 ) (m+12 ) = ( m ) 2 +(8+12 ) ( m ) +[(8 ) × 12 ] = m 2 +4m96
  17. ( x 2 1 ) ( x 2 +3 ) = ( x 2 ) 2 +(1+3 ) ( x 2 ) +[(1 ) × 3 ] = x 4 +2 x 2 3
  18. ( x 3 +6 ) ( x 3 8 ) = ( x 3 ) 2 +(68 ) ( x 3 ) +[6(8 ) ] = x 6 2 x 3 48
  19. (5 x 3 +6 m 4 ) 2 = (5 x 3 ) 2 +2(5 x 3 ) (6 m 4 ) + (6 m 4 ) 2 =25 x 6 +60 x 3 m 4 +36 m 8
  20. ( x 4 2 ) ( x 4 +5 ) = ( x 4 ) 2 +(2+5 ) ( x 4 ) +[(2 ) × 5 ] = x 8 +3 x 4 10
  21. (1a+b ) (ba1 ) =[(ba ) +1 ] [(ba ) 1 ] = (ba ) 2 ( 1 ) 2 = b 2 2ab+ a 2 1
  22. ( a x + b n ) ( a x b n ) = ( a x ) 2 ( b n ) 2 = a 2x b 2n
  23. ( x a+1 8 ) ( x a+1 +9 ) = ( x a+1 ) 2 +(8+9 ) ( x a+1 ) +[(8 ) × 9 ] = x 2a+2 + x a+1 72
  24. ( a 2 b 2 + c 2 ) ( a 2 b 2 c 2 ) = ( a 2 b 2 ) 2 ( c 2 ) 2 = a 4 b 2 c 4
  25. (2a+x ) 3 = (2a ) 3 +3 (2a ) 2 ( x ) +3(2a ) ( x ) 2 + ( x ) 3 =8 a 3 +12 a 2 x+6a x 2 + x 3
  26. ( x 2 11 ) ( x 2 2 ) = ( x 2 ) 2 +(112 ) ( x 2 ) +[(11 ) (2 ) ] = x 4 13 x 2 +22
  27. (2 a 3 5 b 4 ) 2 = (2 a 3 ) 2 +2(2 a 3 ) (5 b 4 ) + (5 b 4 ) 2 =4 a 6 20 a 3 b 4 +25 b 8
  28. ( a 3 +12 ) ( a 3 15 ) = ( a 3 ) 2 +(1215 ) ( a 3 ) +[12(15 ) ] = a 6 3 a 3 180
  29. ( m 2 m+n ) (n+m+ m 2 ) =[( m 2 +n ) m ] [( m 2 +n ) +m ] = ( m 2 +n ) 2 ( m ) 2 = m 4 +2mn+ n 2 m 2
  30. ( x 4 +7 ) ( x 4 11 ) = ( x 4 ) 2 +(711 ) ( x 4 ) +[7(11 ) ] = x 8 4 x 4 77
  31. (11ab ) 2 = ( 11 ) 2 +2( 11 ) (ab ) + (ab ) 2 =12122ab+ a 2 b 2
  32. ( x 2 y 3 8 ) ( x 2 y 3 +6 ) = ( x 2 y 3 ) 2 +(8+6 ) ( x 2 y 3 ) +[(8 ) × 6 ] = x 4 y 6 2 x 2 y 3 48
  33. (a+b ) (ab ) ( a 2 b 2 ) =( a 2 b 2 ) ( a 2 b 2 ) = ( a 2 b 2 ) 2 = ( a 2 ) 2 2( a 2 ) ( b 2 ) + ( b 2 ) 2 = a 4 2 a 2 b 2 + b 4
  34. (x+1 ) (x1 ) ( x 2 2 ) =( x 2 1 ) ( x 2 2 ) = ( x 2 ) 2 +(12 ) ( x 2 ) +[(1 ) (2 ) ] = x 4 3 x 2 +2
  35. (a+3 ) ( a 2 +9 ) (a3 ) =( a 2 9 ) ( a 2 +9 ) = ( a 2 ) 2 ( 9 ) 2 = a 4 81
  36. (x+5 ) (x5 ) ( x 2 +1 ) =( x 2 25 ) ( x 2 +1 ) = ( x 2 ) 2 +(25+1 ) ( x 2 ) +[(25 ) × 1 ] = x 4 24x25
  37. (a+1 ) (a1 ) (a+2 ) (a2 ) =( a 2 1 ) ( a 2 4 ) = ( a 2 ) 2 +(14 ) ( a 2 ) +[(1 ) (4 ) ] = a 4 5 a 2 +4
  38. (a+2 ) (a3 ) (a2 ) (a+3 ) =( a 2 4 ) ( a 2 9 ) = ( a 2 ) 2 +(49 ) ( a 2 ) +[(4 ) (9 ) ] = a 4 13 a 2 +36

