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Ejercicio 112

Gestor Baldor
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DESCARGA
CAPITULO XI

Máximo común divisor de polinomios por descomposición en factores
Ejercicio 112
Hallar, por descomposición en factores, el m.c.d. de:
  1. 2 a 2 +2ab,4 a 2 –4ab
    2 a 2 +2ab=2a( a+b ) 4 a 2 –4ab= 2 2 a( a–b ) m.c.d=2a
  2. 6 x 3 y–6 x 2 y,9 x 3 y 2 +18 x 2 y 2
    6 x 3 y–6 x 2 y=2.3 x 2 y( x–1 ) 9 x 3 y 2 +18 x 2 y 2 = 3 2 x 2 y 2 ( x+2 ) m.c.d=3 x 2 y
  3. 12 a 2 b 3 ,4 a 3 b 2 –8 a 2 b 3
    12 a 2 b 3 =3.4 a 2 b 3 4 a 3 b 2 –8 a 2 b 3 =4 a 2 b 2 ( a–2b ) m.c.d=4 a 2 b 2
  4. ab+b, a 2 +a
    ab+b=b( a+1 ) a 2 +a=a( a+1 ) m.c.d=a+1
  5. x 2 –x, x 3 – x 2
    x 2 –x=x( x–1 ) x 3 – x 2 = x 2 ( x–1 ) m.c.d=x–1
  6. 30a x 2 –15 x 3 ,10ax y 2 –20 x 2 y 2
    30a x 2 –15 x 3 =5.3 x 2 ( 2a–x ) 10ax y 2 –20 x 2 y 2 =5.2x y 2 ( a–2x ) m.c.d=5x
  7. 18 a 2 x 3 y 4 ,6 a 2 x 2 y 4 –18 a 2 x y 4
    18 a 2 x 3 y 4 =3.6 a 2 x 3 y 4 6 a 2 x 2 y 4 –18 a 2 x y 4 =6 a 2 x y 4 ( x–3 ) m.c.d=6 a 2 x y 4
  8. 5 a 2 –15a, a 3 –3 a 2
    5 a 2 –15a=5a( a–3 ) a 3 –3 a 2 = a 2 ( a–3 ) m.c.d=a( a–3 )
  9. 3 x 3 +15 x 2 ,a x 2 +5ax
    3 x 3 +15 x 2 =3 x 2 ( x+5 ) a x 2 +5ax=ax( x+5 ) m.c.d=x( x+5 )
  10. a 2 – b 2 , a 2 –2ab+ b 2
    a 2 – b 2 =( a–b ) ( a+b ) a 2 –2ab+ b 2 = ( a–b ) 2 m.c.d=a–b
  11. m 3 + n 3 ,3am+3an
    m 3 + n 3 =( m+n ) ( m 2 –mn+ n 2 ) 3am+3an=3a( m+n ) m.c.d=m+n
  12. x 2 –4, x 3 –8
    x 2 –4=( x–2 ) ( x+2 ) x 3 –8=( x–2 ) ( x 2 +2x+4 ) m.c.d=x–2
  13. 2a x 2 +4ax, x 3 – x 2 –6x
    2a x 2 +4ax=2ax( x+2 ) x 3 – x 2 –6x=x( x 2 –x–6 ) =x( x–3 ) ( x+2 ) m.c.d=x( x+2 )
  14. 9 x 2 –1,9 x 2 –6x+1
    9 x 2 –1=( 3x–1 ) ( 3x+1 ) 9 x 2 –6x+1= ( 3x–1 ) 2 m.c.d=3x–1
  15. 4 a 2 +4ab+ b 2 ,2 a 2 –2ab+ab– b 2
    4 a 2 +4ab+ b 2 = ( 2a+b ) 2 2 a 2 –2ab+ab– b 2 =2a( a–b ) +b( a–b ) =( a–b ) ( 2a+b ) m.c.d=2a+b
  16. 3 x 2 +3x–60,6 x 2 –18x–24
