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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(ab ) m.c.d=2a
  2. 6 x 3 y6 x 2 y,9 x 3 y 2 +18 x 2 y 2
    6 x 3 y6 x 2 y=2.3 x 2 y(x1 ) 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 (a2b ) 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(x1 ) x 3 x 2 = x 2 (x1 ) m.c.d=x1
  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 (2ax ) 10ax y 2 20 x 2 y 2 =5.2x y 2 (a2x ) 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 (x3 ) m.c.d=6 a 2 x y 4
  8. 5 a 2 15a, a 3 3 a 2
    5 a 2 15a=5a(a3 ) a 3 3 a 2 = a 2 (a3 ) m.c.d=a(a3 )
  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 =(ab ) (a+b ) a 2 2ab+ b 2 = (ab ) 2 m.c.d=ab
  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=(x2 ) (x+2 ) x 3 8=(x2 ) ( x 2 +2x+4 ) m.c.d=x2
  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 x6 ) =x(x3 ) (x+2 ) m.c.d=x(x+2 )
  14. 9 x 2 1,9 x 2 6x+1
    9 x 2 1=(3x1 ) (3x+1 ) 9 x 2 6x+1= (3x1 ) 2 m.c.d=3x1
  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(ab ) +b(ab ) =(ab ) (2a+b ) m.c.d=2a+b
  16. 3 x 2 +3x60,6 x 2 18x24
    3 x 2 +3x60=3( x 2 +x20 ) =3(x+5 ) (x4 ) 6 x 2 18x24=6( x 2 3x4 ) =6(x4 ) (x+1 ) m.c.d=3(x4 )
  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(2xy ) (2x+y ) m.c.d=2x+y
  18. 2 a 3 12 a 2 b+18a b 2 , a 3 x9a b 2 x
    2 a 3 12 a 2 b+18a b 2 = 2a( a 2 6ab+9 b 2 ) = 2a (a3b ) 2 a 3 x9a b 2 x =ax( a 2 9 b 2 ) =ax(a3b ) (a+3b ) m.c.d=a(a3b )
  19. ac+ad2bc2bd,2 c 2 +4cd+2 d 2
    ac+ad2bc2bd =a(c+d ) 2b(c+d ) =(c+d ) (a2b ) 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 m45 a 2 ,6a m 2 x+24amx30ax
    3 a 2 m 2 +6 a 2 m45 a 2 =3 a 2 ( m 2 +2m15 ) =3 a 2 (m+5 ) (m3 ) 6a m 2 x+24amx30ax =6ax( m 2 +4m5 ) =6ax(m+5 ) (m1 ) 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(x1 ) (x+1 ) ( x 2 +1 ) 9 x 3 9x =9x( x 2 1 ) =9x(x1 ) (x+1 ) m.c.d=3x(x1 )
  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 (x1 ) 3 x 2 3x =3x(x1 ) 4 x 3 4 x 2 =4 x 2 (x1 ) m.c.d=x(x1 )
  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 (x3 ) (x+3 ) x 4 5 x 3 +6 x 2 = x 2 ( x 2 5x+6 ) = x 2 (x3 ) (x2 ) x 4 6 x 3 +9 x 2 = x 2 ( x 2 6x+9 ) = x 2 (x3 ) 2 m.c.d= x 2 (x3 )
  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 ) (ab ) m.c.d=ab(a+b )
  27. 2 x 2 +2x4,2 x 2 8x+6,2 x 3 2
    2 x 2 +2x4 =2( x 2 +x2 ) =2(x+2 ) (x1 ) 2 x 2 8x+6 =2( x 2 4x+3 ) =2(x3 ) (x1 ) 2 x 3 2 =2( x 3 1 ) =2(x1 ) ( x 2 +x+1 ) m.c.d=2(x1 )
  28. a x 3 2a x 2 8ax,a x 2 ax6a, a 2 x 3 3 a 2 x 2 10 a 2 x
    a x 3 2a x 2 8ax =ax( x 2 2x8 ) =ax(x4 ) (x+2 ) a x 2 ax6a =a( x 2 x6 ) =a(x3 ) (x+2 ) a 2 x 3 3 a 2 x 2 10 a 2 x = a 2 x( x 2 3x10 ) = a 2 x(x5 ) (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 [(n2 ) (n+2 ) ] 2 =2a (n2 ) 2 (n+2 ) 2 2a n 3 8an =2an( n 2 4 ) =2an(n2 ) (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 +8a12,2 a 2 6a+4,6 a 2 +18a24
    4 a 2 +8a12 =4( a 2 +2a3 ) =4(a+3 ) (a1 ) 2 a 2 6a+4 =2( a 2 3a+2 ) =2(a2 ) (a1 ) 6 a 2 +18a24 =6( a 2 +3a4 ) =6(a+4 ) (a1 ) m.c.d=2(a1 )
  31. 4 a 2 b 2 ,8 a 3 + b 3 ,4 a 2 +4ab+ b 2
    4 a 2 b 2 =(2ab ) (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 2x8, x 2 x12, x 3 9 x 2 +20x