Ejercicio 67

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 67
Escribir, por simple inspección, el resultado de:
  1. (a+1 ) (a+2 ) = ( a ) 2 +(1+2 ) ( a ) +(1 × 2 ) = a 2 +3a+2
  2. (x+2 ) (x+4 ) = ( x ) 2 +(2+4 ) ( x ) +(2 × 4 ) = x 2 +6x+8
  3. (x+5 ) (x2 ) = ( x ) 2 +(52 ) ( x ) +[5(2 ) ] = x 2 +3x10
  4. (m6 ) (m5 ) = ( m ) 2 +(65 ) ( m ) +[(6 ) (5 ) ] = m 2 11m+30
  5. (x+7 ) (x3 ) = ( x ) 2 +(73 ) ( x ) +[7(3 ) ] = x 2 +4x21
  6. (x+2 ) (x1 ) = ( x ) 2 +(21 ) ( x ) +[2(1 ) ] = x 2 +x2
  7. (x3 ) (x1 ) = ( x ) 2 +(31 ) ( x ) +[(3 ) (1 ) ] = x 2 4x+3
  8. (x5 ) (x+4 ) = ( x ) 2 +(5+4 ) ( x ) +[(5 ) × 4 ] = x 2 x20
  9. (a11 ) (a+10 ) = ( a ) 2 +(11+10 ) ( a ) +[(11 ) × 10 ] = a 2 a110
  10. (n19 ) (n+10 ) = ( n ) 2 +(19+10 ) ( n ) +[(19 ) × 10 ] = n 2 9n190
  11. ( a 2 +5 ) ( a 2 9 ) = ( a 2 ) 2 +(59 ) ( a 2 ) +[5(9 ) ] = a 4 4 a 2 45
  12. ( x 2 1 ) ( x 2 7 ) = ( x 2 ) 2 +(17 ) ( x 2 ) +[(1 ) (7 ) ] = x 4 8 x 2 +7
  13. ( n 2 1 ) ( n 2 +20 ) = ( n 2 ) 2 +(1+20 ) ( n 2 ) +[(1 ) × 20 ] = n 4 +19 n 2 20
  14. ( n 3 +3 ) ( n 3 6 ) = ( n 3 ) 2 +(+36 ) ( n 3 ) +[3(6 ) ] = n 6 3 n 3 18
  15. ( x 3 +7 ) ( x 3 6 ) = ( x 3 ) 2 +(76 ) ( x 3 ) +[7(6 ) ] = x 6 + x 3 42
  16. ( a 4 +8 ) ( a 4 1 ) = ( a 4 ) 2 +(81 ) ( a 4 ) +[8(1 ) ] = a 8 +7 a 4 8
  17. ( a 5 2 ) ( a 5 +7 ) = ( a 5 ) 2 +(2+7 ) ( a 5 ) +[(2 ) × 7 ] = a 10 +5 a 5 14
  18. ( a 6 +7 ) ( a 6 9 ) = ( a 6 ) 2 +(79 ) ( a 6 ) +[7(9 ) ] = a 12 2 a 6 63
  19. (ab+5 ) (ab6 ) = (ab ) 2 +(56 ) (ab ) +[5(6 ) ] = a 2 b 2 ab30
  20. (x y 2 9 ) (x y 2 +12 ) = (x y 2 ) 2 +(9+12 ) (x y 2 ) +[(9 ) × 12 ] = x 2 y 4 +3x y 2 108
  21. ( a 2 b 2 1 ) ( a 2 b 2 +7 ) = ( a 2 b 2 ) 2 +(1+7 ) ( a 2 b 2 ) +[(1 ) × 7 ] = a 4 b 4 +6 a 2 b 2 7
  22. ( x 3 y 3 6 ) ( x 3 y 3 +8 ) = ( x 3 y 3 ) 2 +(6+8 ) ( x 3 y 3 ) +[(6 ) × 8 ] = x 6 y 6 +2 x 3 y 3 48
  23. ( a x 3 ) ( a x +8 ) = ( a x ) 2 +(3+8 ) ( a x ) +[(3 ) × 8 ] = a 2x +5 a x 24
  24. ( a x+1 6 ) ( a x+1 5 ) = ( a x+1 ) 2 +(65 ) ( a x+1 ) +[(6 ) (5 ) ] = a 2x+2 11 a x+1 +30