    3 x 2 +3x–60=3( x 2 +x–20 ) =3( x+5 ) ( x–4 ) 6 x 2 –18x–24=6( x 2 –3x–4 ) =6( x–4 ) ( x+1 ) m.c.d=3( x–4 )
  17. 8 x 3 + y 3 ,4a x 2 –a y 2
    8 x 3 + y 3 =( 2x+y ) ( 4 x 2 –2xy+ y 2 ) 4a x 2 –a y 2 =a( 4 x 2 – y 2 ) =a( 2x–y ) ( 2x+y ) m.c.d=2x+y
  18. 2 a 3 –12 a 2 b+18a b 2 , a 3 x–9a b 2 x
    2 a 3 –12 a 2 b+18a b 2 = 2a( a 2 –6ab+9 b 2 ) = 2a ( a–3b ) 2 a 3 x–9a b 2 x =ax( a 2 –9 b 2 ) =ax( a–3b ) ( a+3b ) m.c.d=a( a–3b )
  19. ac+ad–2bc–2bd,2 c 2 +4cd+2 d 2
    ac+ad–2bc–2bd =a( c+d ) –2b( c+d ) =( c+d ) ( a–2b ) 2 c 2 +4cd+2 d 2 =2( c 2 +2cd+ d 2 ) =2 ( c+d ) 2 m.c.d=c+d
  20. 3 a 2 m 2 +6 a 2 m–45 a 2 ,6a m 2 x+24amx–30ax
    3 a 2 m 2 +6 a 2 m–45 a 2 =3 a 2 ( m 2 +2m–15 ) =3 a 2 ( m+5 ) ( m–3 ) 6a m 2 x+24amx–30ax =6ax( m 2 +4m–5 ) =6ax( m+5 ) ( m–1 ) m.c.d=3a( m+5 )
  21. 4 x 4 – y 2 , ( 2 x 2 –y ) 2
    4 x 4 – y 2 =( 2 x 2 –y ) ( 2 x 2 +y ) ( 2 x 2 –y ) 2 = ( 2 x 2 –y ) 2 m.c.d=2 x 2 –y
  22. 3 x 5 –3x,9 x 3 –9x
    3 x 5 –3x =3x( x 4 –1 ) =3x( x 2 –1 ) ( x 2 +1 ) =3x( x–1 ) ( x+1 ) ( x 2 +1 ) 9 x 3 –9x =9x( x 2 –1 ) =9x( x–1 ) ( x+1 ) m.c.d=3x( x–1 )
  23. a 2 +ab,ab+ b 2 , a 3 + a 2 b
    a 2 +ab =a( a+b ) ab+ b 2 =b( a+b ) a 3 + a 2 b = a 2 ( a+b ) m.c.d=a( a+b )
  24. 2 x 3 –2 x 2 ,3 x 2 –3x,4 x 3 –4 x 2
    2 x 3 –2 x 2 =2 x 2 ( x–1 ) 3 x 2 –3x =3x( x–1 ) 4 x 3 –4 x 2 =4 x 2 ( x–1 ) m.c.d=x( x–1 )
  25. x 4 –9 x 2 , x 4 –5 x 3 +6 x 2 , x 4 –6 x 3 +9 x 2
    x 4 –9 x 2 = x 2 ( x 2 –9 ) = x 2 ( x–3 ) ( x+3 ) x 4 –5 x 3 +6 x 2 = x 2 ( x 2 –5x+6 ) = x 2 ( x–3 ) ( x–2 ) x 4 –6 x 3 +9 x 2 = x 2 ( x 2 –6x+9 ) = x 2 ( x–3 ) 2 m.c.d= x 2 ( x–3 )
  26. a 3 b+2 a 2 b 2 +a b 3 , a 4 b– a 2 b 3
    a 3 b+2 a 2 b 2 +a b 3 =ab( a 2 +2ab+ b 2 ) =ab ( a+b ) 2 a 4 b– a 2 b 3 = a 2 b( a 2 – b 2 ) = a 2 b( a+b ) ( a–b ) m.c.d=ab( a+b )
  27. 2 x 2 +2x–4,2 x 2 –8x+6,2 x 3 –2
    2 x 2 +2x–4 =2( x 2 +x–2 ) =2( x+2 ) ( x–1 ) 2 x 2 –8x+6 =2( x 2 –4x+3 ) =2( x–3 ) ( x–1 ) 2 x 3 –2 =2( x 3 –1 ) =2( x–1 ) ( x 2 +x+1 ) m.c.d=2( x–1 )
