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CAPITULO XII

Mínimo Común Múltiplo
Ejercicio 117
Hallar el m.c.m de:
  1. 3x+3,6x6 3x+3=3(x+1 ) 6x6=6(x1 ) =2.3(x1 ) m.c.m=2.3(x1 ) (x+1 ) m.c.m=6(x1 ) (x+1 )
  2. 5x+10,10 x 2 40 5x+10=5(x+2 ) 10 x 2 40=10( x 2 4 ) =2.5(x+2 ) (x2 ) m.c.m=2.5(x+2 ) (x2 ) m.c.m=10(x+2 ) (x2 )
  3. x 3 +2 x 2 y, x 2 4 y 2 x 3 +2 x 2 y= x 2 (x+2y ) x 2 4 y 2 =(x2y ) (x+2y ) m.c.m= x 2 (x+2y ) (x2y )
  4. 3 a 2 x9 a 2 , x 2 6x+9 3 a 2 x9 a 2 =3 a 2 (x3 ) x 2 6x+9= (x3 ) 2 m.c.m=3 a 2 (x3 ) 2
  5. 4 a 2 9 b 2 ,4 a 2 12ab+9 b 2 4 a 2 9 b 2 =(2a3b ) (2a+3b ) 4 a 2 12ab+9 b 2 = (2a3b ) 2 m.c.m= (2a3b ) 2 (2a+3b )
  6. a 3 + a 2 b, a 3 +2 a 2 b+a b 2 a 3 + a 2 b= a 2 (a+b ) a 3 +2 a 2 b+a b 2 =a( a 2 +2ab+ b 2 ) =a (a+b ) 2 m.c.m= a 2 (a+b ) 2
  7. 3ax+12a,2b x 2 +6bx8b 3ax+12a=3a(x+4 ) 2b x 2 +6bx8b=2b( x 2 +3x4 ) =2b(x+4 ) (x1 ) m.c.m=2.3ab(x+4 ) (x1 ) m.c.m=6ab(x+4 ) (x1 )
  8. x 3 25x, x 2 +2x15 x 3 25x=x( x 2 25 ) =x(x5 ) (x+5 ) x 2 +2x15=(x+5 ) (x3 ) m.c.m=x(x5 ) (x+5 ) (x3 )
  9. (x1 ) 2 , x 2 1 (x1 ) 2 x 2 1=(x+1 ) (x1 ) m.c.m=(x+1 ) (x1 ) 2
  10. (x+1 ) 2 , x 2 +1 (x+1 ) 2 x 2 +1 m.c.m=( x 2 +1 ) (x+1 ) 2
  11. x 3 + y 3 , (x+y ) 3 x 3 + y 3 =(x+y ) ( x 2 xy+ y 2 ) (x+y ) 3 m.c.m= (x+y ) 3 ( x 2 xy+ y 2 )
  12. x 3 y 3 , (xy ) 3 x 3 y 3 =(xy ) ( x 2 +xy+ y 2 ) (xy ) 3 m.c.m= (xy ) 3 ( x 2 +xy+ y 2 )
  13. x 2 +3x10,4 x 2 7x2 x 2 +3x10=(x+5 ) (x2 ) 4 x 2 7x2=4 x 2 8x+x2=4x(x2 ) +(x2 ) =(4x+1 ) (x2 ) m.c.m=(4x+1 ) (x2 ) (x+5 )
  14. a 2 +a30, a 2 +3a18 a 2 +a30=(a+6 ) (a5 ) a 2 +3a18=(a+6 ) (a3 ) m.c.m=(a+6 ) (a5 ) (a3 )
  15. x 3 9x+5 x 2 45, x 4 +2 x 3 15 x 2 x 3 9x+5 x 2 45=x( x 2 9 ) +5( x 2 9 ) =( x 2 9 ) (x+5 ) =(x3 ) (x+3 ) (x+5 ) x 4 +2 x 3 15 x 2 = x 2 ( x 2 +2x15 ) =(x+5 ) (x3 ) m.c.m=(x3 ) (x+3 ) (x+5 )