Ejercicio 66

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 66
Desarrollar:
  1. (a+2 ) 3 = ( a ) 3 +3 ( a ) 2 ( 2 ) +3( a ) ( 2 ) 2 + ( 2 ) 3 = a 3 +6 a 2 +12a+8
  2. (x1 ) 3 = ( x ) 3 3 ( x ) 2 ( 1 ) +3( x ) ( 1 ) 2 ( 1 ) 3 = x 3 3 x 2 +3x1
  3. (m+3 ) 3 = ( m ) 3 +3 ( m ) 2 ( 3 ) +3( m ) ( 3 ) 2 + ( 3 ) 3 = m 3 +9 m 2 +27m+27
  4. (n4 ) 3 = ( n ) 3 3 ( n ) 2 ( 4 ) +3( n ) ( 4 ) 2 ( 4 ) 3 = n 3 12 n 2 +48n64
  5. (2x+1 ) 3 = (2x ) 3 +3 (2x ) 2 ( 1 ) +3(2x ) ( 1 ) 2 + ( 1 ) 3 =8 x 3 +12 x 2 +6x+1
  6. (13y ) 3 = ( 1 ) 3 3 ( 1 ) 2 (3y ) +3( 1 ) (3y ) 2 (3y ) 3 =19y+27 y 2 27 y 3
  7. (2+ y 2 ) 3 = ( 2 ) 3 +3 ( 2 ) 2 ( y 2 ) +3( 2 ) ( y 2 ) 2 + ( y 2 ) 3 =8+12 y 2 +6 y 4 + y 6
  8. (12n ) 3 = ( 1 ) 3 3 ( 1 ) 2 (2n ) +3( 1 ) (2n ) 2 (2n ) 3 =16n+12 n 2 8 n 3
  9. (4n+3 ) 3 = (4n ) 3 +3 (4n ) 2 ( 3 ) +3(4n ) ( 3 ) 2 + ( 3 ) 3 =64 n 3 +144 n 2 +108n+27
  10. ( a 2 2b ) 3 = ( a 2 ) 3 3 ( a 2 ) 2 (2b ) +3( a 2 ) (2b ) 2 (2b ) 3 = a 6 6 a 4 b+12 a 2 b 2 8 b 3
  11. (2x+3y ) 3 = (2x ) 3 +3 (2x ) 2 (3y ) +3(2x ) (3y ) 2 + (3y ) 3 =8 x 3 +36 x 2 y+54x y 2 +27 y 3
  12. (1 a 2 ) 3 = ( 1 ) 3 3 ( 1 ) 2 ( a 2 ) +3( 1 ) ( a 2 ) 2 ( a 2 ) 3 =13 a 2 +3 a 4 a 6