  28. a x 3 –2a x 2 –8ax,a x 2 –ax–6a, a 2 x 3 –3 a 2 x 2 –10 a 2 x
    a x 3 –2a x 2 –8ax =ax( x 2 –2x–8 ) =ax( x–4 ) ( x+2 ) a x 2 –ax–6a =a( x 2 –x–6 ) =a( x–3 ) ( x+2 ) a 2 x 3 –3 a 2 x 2 –10 a 2 x = a 2 x( x 2 –3x–10 ) = a 2 x( x–5 ) ( x+2 ) m.c.d=a( x+2 )
  29. 2a n 4 –16a n 2 +32a,2a n 3 –8an,2 a 2 n 3 +16 a 2
    2a n 4 –16a n 2 +32a =2a( n 4 –8 n 2 +16 ) =2a ( n 2 –4 ) 2 =2a [ ( n–2 ) ( n+2 ) ] 2 =2a ( n–2 ) 2 ( n+2 ) 2 2a n 3 –8an =2an( n 2 –4 ) =2an( n–2 ) ( n+2 ) 2 a 2 n 3 +16 a 2 =2 a 2 ( n 3 +8 ) =2 a 2 ( n+2 ) ( n 2 –2n+4 ) m.c.d=2a( n+2 )
  30. 4 a 2 +8a–12,2 a 2 –6a+4,6 a 2 +18a–24
    4 a 2 +8a–12 =4( a 2 +2a–3 ) =4( a+3 ) ( a–1 ) 2 a 2 –6a+4 =2( a 2 –3a+2 ) =2( a–2 ) ( a–1 ) 6 a 2 +18a–24 =6( a 2 +3a–4 ) =6( a+4 ) ( a–1 ) m.c.d=2( a–1 )
  31. 4 a 2 – b 2 ,8 a 3 + b 3 ,4 a 2 +4ab+ b 2
    4 a 2 – b 2 =( 2a–b ) ( 2a+b ) 8 a 3 + b 3 =( 2a+b ) ( 4 a 2 –2ab+ b 2 ) 4 a 2 +4ab+ b 2 = ( 2a+b ) 2 m.c.d=2a+b
  32. x 2 –2x–8, x 2 –x–12, x 3 –9 x 2 +20x
    x 2 –2x–8 =( x–4 ) ( x+2 ) x 2 –x–12 =( x–4 ) ( x+3 ) x 3 –9 x 2 +20x =x( x 2 –9x+20 ) =x( x–5 ) ( x–4 ) m.c.d=x–4
  33. a 2 +a, a 3 –6 a 2 –7a, a 6 +a
    a 2 +a =a( a+1 ) a 3 –6 a 2 –7a =a( a 2 –6a–7 ) =a( a–7 ) ( a+1 ) a 6 +a =a( a 5 +1 ) =a( a+1 ) ( a 4 – a 3 + a 2 –a+1 ) m.c.d=a( a+1 )
  34. x 3 +27,2 x 2 –6x+18, x 4 –3 x 3 +9 x 2
    x 3 +27 =( x+3 ) ( x 2 –3x+9 ) 2 x 2 –6x+18 =2( x 2 –3x+9 ) x 4 –3 x 3 +9 x 2 = x 2 ( x 2 –3x+9 ) m.c.d= x 2 –3x+9
  35. x 2 +ax–6 a 2 , x 2 +2ax–3 a 2 , x 2 +6ax+9 a 2
    x 2 +ax–6 a 2 =( x+3a ) ( x–2a ) x 2 +2ax–3 a 2 =( x+3a ) ( x–a ) x 2 +6ax+9 a 2 = ( x+3a ) 2 m.c.d=x+3a
  36. 54 x 3 +250,18a x 2 –50a,50+60x+18 x 2
    54 x 3 +250 =2( 27 x 3 +125 ) =2( 3x+5 ) ( 9 x 2 –15x+25 ) 18a x 2 –50a =2a( 9 x 2 –25 ) =2a( 3x+5 ) ( 3x–5 ) 50+60x+18 x 2 =2( 9 x 2 +30x+25 ) =2 ( 3x+5 ) 2 m.c.d=2( 3x+5 )
  37. ( x 2 –1 ) 2 , x 2 –4x–5, x 4 –1
    ( x 2 –1 ) 2 = [ ( x+1 ) ( x–1 ) ] 2 = ( x+1 ) 2 ( x–1 ) 2 x 2 –4x–5 =( x–4 ) ( x+1 ) x 4 –1 =( x 2 +1 ) ( x 2 –1 ) =( x 2 +1 ) ( x+1 ) ( x–1 ) m.c.d=x+1