  16. x 6 4 x 3 32,a x 4 +2a x 3 +4a x 2 x 6 4 x 3 32= x 6 8 x 3 +4 x 3 32= x 3 ( x 3 8 ) +4( x 3 8 ) =( x 3 8 ) ( x 3 +4 ) =(x2 ) ( x 2 +2x+4 ) ( x 3 +4 ) a x 4 +2a x 3 +4a x 2 =a x 2 ( x 2 +2x+4 ) m.c.m=a x 2 (x2 ) ( x 2 +2x+4 ) ( x 3 +4 )
  17. 8 (xy ) 2 ,12( x 2 y 2 ) 8 (xy ) 2 = 2 3 (xy ) 2 12( x 2 y 2 ) = 2 2 .3(x+y ) (xy ) m.c.m= 2 3 .3(x+y ) (xy ) 2 m.c.m=24(x+y ) (xy ) 2
  18. 5 (x+y ) 2 ,10( x 2 + y 2 ) 5 (x+y ) 2 10( x 2 + y 2 ) =2.5( x 2 + y 2 ) m.c.m=2.5( x 2 + y 2 ) (x+y ) 2 m.c.m=10( x 2 + y 2 ) (x+y ) 2
  19. 6a (m+n ) 3 ,4 a 2 b( m 3 + n 3 ) 6a (m+n ) 3 =2.3a (m+n ) 3 4 a 2 b( m 3 + n 3 ) = 2 2 a 2 b( m 3 + n 3 ) = 2 2 a 2 b(m+n ) ( m 2 mn+ n 2 ) m.c.m= 2 2 .3 a 2 b (m+n ) 3 ( m 2 mn+ n 2 ) m.c.m=12 a 2 b (m+n ) 3 ( m 2 mn+ n 2 )
  20. ax (mn ) 3 , x 3 ( m 3 n 3 ) ax (mn ) 3 x 3 ( m 3 n 3 ) = x 3 (mn ) ( m 2 +mn+ n 2 ) m.c.m=a x 3 (mn ) 3 ( m 2 +mn+ n 2 )
  21. 2 a 2 +2a,3 a 2 3a, a 4 a 2 2 a 2 +2a=2a(a+1 ) 3 a 2 3a=3a(a1 ) a 4 a 2 = a 2 ( a 2 1 ) = a 2 (a1 ) (a+1 ) m.c.m=2.3 a 2 (a1 ) (a+1 ) m.c.m=6 a 2 (a1 ) (a+1 )
  22. x 2 +2x, x 3 2 x 2 , x 2 4 x 2 +2x=x(x+2 ) x 3 2 x 2 = x 2 (x2 ) x 2 4=(x2 ) (x+2 ) m.c.m= x 2 (x2 ) (x+2 )
  23. x 2 +x2, x 2 4x+3, x 2 x6 x 2 +x2=(x+2 ) (x1 ) x 2 4x+3=(x3 ) (x1 ) x 2 x6=(x3 ) (x+2 ) m.c.m=(x3 ) (x+2 ) (x1 )
  24. 6 a 2 +13a+6,3 a 2 +14a+8,4+12a+9 a 2 6 a 2 +13a+6=6 a 2 +9a+4a+6=3a(2a+3 ) +2(2a+3 ) =(2a+3 ) (3a+2 ) 3 a 2 +14a+8=3 a 2 +12a+2a+8=3a(a+4 ) +2(a+4 ) =(a+4 ) (3a+2 ) 4+12a+9 a 2 = (2+3a ) 2 = (3a+2 ) 2 m.c.m= (3a+2 ) 2 (a+4 ) (2a+3 )
  25. 10 x 2 +10,15x+15,5 x 2 5 10 x 2 +10=10( x 2 +1 ) =2.5( x 2 +1 ) 15x+15=15(x+1 ) =3.5(x+1 ) 5 x 2 5=5( x 2 1 ) =5(x1 ) (x+1 ) m.c.m=2.3.5( x 2 +1 ) (x1 ) (x+1 ) m.c.m=30( x 2 +1 ) (x1 ) (x+1 )
