Ejercicio 95

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CAPITULO X

Descomposición Factorial
Ejercicio 95
Factorar o descomponer en dos factores:
  1. a 2 +2ab+ b 2 x 2 = (a+b ) 2 x 2 =[(a+b ) +x ] [(a+b ) x ] =(a+b+x ) (a+bx )
  2. x 2 2xy+ y 2 m 2 = (xy ) 2 m 2 =[(xy ) +m ] [(xy ) m ] =(xy+m ) (xym )
  3. m 2 +2mn+ n 2 1 = (m+n ) 2 1 =[(m+n ) 1 ] [(m+n ) +1 ] =(m+n1 ) (m+n+1 )
  4. a 2 2a+1 b 2 = (a1 ) 2 b 2 =[(a1 ) +b ] [(a1 ) b ] =(a+b1 ) (ab1 )
  5. n 2 +6n+9 c 2 = (n+3 ) 2 c 2 =[(n+3 ) c ] [(n+3 ) +c ] =(nc+3 ) (n+c+3 )
  6. a 2 + x 2 +2ax4 = a 2 +2ax+ x 2 4 = (a+x ) 2 4 =[(a+x ) 2 ] [(a+x ) +2 ] =(a+x2 ) (a+x+2 )
  7. a 2 +44a9 b 2 = a 2 4a+49 b 2 = (a2 ) 2 9 b 2 =[(a2 ) +3b ] [(a2 ) 3b ] =(a+3b2 ) (a3b2 )
  8. x 2 +4 y 2 4xy1 = x 2 4xy+4 y 2 1 = (x2y ) 2 1 =[(x2y ) 1 ] [(x2y ) +1 ] =(x2y1 ) (x2y+1 )
  9. a 2 6ay+9 y 2 4 x 2 = (a3y ) 2 4 x 2 =[(a3y ) 2x ] [(a3y ) +2x ] =(a2x3y ) (a+2x3y )
  10. 4 x 2 +25 y 2 36+20xy =4 x 2 +20xy+25 y 2 36 = (2x+5y ) 2 36 =[(2x+5y ) 6 ] [(2x+5y ) +6 ] =(2x+5y6 ) (2x+5y+6 )
  11. 9 x 2 1+16 a 2 24ax =9 x 2 24ax+16 a 2 1 = (3x4a ) 2 1 =[(3x4a ) 1 ] [(3x4a ) +1 ] =(3x4a1 ) (3x4a+1 )
  12. 1+64 a 2 b 2 x 4 16ab =116ab+64 a 2 b 2 x 4 = (18ab ) 2 x 4 =[(18ab ) + x 2 ] [(18ab ) x 2 ] =(18ab+ x 2 ) (18ab x 2 )
  13. a 2 b 2 2bc c 2 = a 2 ( b 2 +2bc+ c 2 ) = a 2 (b+c ) 2 =[a+(b+c ) ] [a(b+c ) ] =(a+b+c ) (abc )
  14. 1 a 2 +2ax x 2 =1( a 2 2ax+ x 2 ) =1 (ax ) 2 =[1(ax ) ] [1+(ax ) ] =(1a+x ) (1+ax )
  15. m 2 x 2 2xy y 2 = m 2 ( x 2 +2xy+ y 2 ) = m 2 (x+y ) 2 =[m(x+y ) ] [m+(x+y ) ] =(mxy ) (m+x+y )
  16. c 2 a 2 +2a1 = c 2 ( a 2 2a+1 ) = c 2 (a1 ) 2 =[c(a1 ) ] [c+(a1 ) ] =(ca+1 ) (c+a1 )
  17. 9 n 2 2510n =9 n 2 10n25 =9( n 2 +10n+25 ) =9 (n+5 ) 2 =[3(n+5 ) ] [3+(n+5 ) ] =(3n5 ) (3+n+5 ) =(2n ) (n+8 ) =(n+2 ) (n+8 )
  18. 4 a 2 x 2 +4x4 =4 a 2 ( x 2 4x+4 ) =4 a 2 (x2 ) 2 =[2a(x2 ) ] [2a+(x2 ) ] =(2ax+2 ) (2a+x2 )
  19. 1 a 2 9 n 2 6an =1 a 2 6an9 n 2 =1( a 2 +6an+9 n 2 ) =1 (a+3n ) 2 =[1(a+3n ) ] [1+(a+3n ) ] =(1a3n ) (1+a+3n )
  20. 25 x 2 16 y 2 +8xy =25 x 2 +8xy16 y 2 =25( x 2 8xy+16 y 2 ) =25 (x4y ) 2 =[5+(x4y ) ] [5(x4y ) ] =(5+x4y ) (5x+4y )
  21. 9 x 2 a 2 4 m 2 +4am =9 x 2 a 2 +4am4 m 2 =9 x 2 ( a 2 4am+4 m 2 ) =9 x 2 (a2m ) 2 =[3x(a2m ) ] [3x+(a2m ) ] =(3xa+2m ) (3x+a2m )
  22. 16 x 2 y 2 +12ab4 a 2 9 b 2 =16 x 2 y 2 4 a 2 +12ab9 b 2 =16 x 2 y 2 (4 a 2 12ab+9 b 2 ) =16 x 2 y 2 (2a3b ) 2 =[4xy(2a3b ) ] [4xy+(2a3b ) ] =(4xy2a+3b ) (4xy+2a3b )