  38. 4a x 2 –28ax, a 2 x 3 –8 a 2 x 2 +7 a 2 x,a x 4 –15a x 3 +56a x 2
    4a x 2 –28ax =4ax( x–7 ) a 2 x 3 –8 a 2 x 2 +7 a 2 x = a 2 x( x 2 –8x+7 ) = a 2 x( x–7 ) ( x–1 ) a x 4 –15a x 3 +56a x 2 =a x 2 ( x 2 –15x+56 ) =a x 2 ( x–8 ) ( x–7 ) m.c.d=ax( x–7 )
  39. 3 a 2 –6a, a 3 –4a, a 2 b–2ab, a 2 –a–2
    3 a 2 –6a =3a( a–2 ) a 3 –4a =a( a 2 –4 ) =a( a–2 ) ( a+2 ) a 2 b–2ab =ab( a–2 ) a 2 –a–2 =( a–2 ) ( a+1 ) m.c.d=a–2
  40. 3 x 2 –x,27 x 3 –1,9 x 2 –6x+1,3ax–a+6x–2
    3 x 2 –x =x( 3x–1 ) 27 x 3 –1 =( 3x–1 ) ( 9 x 2 +3x+1 ) 9 x 2 –6x+1 = ( 3x–1 ) 2 3ax–a+6x–2 =a( 3x–1 ) +2( 3x–1 ) =( 3x–1 ) ( a+2 ) m.c.d=3x–1
  41. a 4 –1, a 3 + a 2 +a+1, a 3 x+ a 2 x+ax+x, a 5 + a 3 + a 2 +1
    a 4 –1 =( a 2 +1 ) ( a 2 –1 ) =( a 2 +1 ) ( a+1 ) ( a–1 ) a 3 + a 2 +a+1 = a 2 ( a+1 ) +( a+1 ) =( a+1 ) ( a 2 +1 ) a 3 x+ a 2 x+ax+x = a 2 x( a+1 ) +x( a+1 ) =( a+1 ) ( a 2 x+x ) =x( a+1 ) ( a 2 +1 ) a 5 + a 3 + a 2 +1 = a 3 ( a 2 +1 ) +( a 2 +1 ) =( a 2 +1 ) ( a 3 +1 ) =( a 2 +1 ) ( a+1 ) ( a 2 –a+1 ) m.c.d=( a+1 ) ( a 2 +1 )
  42. 2 m 2 +4mn+2 n 2 , m 3 + m 2 n+m n 2 + n 3 , m 3 + n 3 , m 3 –m n 2
    2 m 2 +4mn+2 n 2 =2( m 2 +2mn+ n 2 ) =2 ( m+n ) 2 m 3 + m 2 n+m n 2 + n 3 = m 2 ( m+n ) + n 2 ( m+n ) =( m+n ) ( m 2 + n 2 ) m 3 + n 3 =( m+n ) ( m 2 –mn+ n 2 ) m 3 –m n 2 =m( m 2 – n 2 ) =m( m–n ) ( m+n ) m.c.d=m+n
  43. a 3 –3 a 2 +3a–1, a 2 –2a+1, a 3 –a, a 2 –4a+3
    a 3 –3 a 2 +3a–1 =( a 3 –1 ) –3a( a–1 ) =( a–1 ) ( a 2 +a+1 ) –3a( a–1 ) =( a–1 ) ( a 2 +a+1–3a ) =( a–1 ) ( a 2 –2a+1 ) a 2 –2a+1 = ( a–1 ) 2 a 3 –a =a( a 2 –1 ) =a( a–1 ) ( a+1 ) a 2 –4a+3 =( a–3 ) ( a–1 ) m.c.d=a–1
  44. 16 a 3 x+54x,12 a 2 x 2 –42a x 2 –90 x 2 ,32 a 3 x+24 a 2 x–36ax,32 a 4 x–144 a 2 x+162x