Ejercicio 65

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 65
Escribir, por simple inspección, el resultado de:
  1. (x+y+z ) (x+yz ) =[(x+y ) +z ] [(x+y ) z ] = (x+y ) 2 ( z ) 2 = x 2 +2xy+ y 2 z 2
  2. (xy+z ) (x+yz ) =[x(yz ) ] [x+(yz ) ] = ( x ) 2 (yz ) 2 = x 2 ( y 2 2yz+ z 2 ) = x 2 y 2 +2yz z 2
  3. (x+y+z ) (xyz ) =[x+(y+z ) ] [x(y+z ) ] = ( x ) 2 (y+z ) 2 = x 2 ( y 2 +2yz+ z 2 ) = x 2 y 2 2yz z 2
  4. (m+n+1 ) (m+n1 ) =[(m+n ) +1 ] [(m+n ) 1 ] = (m+n ) 2 ( 1 ) 2 = m 2 +2m+ n 2 1
  5. (mn1 ) (mn+1 ) =[(mn ) 1 ] [(mn ) +1 ] = (mn ) 2 ( 1 ) 2 = m 2 2m+ n 2 1
  6. (x+y2 ) (xy+2 ) =[x+(y2 ) ] [x(y2 ) ] = ( x ) 2 (y2 ) 2 = x 2 ( y 2 4y+4 ) = x 2 y 2 +4y4
  7. ( n 2 +2n+1 ) ( n 2 2n1 ) =[ n 2 +(2n+1 ) ] [ n 2 (2n+1 ) ] = ( n 2 ) 2 (2n+1 ) 2 = n 4 (4 n 2 +4n+1 ) = n 4 4 n 2 4n1
  8. ( a 2 2a+3 ) ( a 2 +2a+3 ) =[( a 2 +3 ) 2a ] [( a 2 +3 ) +2a ] = ( a 2 +3 ) 2 (2a ) 2 = a 4 +6 a 2 +94 a 2 = a 4 +2 a 2 +9
  9. ( m 2 m+1 ) ( m 2 +m+1 ) =[( m 2 1 ) m ] [( m 2 1 ) +m ] = ( m 2 1 ) 2 ( m ) 2 = m 4 2 m 2 +1 m 2 = m 4 3 m 2 +1
  10. (2abc ) (2ab+c ) =[(2ab ) c ] [(2ab ) +c ] = (2ab ) 2 ( c ) 2 =4 a 2 +4ab+ b 2 c 2
  11. (2x+yz ) (2xy+z ) =[2x+(yz ) ] [2x(yz ) ] = (2x ) 2 (yz ) 2 =4 x 2 ( y 2 2yz+ z 2 ) =4 x 2 y 2 +2yz z 2
  12. ( x 2 5x+6 ) ( x 2 +5x6 ) =[ x 2 (5x6 ) ] [ x 2 +(5x6 ) ] = ( x 2 ) 2 (5x6 ) 2 = x 4 (25 x 2 60x+36 ) = x 4 25 x 2 +60x36
  13. ( a 2 ab+ b 2 ) ( a 2 + b 2 +ab ) =[( a 2 + b 2 ) ab ] [( a 2 + b 2 ) +ab ] = ( a 2 + b 2 ) 2 (ab ) 2 = a 4 +2 a 2 b 2 + b 4 a 2 b 2 = a 4 + a 2 b 2 + b 4
  14. ( x 3 x 2 x ) ( x 3 + x 2 +x ) =[ x 3 ( x 2 +x ) ] [ x 3 +( x 2 +x ) ] = ( x 3 ) 2 ( x 2 +x ) 2 = x 6 ( x 4 +2 x 3 + x 2 ) = x 6 x 4 2 x 3 x 2

Ejercicio 64

CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 64
Escribir, por simple inspección, el resultado de:
  1. (x+y ) (xy ) = x 2 y 2
  2. (mn ) (m+n ) = m 2 n 2
  3. (ax ) (x+a ) =(ax ) (a+x ) = a 2 x 2
  4. ( x 2 + a 2 ) ( x 2 a 2 ) = ( x 2 ) 2 ( a 2 ) 2 = x 4 a 4
  5. (2a1 ) (1+2a ) =(2a1 ) (2a+1 ) = (2a ) 2 ( 1 ) 2 =4 a 2 1
  6. (n1 ) (n+1 ) = n 2 1
  7. (13ax ) (3ax+1 ) =(13ax ) (1+3ax ) =19 a 2 x 2
  8. (2m+9 ) (2m9 ) = (2m ) 2 ( 9 ) 2 =4 m 2 81
  9. ( a 3 b 2 ) ( a 3 + b 2 ) = ( a 3 ) 2 ( b 2 ) 2 = a 6 b 4
  10. ( y 2 3y ) ( y 2 +3y ) = ( y 2 ) 2 (3y ) 2 = y 4 9 y 2
  11. (18xy ) (8xy+1 ) =(18xy ) (1+8xy ) = 1 2 (8xy ) 2 =164 x 2 y 2
  12. (6 x 2 m 2 x ) (6 x 2 + m 2 x ) = (6 x 2 ) 2 ( m 2 x ) 2 =36 x 4 m 4 x 2
  13. ( a m + b n ) ( a m b n ) = ( a m ) 2 ( b n ) 2 = a 2m b 2n
  14. (3 x a 5 y m ) (5 y m +3 x a ) =(3 x a 5 y m ) (3 x a +5 y m ) = (3 x a ) 2 (5 y m ) 2 =9 x 2a 25 y 2m
  15. ( a x+1 2 b x1 ) (2 b x1 + a x+1 ) =( a x+1 2 b x1 ) ( a x+1 +2 b x1 ) = ( a x+1 ) 2 (2 b x1 ) 2 = a 2x+2 4 b 2x2