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Ejercicio 117

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CAPITULO XII

Mínimo Común Múltiplo
Ejercicio 117
Hallar el m.c.m de:
  1. 3x+3,6x–6 3x+3=3( x+1 ) 6x–6=6( x–1 ) =2.3( x–1 ) m.c.m=2.3( x–1 ) ( x+1 ) m.c.m=6( x–1 ) ( 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 ) ( x–2 ) m.c.m=2.5( x+2 ) ( x–2 ) m.c.m=10( x+2 ) ( x–2 )
  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 =( x–2y ) ( x+2y ) m.c.m= x 2 ( x+2y ) ( x–2y )
  4. 3 a 2 x–9 a 2 , x 2 –6x+9 3 a 2 x–9 a 2 =3 a 2 ( x–3 ) x 2 –6x+9= ( x–3 ) 2 m.c.m=3 a 2 ( x–3 ) 2
  5. 4 a 2 –9 b 2 ,4 a 2 –12ab+9 b 2 4 a 2 –9 b 2 =( 2a–3b ) ( 2a+3b ) 4 a 2 –12ab+9 b 2 = ( 2a–3b ) 2 m.c.m= ( 2a–3b ) 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 +6bx–8b 3ax+12a=3a( x+4 ) 2b x 2 +6bx–8b=2b( x 2 +3x–4 ) =2b( x+4 ) ( x–1 ) m.c.m=2.3ab( x+4 ) ( x–1 ) m.c.m=6ab( x+4 ) ( x–1 )
  8. x 3 –25x, x 2 +2x–15 x 3 –25x=x( x 2 –25 ) =x( x–5 ) ( x+5 ) x 2 +2x–15=( x+5 ) ( x–3 ) m.c.m=x( x–5 ) ( x+5 ) ( x–3 )
  9. ( x–1 ) 2 , x 2 –1 ( x–1 ) 2 x 2 –1=( x+1 ) ( x–1 ) m.c.m=( x+1 ) ( x–1 ) 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 , ( 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 )
  13. x 2 +3x–10,4 x 2 –7x–2 x 2 +3x–10=( x+5 ) ( x–2 ) 4 x 2 –7x–2=4 x 2 –8x+x–2=4x( x–2 ) +( x–2 ) =( 4x+1 ) ( x–2 ) m.c.m=( 4x+1 ) ( x–2 ) ( x+5 )
  14. a 2 +a–30, a 2 +3a–18 a 2 +a–30=( a+6 ) ( a–5 ) a 2 +3a–18=( a+6 ) ( a–3 ) m.c.m=( a+6 ) ( a–5 ) ( a–3 )
  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 ) =( x–3 ) ( x+3 ) ( x+5 ) x 4 +2 x 3 –15 x 2 = x 2 ( x 2 +2x–15 ) =( x+5 ) ( x–3 ) m.c.m=( x–3 ) ( 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 ) =( x–2 ) ( 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 ( x–2 ) ( x 2 +2x+4 ) ( x 3 +4 )
  17. 8 ( x–y ) 2 ,12( x 2 – y 2 ) 8 ( x–y ) 2 = 2 3 ( x–y ) 2 12( x 2 – y 2 ) = 2 2 .3( x+y ) ( x–y ) m.c.m= 2 3 .3( x+y ) ( x–y ) 2 m.c.m=24( x+y ) ( x–y ) 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 ( m–n ) 3 , x 3 ( m 3 – n 3 ) ax ( m–n ) 3 x 3 ( m 3 – n 3 ) = x 3 ( m–n ) ( m 2 +mn+ n 2 ) m.c.m=a x 3 ( m–n ) 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( a–1 ) a 4 – a 2 = a 2 ( a 2 –1 ) = a 2 ( a–1 ) ( a+1 ) m.c.m=2.3 a 2 ( a–1 ) ( a+1 ) m.c.m=6 a 2 ( a–1 ) ( 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 ( x–2 ) x 2 –4=( x–2 ) ( x+2 ) m.c.m= x 2 ( x–2 ) ( x+2 )
  23. x 2 +x–2, x 2 –4x+3, x 2 –x–6 x 2 +x–2=( x+2 ) ( x–1 ) x 2 –4x+3=( x–3 ) ( x–1 ) x 2 –x–6=( x–3 ) ( x+2 ) m.c.m=( x–3 ) ( x+2 ) ( x–1 )
  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( x–1 ) ( x+1 ) m.c.m=2.3.5( x 2 +1 ) ( x–1 ) ( x+1 ) m.c.m=30( x 2 +1 ) ( x–1 ) ( x+1 )