    x 2 2x8 =(x4 ) (x+2 ) x 2 x12 =(x4 ) (x+3 ) x 3 9 x 2 +20x =x( x 2 9x+20 ) =x(x5 ) (x4 ) m.c.d=x4
  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 6a7 ) =a(a7 ) (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 +ax6 a 2 , x 2 +2ax3 a 2 , x 2 +6ax+9 a 2
    x 2 +ax6 a 2 =(x+3a ) (x2a ) x 2 +2ax3 a 2 =(x+3a ) (xa ) 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 ) (3x5 ) 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 4x5, x 4 1
    ( x 2 1 ) 2 = [(x+1 ) (x1 ) ] 2 = (x+1 ) 2 (x1 ) 2 x 2 4x5 =(x4 ) (x+1 ) x 4 1 =( x 2 +1 ) ( x 2 1 ) =( x 2 +1 ) (x+1 ) (x1 ) 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(x7 ) a 2 x 3 8 a 2 x 2 +7 a 2 x = a 2 x( x 2 8x+7 ) = a 2 x(x7 ) (x1 ) a x 4 15a x 3 +56a x 2 =a x 2 ( x 2 15x+56 ) =a x 2 (x8 ) (x7 ) m.c.d=ax(x7 )
  39. 3 a 2 6a, a 3 4a, a 2 b2ab, a 2 a2
    3 a 2 6a =3a(a2 ) a 3 4a =a( a 2 4 ) =a(a2 ) (a+2 ) a 2 b2ab =ab(a2 ) a 2 a2 =(a2 ) (a+1 ) m.c.d=a2
  40. 3 x 2 x,27 x 3 1,9 x 2 6x+1,3axa+6x2
    3 x 2 x =x(3x1 ) 27 x 3 1 =(3x1 ) (9 x 2 +3x+1 ) 9 x 2 6x+1 = (3x1 ) 2 3axa+6x2 =a(3x1 ) +2(3x1 ) =(3x1 ) (a+2 ) m.c.d=3x1
  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 ) (a1 ) 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(mn ) (m+n ) m.c.d=m+n
  43. a 3 3 a 2 +3a1, a 2 2a+1, a 3 a, a 2 4a+3
    a 3 3 a 2 +3a1 =( a 3 1 ) 3a(a1 ) =(a1 ) ( a 2 +a+1 ) 3a(a1 ) =(a1 ) ( a 2 +a+13a ) =(a1 ) ( a 2 2a+1 ) a 2 2a+1 = (a1 ) 2 a 3 a =a( a 2 1 ) =a(a1 ) (a+1 ) a 2 4a+3 =(a3 ) (a1 ) m.c.d=a1
  44. 16 a 3 x+54x,12 a 2 x 2 42a x 2 90 x 2 ,32 a 3 x+24 a 2 x36ax,32 a 4 x144 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 7a15 ) =6 x 2 (2 a 2 10a+3a15 ) =6 x 2 [2a(a5 ) +3(a5 ) ] =6 x 2 (a5 ) (2a+3 ) 32 a 3 x+24 a 2 x36ax =4ax(8 a 2 +6a9 ) =4ax(8 a 2 +12a6a9 ) =4ax[4a(2a+3 ) 3(2a+3 ) ] =4ax(2a+3 ) (4a3 ) 32 a 4 x144 a 2 x+162x =2x(16 a 4 72 a 2 +81 ) =2x (4 a 2 9 ) 2 =2x [(2a3 ) (2a+3 ) ] 2 =2x (2a3 ) 2 (2a+3 ) 2 m.c.d=2x(2a+3 )
  45. (xy+ y 2 ) 2 , x 2 y2x 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 y2x y 2 3 y 3 =y( x 2 2xy3 y 2 ) =y(x3 ) (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(xy ) (x+y ) m.c.d=y
  46. 2 a 2 am+4a2m,2a m 2 m 3 ,6 a 2 +5am4 m 2 ,16 a 2 +72am40 m 2
    2 a 2 am+4a2m =a(2am ) +2(2am ) =(2am ) (a+2 ) 2a m 2 m 3 = m 2 (2am ) 6 a 2 +5am4 m 2 =6 a 2 3am+8am4 m 2 =3a(2am ) +4m(2am ) =(2am ) (3a+4m ) 16 a 2 +72am40 m 2 =8(2 a 2 +9am5 m 2 ) =8(2 a 2 am+10am5 m 2 ) =8[a(2am ) +5m(2am ) ] =8(2am ) (a+5m ) m.c.d=2am
  47. 12ax6ay+24bx12by,3 a 3 +24 b 3 ,9 a 2 +9ab18 b 2 ,12 a 2 +24ab
    12ax6ay+24bx12by =6a(2xy ) +12b(2xy ) =(2xy ) (6a+12b ) =6(2xy ) (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 +9ab18 b 2 =9( a 2 +ab2 b 2 ) =9(a+2b ) (ab ) 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 y15 x 2 y,20 a 3 20a y 2 +20 a 2 x20x 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 y15 x 2 y =15 a 3 +15 a 2 y15a 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 ) (ax ) 20 a 3 20a y 2 +20 a 2 x20x y 2 =20a( a 2 y 2 ) +20x( a 2 y 2 ) =( a 2 y 2 ) (20a+20x ) =20(a+y ) (ay ) (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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