  26. ax2bx+ay2by, x 2 +xy, x 2 xy ax2bx+ay2by=x(a2b ) +y(a2b ) =(a2b ) (x+y ) x 2 +xy=x(x+y ) x 2 xy=x(xy ) m.c.m=x(x+y ) (xy ) (a2b )
  27. 4 a 2 b+4a b 2 ,6a6b,15 a 2 15 b 2 4 a 2 b+4a b 2 =4ab(a+b ) = 2 2 ab(a+b ) 6a6b=6(ab ) =2.3(ab ) 15 a 2 15 b 2 =15( a 2 b 2 ) =3.5(ab ) (a+b ) m.c.m= 2 2 .3.5ab(a+b ) (ab ) m.c.m=60ab(a+b ) (ab )
  28. x 2 25, x 3 125,2x+10 x 2 25=(x5 ) (x+5 ) x 3 125=(x5 ) ( x 2 +5x+25 ) 2x+10=2(x+5 ) m.c.m=2(x+5 ) (x5 ) ( x 2 +5x+25 )
  29. a 2 2ab3 b 2 , a 3 b6 a 2 b 2 +9a b 3 ,a b 2 + b 3 a 2 2ab3 b 2 = a 2 +ab3ab3 b 2 =a(a+b ) 3b(a+b ) =(a+b ) (a3b ) a 3 b6 a 2 b 2 +9a b 3 =ab( a 2 6ab+9 b 2 ) =ab (a3b ) 2 a b 2 + b 3 = b 2 (a+b ) m.c.m=a b 2 (a+b ) (a3b ) 2
  30. 2 m 2 +2mn,4mn4 n 2 ,6 m 3 n6m n 3 2 m 2 +2mn=2m(m+n ) 4mn4 n 2 =4n(mn ) = 2 2 n(mn ) 6 m 3 n6m n 3 =6mn( m 2 n 2 ) =2.3mn(mn ) (m+n ) m.c.m= 2 2 .3mn(mn ) (m+n ) m.c.m=12mn(mn ) (m+n )
  31. 20( x 2 y 2 ) ,15 (xy ) 2 ,12 (x+y ) 2 20( x 2 y 2 ) = 2 2 .5(xy ) (x+y ) 15 (xy ) 2 =3.5 (xy ) 2 12 (x+y ) 2 = 2 2 .3 (x+y ) 2 m.c.m= 2 2 .3.5 (x+y ) 2 (xy ) 2 m.c.m=60 (x+y ) 2 (xy ) 2
  32. a x 2 +5ax14a, x 3 +14 x 2 +49x, x 4 +7 x 3 18 x 2 a x 2 +5ax14a=a( x 2 +5x14 ) =a(x+7 ) (x2 ) x 3 +14 x 2 +49x=x( x 2 +14x+49 ) =x (x+7 ) 2 x 4 +7 x 3 18 x 2 = x 2 ( x 2 +7x18 ) = x 2 (x+9 ) (x2 ) m.c.m=a x 2 (x+9 ) (x2 ) (x+7 ) 2
  33. 2 x 3 12 x 2 +18x,3 x 4 27 x 2 ,5 x 3 +30 x 2 +45x 2 x 3 12 x 2 +18x=2x( x 2 6x+9 ) =2x (x3 ) 2 3 x 4 27 x 2 =3 x 2 ( x 2 9 ) =3 x 2 (x+3 ) (x3 ) 5 x 3 +30 x 2 +45x=5x( x 2 +6x+9 ) =5x (x+3 ) 2 m.c.m=2.3.5 x 2 (x+3 ) 2 (x3 ) 2 m.c.m=30 x 2 (x+3 ) 2 (x3 ) 2
  34. 33 a 2 ,6+6a,99a,12+12 a 2 33 a 2 =3(1 a 2 ) =3(1a ) (1+a ) 6+6a=6(1+a ) =2.3(1+a ) 12+12 a 2 =12(1+ a 2 ) = 2 2 .3(1+ a 2 ) m.c.m= 2 2 .3(1+ a 2 ) (1a ) (1+a ) m.c.m=12(1+ a 2 ) (1a ) (1+a )