  26. ax–2bx+ay–2by, x 2 +xy, x 2 –xy ax–2bx+ay–2by=x( a–2b ) +y( a–2b ) =( a–2b ) ( x+y ) x 2 +xy=x( x+y ) x 2 –xy=x( x–y ) m.c.m=x( x+y ) ( x–y ) ( a–2b )
  27. 4 a 2 b+4a b 2 ,6a–6b,15 a 2 –15 b 2 4 a 2 b+4a b 2 =4ab( a+b ) = 2 2 ab( a+b ) 6a–6b=6( a–b ) =2.3( a–b ) 15 a 2 –15 b 2 =15( a 2 – b 2 ) =3.5( a–b ) ( a+b ) m.c.m= 2 2 .3.5ab( a+b ) ( a–b ) m.c.m=60ab( a+b ) ( a–b )
  28. x 2 –25, x 3 –125,2x+10 x 2 –25=( x–5 ) ( x+5 ) x 3 –125=( x–5 ) ( x 2 +5x+25 ) 2x+10=2( x+5 ) m.c.m=2( x+5 ) ( x–5 ) ( x 2 +5x+25 )
  29. a 2 –2ab–3 b 2 , a 3 b–6 a 2 b 2 +9a b 3 ,a b 2 + b 3 a 2 –2ab–3 b 2 = a 2 +ab–3ab–3 b 2 =a( a+b ) –3b( a+b ) =( a+b ) ( a–3b ) a 3 b–6 a 2 b 2 +9a b 3 =ab( a 2 –6ab+9 b 2 ) =ab ( a–3b ) 2 a b 2 + b 3 = b 2 ( a+b ) m.c.m=a b 2 ( a+b ) ( a–3b ) 2
  30. 2 m 2 +2mn,4mn–4 n 2 ,6 m 3 n–6m n 3 2 m 2 +2mn=2m( m+n ) 4mn–4 n 2 =4n( m–n ) = 2 2 n( m–n ) 6 m 3 n–6m n 3 =6mn( m 2 – n 2 ) =2.3mn( m–n ) ( m+n ) m.c.m= 2 2 .3mn( m–n ) ( m+n ) m.c.m=12mn( m–n ) ( m+n )
  31. 20( x 2 – y 2 ) ,15 ( x–y ) 2 ,12 ( x+y ) 2 20( x 2 – y 2 ) = 2 2 .5( x–y ) ( x+y ) 15 ( x–y ) 2 =3.5 ( x–y ) 2 12 ( x+y ) 2 = 2 2 .3 ( x+y ) 2 m.c.m= 2 2 .3.5 ( x+y ) 2 ( x–y ) 2 m.c.m=60 ( x+y ) 2 ( x–y ) 2
  32. a x 2 +5ax–14a, x 3 +14 x 2 +49x, x 4 +7 x 3 –18 x 2 a x 2 +5ax–14a=a( x 2 +5x–14 ) =a( x+7 ) ( x–2 ) 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 +7x–18 ) = x 2 ( x+9 ) ( x–2 ) m.c.m=a x 2 ( x+9 ) ( x–2 ) ( 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 ( x–3 ) 2 3 x 4 –27 x 2 =3 x 2 ( x 2 –9 ) =3 x 2 ( x+3 ) ( x–3 ) 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 ( x–3 ) 2 m.c.m=30 x 2 ( x+3 ) 2 ( x–3 ) 2
  34. 3–3 a 2 ,6+6a,9–9a,12+12 a 2 3–3 a 2 =3( 1– a 2 ) =3( 1–a ) ( 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 ) ( 1–a ) ( 1+a ) m.c.m=12( 1+ a 2 ) ( 1–a ) ( 1+a )
  35. 2 ( 3n–2 ) 2 ,135 n 3 –40,12n–8 2 ( 3n–2 ) 2 135 n 3 –40=5( 27 n 3 –8 ) =5( 3n–2 ) ( 9 n 2 +6n+4 ) 12n–8=4( 3n–2 ) = 2 2 ( 3n–2 ) m.c.m= 2 2 .5 ( 3n–2 ) 2 ( 9 n 2 +6n+4 ) m.c.m=20 ( 3n–2 ) 2 ( 9 n 2 +6n+4 )
  36. 12mn+8m–3n–2,48 m 2 n–3n+32 m 2 –2,6 n 2 –5n–6 12mn+8m–3n–2=4m( 3n+2 ) –( 3n+2 ) =( 3n+2 ) ( 4m–1 ) 48 m 2 n–3n+32 m 2 –2=3n( 16 m 2 –1 ) +2( 16 m 2 –1 ) =( 16 m 2 –1 ) ( 3n+2 ) =( 4m–1 ) ( 4m+1 ) ( 3n+2 ) 6 n 2 –5n–6=6 n 2 –9n+4n–6=3n( 2n–3 ) +2( 2n–3 ) =( 2n–3 ) ( 3n+2 ) m.c.m=( 4m–1 ) ( 4m+1 ) ( 3n+2 ) ( 2n–3 )
  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 +16x–15 ) = a 2 x 3 ( 15 x 2 –9x+25x–15 ) = a 2 x 3 [ 3x( 5x–3 ) +5( 5x–3 ) ] = a 2 x 3 ( 5x–3 ) ( 3x+5 ) m.c.m= 2 2 a 2 x 3 ( 5x–3 ) ( 3x+5 ) 2 m.c.m=4 a 2 x 3 ( 5x–3 ) ( 3x+5 ) 2