    16 a 3 x+54x =2x( 8 a 3 +27 ) =2x( 2a+3 ) ( 4 a 2 –6a+9 ) 12 a 2 x 2 –42a x 2 –90 x 2 =6 x 2 ( 2 a 2 –7a–15 ) =6 x 2 ( 2 a 2 –10a+3a–15 ) =6 x 2 [ 2a( a–5 ) +3( a–5 ) ] =6 x 2 ( a–5 ) ( 2a+3 ) 32 a 3 x+24 a 2 x–36ax =4ax( 8 a 2 +6a–9 ) =4ax( 8 a 2 +12a–6a–9 ) =4ax[ 4a( 2a+3 ) –3( 2a+3 ) ] =4ax( 2a+3 ) ( 4a–3 ) 32 a 4 x–144 a 2 x+162x =2x( 16 a 4 –72 a 2 +81 ) =2x ( 4 a 2 –9 ) 2 =2x [ ( 2a–3 ) ( 2a+3 ) ] 2 =2x ( 2a–3 ) 2 ( 2a+3 ) 2 m.c.d=2x( 2a+3 )
  45. ( xy+ y 2 ) 2 , x 2 y–2x y 2 –3 y 3 ,a x 3 y+a y 4 , x 2 y– y 3
    ( xy+ y 2 ) 2 = [ y( x+y ) ] 2 = y 2 ( x+y ) 2 x 2 y–2x y 2 –3 y 3 =y( x 2 –2xy–3 y 2 ) =y( x–3 ) ( x+1 ) a x 3 y+a y 4 =ay( x 3 + y 4 ) x 2 y– y 3 =y( x 2 – y 2 ) =y( x–y ) ( x+y ) m.c.d=y
  46. 2 a 2 –am+4a–2m,2a m 2 – m 3 ,6 a 2 +5am–4 m 2 ,16 a 2 +72am–40 m 2
    2 a 2 –am+4a–2m =a( 2a–m ) +2( 2a–m ) =( 2a–m ) ( a+2 ) 2a m 2 – m 3 = m 2 ( 2a–m ) 6 a 2 +5am–4 m 2 =6 a 2 –3am+8am–4 m 2 =3a( 2a–m ) +4m( 2a–m ) =( 2a–m ) ( 3a+4m ) 16 a 2 +72am–40 m 2 =8( 2 a 2 +9am–5 m 2 ) =8( 2 a 2 –am+10am–5 m 2 ) =8[ a( 2a–m ) +5m( 2a–m ) ] =8( 2a–m ) ( a+5m ) m.c.d=2a–m
  47. 12ax–6ay+24bx–12by,3 a 3 +24 b 3 ,9 a 2 +9ab–18 b 2 ,12 a 2 +24ab
    12ax–6ay+24bx–12by =6a( 2x–y ) +12b( 2x–y ) =( 2x–y ) ( 6a+12b ) =6( 2x–y ) ( a+2b ) 3 a 3 +24 b 3 =3( a 3 +8 b 3 ) =3( a+2b ) ( a 2 –2ab+4 b 2 ) 9 a 2 +9ab–18 b 2 =9( a 2 +ab–2 b 2 ) =9( a+2b ) ( a–b ) 12 a 2 +24ab =12a( a+2b ) m.c.d=a+2b
  48. 5 a 2 +5ax+5ay+5xy,15 a 3 –15a x 2 +15 a 2 y–15 x 2 y,20 a 3 –20a y 2 +20 a 2 x–20x y 2 ,5 a 5 +5 a 4 x+5 a 2 y 3 +5ax y 3
    5 a 2 +5ax+5ay+5xy =5a( a+x ) +5y( a+x ) =( a+x ) ( 5a+5y ) =5( a+x ) ( a+y ) 15 a 3 –15a x 2 +15 a 2 y–15 x 2 y =15 a 3 +15 a 2 y–15a x 2 –15 x 2 y =15 a 2 ( a+y ) –15 x 2 ( a+y ) =( a+y ) ( 15 a 2 –15 x 2 ) =15( a+y ) ( a 2 – x 2 ) =15( a+y ) ( a+x ) ( a–x ) 20 a 3 –20a y 2 +20 a 2 x–20x y 2 =20a( a 2 – y 2 ) +20x( a 2 – y 2 ) =( a 2 – y 2 ) ( 20a+20x ) =20( a+y ) ( a–y ) ( a+x ) 5 a 5 +5 a 4 x+5 a 2 y 3 +5ax y 3 =5 a 4 ( a+x ) +5a y 3 ( a+x ) =( a+x ) ( 5 a 4 +5a y 3 ) =( a+x ) 5a( a 3 + y 3 ) =5a( a+x ) ( a+y ) ( a 2 –ay+ y 2 ) m.c.d=5( a+x ) ( a+y )
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