  35. 2 (3n2 ) 2 ,135 n 3 40,12n8 2 (3n2 ) 2 135 n 3 40=5(27 n 3 8 ) =5(3n2 ) (9 n 2 +6n+4 ) 12n8=4(3n2 ) = 2 2 (3n2 ) m.c.m= 2 2 .5 (3n2 ) 2 (9 n 2 +6n+4 ) m.c.m=20 (3n2 ) 2 (9 n 2 +6n+4 )
  36. 12mn+8m3n2,48 m 2 n3n+32 m 2 2,6 n 2 5n6 12mn+8m3n2=4m(3n+2 ) (3n+2 ) =(3n+2 ) (4m1 ) 48 m 2 n3n+32 m 2 2=3n(16 m 2 1 ) +2(16 m 2 1 ) =(16 m 2 1 ) (3n+2 ) =(4m1 ) (4m+1 ) (3n+2 ) 6 n 2 5n6=6 n 2 9n+4n6=3n(2n3 ) +2(2n3 ) =(2n3 ) (3n+2 ) m.c.m=(4m1 ) (4m+1 ) (3n+2 ) (2n3 )
  37. 18 x 3 +60 x 2 +50x,12a x 3 +20a x 2 ,15 a 2 x 5 +16 a 2 x 4 15 a 2 x 3 18 x 3 +60 x 2 +50x=2x(9 x 2 +30x+25 ) =2x (3x+5 ) 2 12a x 3 +20a x 2 =4a x 2 (3x+5 ) = 2 2 a x 2 (3x+5 ) 15 a 2 x 5 +16 a 2 x 4 15 a 2 x 3 = a 2 x 3 (15 x 2 +16x15 ) = a 2 x 3 (15 x 2 9x+25x15 ) = a 2 x 3 [3x(5x3 ) +5(5x3 ) ] = a 2 x 3 (5x3 ) (3x+5 ) m.c.m= 2 2 a 2 x 3 (5x3 ) (3x+5 ) 2 m.c.m=4 a 2 x 3 (5x3 ) (3x+5 ) 2
  38. 16 x 4 ,16+8 x 2 + x 4 ,168 x 2 + x 4 16 x 4 =(4+ x 2 ) (4 x 2 ) =(4+ x 2 ) (2x ) (2+x ) 16+8 x 2 + x 4 = (4+ x 2 ) 2 168 x 2 + x 4 = (4 x 2 ) 2 = [(2x ) (2+x ) ] 2 = (2x ) 2 (2+x ) 2 m.c.m= (4+ x 2 ) 2 (2x ) 2 (2+x ) 2
  39. 1+ a 2 , (1+a ) 2 ,1+ a 3 1+ a 2 (1+a ) 2 1+ a 3 =(1+a ) (1a+ a 2 ) m.c.m=(1+ a 2 ) (1+a ) 2 (1a+ a 2 )
  40. 8 n 2 10n3,20 n 2 +13n+2,10 n 2 11n6 8 n 2 10n3=8 n 2 12n+2n3=4n(2n3 ) +(2n3 ) =(2n3 ) (4n+1 ) 20 n 2 +13n+2=20 n 2 +5n+8n+2=5n(4n+1 ) +2(4n+1 ) =(4n+1 ) (5n+2 ) 10 n 2 11n6=10 n 2 15n+4n6=5n(2n3 ) +2(2n3 ) =(2n3 ) (5n+2 ) m.c.m=(2n3 ) (4n+1 ) (5n+2 )
  41. 6 a 2 +ab2 b 2 ,15 a 2 +22ab+8 b 2 ,10 a 2 +3ab4 b 2 6 a 2 +ab2 b 2 =6 a 2 3ab+4ab2 b 2 =3a(2ab ) +2b(2ab ) =(2ab ) (3a+2b ) 15 a 2 +22ab+8 b 2 =15 a 2 +10ab+12ab+8 b 2 =5a(3a+2b ) +4b(3a+2b ) =(3a+2b ) (5a+4b ) 10 a 2 +3ab4 b 2 =10 a 2 5ab+8ab4 b 2 =5a(2ab ) +4b(2ab ) =(2ab ) (5a+4b ) m.c.m=(2ab ) (3a+2b ) (5a+4b )