  38. 16– x 4 ,16+8 x 2 + x 4 ,16–8 x 2 + x 4 16– x 4 =( 4+ x 2 ) ( 4– x 2 ) =( 4+ x 2 ) ( 2–x ) ( 2+x ) 16+8 x 2 + x 4 = ( 4+ x 2 ) 2 16–8 x 2 + x 4 = ( 4– x 2 ) 2 = [ ( 2–x ) ( 2+x ) ] 2 = ( 2–x ) 2 ( 2+x ) 2 m.c.m= ( 4+ x 2 ) 2 ( 2–x ) 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 ) ( 1–a+ a 2 ) m.c.m=( 1+ a 2 ) ( 1+a ) 2 ( 1–a+ a 2 )
  40. 8 n 2 –10n–3,20 n 2 +13n+2,10 n 2 –11n–6 8 n 2 –10n–3=8 n 2 –12n+2n–3=4n( 2n–3 ) +( 2n–3 ) =( 2n–3 ) ( 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 –11n–6=10 n 2 –15n+4n–6=5n( 2n–3 ) +2( 2n–3 ) =( 2n–3 ) ( 5n+2 ) m.c.m=( 2n–3 ) ( 4n+1 ) ( 5n+2 )
  41. 6 a 2 +ab–2 b 2 ,15 a 2 +22ab+8 b 2 ,10 a 2 +3ab–4 b 2 6 a 2 +ab–2 b 2 =6 a 2 –3ab+4ab–2 b 2 =3a( 2a–b ) +2b( 2a–b ) =( 2a–b ) ( 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 +3ab–4 b 2 =10 a 2 –5ab+8ab–4 b 2 =5a( 2a–b ) +4b( 2a–b ) =( 2a–b ) ( 5a+4b ) m.c.m=( 2a–b ) ( 3a+2b ) ( 5a+4b )
  42. 12 x 2 +5xy–2 y 2 ,15 x 2 +13xy+2 y 2 ,20 x 2 –xy– y 2 12 x 2 +5xy–2 y 2 =12 x 2 –3xy+8xy–2 y 2 =3x( 4x–y ) +2y( 4x–y ) =( 4x–y ) ( 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( 4x–y ) +y( 4x–y ) =( 4x–y ) ( 5x+y ) m.c.m=( 4x–y ) ( 5x+y ) ( 3x+2y )
  43. 6 b 2 x 2 +6 b 2 x 3 ,3 a 2 x–3 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 x–3 a 2 x 2 =3 a 2 x( 1–x ) 1– x 4 =( 1+ x 2 ) ( 1– x 2 ) =( 1+ x 2 ) ( 1+x ) ( 1–x ) m.c.m=2.3 a 2 b 2 x 2 ( 1+ x 2 ) ( 1+x ) ( 1–x ) m.c.m=6 a 2 b 2 x 2 ( 1+ x 2 ) ( 1+x ) ( 1–x )
  44. x 4 +8x–4 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 +8x–4 x 3 –32= x 3 ( x–4 ) +8( x–4 ) =( x–4 ) ( x 3 +8 ) =( x–4 ) ( 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 –2x–8 ) = a 2 x 2 ( x–4 ) ( 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 ( x–4 ) ( 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 ) =( x–3 ) ( 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 ) ( x–1 ) ( x–3 ) ( x+3 ) x 2 –4x+3=( x–3 ) ( x–1 ) m.c.m=( x+1 ) ( x–1 ) ( x–3 ) ( x+3 )
  46. 1– a 3 ,1–a,1– a 2 ,1–2a+ a 2 1– a 3 =( 1–a ) ( 1+a+ a 2 ) 1–a 1– a 2 =( 1+a ) ( 1–a ) 1–2a+ a 2 = ( 1–a ) 2 m.c.m=( 1+a ) ( 1–a ) 2 ( 1+a+ a 2 )
  47. a 2 b–a b 2 , a 4 b 2 – a 2 b 4 ,a ( ab– b 2 ) 2 ,b ( a 2 +ab ) 2 a 2 b–a b 2 =ab( a–b ) a 4 b 2 – a 2 b 4 = a 2 b 2 ( a 2 – b 2 ) = a 2 b 2 ( a+b ) ( a–b ) a ( ab– b 2 ) 2 =a [ b( a–b ) ] 2 =a b 2 ( a–b ) 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 ( a–b ) 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 =( m–3n ) ( m 2 +3mn+9 n 2 ) m 2 –6mn+9 n 2 = ( m–3n ) 2 m 2 +3mn+9 n 2 m.c.m= ( m–3n ) 2 ( m 2 +3mn+9 n 2 )
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