  23. a 2 +25 m 2 12a =25 m 2 a 2 2a1 =25 m 2 ( a 2 +2a+1 ) =25 m 2 (a+1 ) 2 =[5m(a+1 ) ] [5m+(a+1 ) ] =(5ma1 ) (5m+a+1 )
  24. 49 x 4 25 x 2 9 y 2 +30xy =49 x 4 25 x 2 +30xy9 y 2 =49 x 4 (25 x 2 30xy+9 y 2 ) =49 x 4 (5x3y ) 2 =[7 x 2 (5x3y ) ] [7 x 2 +(5x3y ) ] =(7 x 2 5x+3y ) (7 x 2 +5x3y )
  25. a 2 2ab+ b 2 c 2 2cd d 2 =( a 2 2ab+ b 2 ) ( c 2 +2cd+ d 2 ) = (ab ) 2 (c+d ) 2 =[(ab ) +(c+d ) ] [(ab ) (c+d ) ] =(ab+c+d ) (abcd )
  26. x 2 +2xy+ y 2 m 2 +2mn n 2 =( x 2 +2xy+ y 2 ) ( m 2 2mn+ n 2 ) = (x+y ) 2 (mn ) 2 =[(x+y ) (mn ) ] [(x+y ) +(mn ) ] =(x+ym+n ) (x+y+mn )
  27. a 2 +4 b 2 +4ab x 2 2ax a 2 =( a 2 +4ab+4 b 2 ) ( x 2 +2ax+ a 2 ) = (a+2b ) 2 (x+a ) 2 =[(a+2b ) (x+a ) ] [(a+2b ) +(x+a ) ] =( a +2bx a ) (a+2b+x+a ) =(2bx ) (2a+2b+x )
  28. x 2 +4 a 2 4ax y 2 9 b 2 +6by =( x 2 4ax+4 a 2 ) ( y 2 6by+9 b 2 ) = (x2a ) 2 (y3b ) 2 =[(x2a ) (y3b ) ] [(x2a ) +(y3b ) ] =(x2ay+3b ) (x2a+y3b )
  29. m 2 x 2 +9 n 2 +6mn4ax4 a 2 =( m 2 +6mn+9 n 2 ) ( x 2 +4ax+4 a 2 ) = (m+3n ) 2 (x+2a ) 2 =[(m+3n ) (x+2a ) ] [(m+3n ) +(x+2a ) ] =(m+3nx2a ) (m+3n+x+2a )
  30. 9 x 2 +4 y 2 a 2 12xy25 b 2 10ab =(9 x 2 12xy+4 y 2 ) ( a 2 +10ab+25 b 2 ) = (3x2y ) 2 (a+5b ) 2 =[(3x2y ) (a+5b ) ] [(3x2y ) +(a+5b ) ] =(3x2ya5b ) (3x2y+a+5b )
  31. 2am x 2 9+ a 2 + m 2 6x =( a 2 +2am+ m 2 ) ( x 2 +6x+9 ) = (a+m ) 2 (x+3 ) 2 =[(a+m ) (x+3 ) ] [(a+m ) +(x+3 ) ] =(a+mx3 ) (a+m+x+3 )
  32. x 2 9 a 4 +6 a 2 b+1+2x b 2 =( x 2 +2x+1 ) (9 a 4 6 a 2 b+ b 2 ) = (x+1 ) 2 (3 a 2 b ) 2 =[(x+1 ) (3 a 2 b ) ] [(x+1 ) +(3 a 2 b ) ] =(x+13 a 2 +b ) (x+1+3 a 2 b )
  33. 16 a 2 110m+9 x 2 24ax25 m 2 =(16 a 2 24ax+9 x 2 ) (25 m 2 +10m+1 ) = (4a3x ) 2 (5m+1 ) 2 =[(4a3x ) +(5m+1 ) ] [(4a3x ) (5m+1 ) ] =(4a3x+5m+1 ) (4a3x5m1 )
  34. 9 m 2 a 2 +2acd c 2 d 2 +10060m =(9 m 2 60m+100 ) ( a 2 2acd+ c 2 d 2 ) = (3m+10 ) 2 (acd ) 2 =[(3m+10 ) (acd ) ] [(3m+10 ) +(acd ) ] =(3m+10a+cd ) (3m+10+acd )
  35. 4 a 2 9 x 2 +49 b 2 30xy25 y 2 28ab =(4 a 2 28ab+49 b 2 ) (9 x 2 +30xy+25 y 2 ) = (2a7b ) 2 (3x+5y ) 2 =[(2a7b ) (3x+5y ) ] [(2a7b ) +(3x+5y ) ] =(2a7b3x5y ) (2a7b+3x+5y )
  36. 225 a 2 169 b 2 +1+30a+26bc c 2 =(225 a 2 +30a+1 ) (169 b 2 26bc+ c 2 ) = (15a+1 ) 2 (13bc ) 2 =[(15a+1 ) (13bc ) ] [(15a+1 ) +(13bc ) ] =(15a+113b+c ) (15a+1+13bc ) =(15a13b+c+1 ) (15a+13bc+1 )
  37. x 2 y 2 +4+4x12y =( x 2 +4x+4 ) ( y 2 +2y+1 ) = (x+2 ) 2 (y+1 ) 2 =[(x+2 ) (y+1 ) ] [(x+2 ) +(y+1 ) ] =(x+2y1 ) (x+2+y+1 ) =(xy+1 ) (x+y+3 )
  38. a 2 16 x 2 +36+12a8x =( a 2 +12a+36 ) ( x 2 +8x+16 ) = (a+6 ) 2 (x+4 ) 2 =[(a+6 ) (x+4 ) ] [(a+6 ) +(x+4 ) ] =(a+6x4 ) (a+6+x+4 ) =(ax+2 ) (a+x+10 )