  42. 12 x 2 +5xy2 y 2 ,15 x 2 +13xy+2 y 2 ,20 x 2 xy y 2 12 x 2 +5xy2 y 2 =12 x 2 3xy+8xy2 y 2 =3x(4xy ) +2y(4xy ) =(4xy ) (3x+2y ) 15 x 2 +13xy+2 y 2 =15 x 2 +10xy+3xy+2 y 2 =5x(3x+2y ) +y(3x+2y ) =(3x+2y ) (5x+y ) 20 x 2 xy y 2 =20 x 2 5xy+4xy y 2 =5x(4xy ) +y(4xy ) =(4xy ) (5x+y ) m.c.m=(4xy ) (5x+y ) (3x+2y )
  43. 6 b 2 x 2 +6 b 2 x 3 ,3 a 2 x3 a 2 x 2 ,1 x 4 6 b 2 x 2 +6 b 2 x 3 =6 b 2 x 2 (1+x ) =2.3 b 2 x 2 (1+x ) 3 a 2 x3 a 2 x 2 =3 a 2 x(1x ) 1 x 4 =(1+ x 2 ) (1 x 2 ) =(1+ x 2 ) (1+x ) (1x ) m.c.m=2.3 a 2 b 2 x 2 (1+ x 2 ) (1+x ) (1x ) m.c.m=6 a 2 b 2 x 2 (1+ x 2 ) (1+x ) (1x )
  44. x 4 +8x4 x 3 32, a 2 x 4 2 a 2 x 3 8 a 2 x 2 ,2 x 4 4 x 3 +8 x 2 x 4 +8x4 x 3 32= x 3 (x4 ) +8(x4 ) =(x4 ) ( x 3 +8 ) =(x4 ) (x+2 ) ( x 2 2x+4 ) a 2 x 4 2 a 2 x 3 8 a 2 x 2 = a 2 x 2 ( x 2 2x8 ) = a 2 x 2 (x4 ) (x+2 ) 2 x 4 4 x 3 +8 x 2 =2 x 2 ( x 2 2x+4 ) m.c.m=2 a 2 x 2 (x4 ) (x+2 ) ( x 2 2x+4 )
  45. x 3 9x+ x 2 9, x 4 10 x 2 +9, x 2 +4x+3, x 2 4x+3 x 3 9x+ x 2 9=x( x 2 9 ) +( x 2 9 ) =( x 2 9 ) (x+1 ) =(x3 ) (x+3 ) (x+1 ) x 4 10 x 2 +9= x 4 x 2 9 x 2 +9= x 2 ( x 2 1 ) 9( x 2 1 ) =( x 2 1 ) ( x 2 9 ) =(x+1 ) (x1 ) (x3 ) (x+3 ) x 2 4x+3=(x3 ) (x1 ) m.c.m=(x+1 ) (x1 ) (x3 ) (x+3 )
  46. 1 a 3 ,1a,1 a 2 ,12a+ a 2 1 a 3 =(1a ) (1+a+ a 2 ) 1a 1 a 2 =(1+a ) (1a ) 12a+ a 2 = (1a ) 2 m.c.m=(1+a ) (1a ) 2 (1+a+ a 2 )
  47. a 2 ba b 2 , a 4 b 2 a 2 b 4 ,a (ab b 2 ) 2 ,b ( a 2 +ab ) 2 a 2 ba b 2 =ab(ab ) a 4 b 2 a 2 b 4 = a 2 b 2 ( a 2 b 2 ) = a 2 b 2 (a+b ) (ab ) a (ab b 2 ) 2 =a [b(ab ) ] 2 =a b 2 (ab ) 2 b ( a 2 +ab ) 2 =b [a(a+b ) ] 2 =b a 2 (a+b ) 2 m.c.m= a 2 b 2 (a+b ) 2 (ab ) 2
  48. m 3 27 n 3 , m 2 9 n 2 , m 2 6mn+9 n 2 , m 2 +3mn+9 n 2 m 3 27 n 3 =(m3n ) ( m 2 +3mn+9 n 2 ) m 2 6mn+9 n 2 = (m3n ) 2 m 2 +3mn+9 n 2 m.c.m= (m3n ) 2 ( m 2 +3mn+9 n 2 )