Ejercicio 108

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DESCARGA
CAPITULO X

Descomposición Factorial
Ejercicio 108
Descomponer en cuatro factores:
  1. 1– a 8 =(1+ a 4 ) (1– a 4 ) =(1+ a 4 ) (1+ a 2 ) (1– a 2 ) =(1+ a 4 ) (1+ a 2 ) (1+a ) (1–a )
  2. a 6 –1 =( a 3 –1 ) ( a 3 +1 ) =(a–1 ) ( a 2 +a+1 ) (a+1 ) ( a 2 –a+1 )
  3. x 4 –41 x 2 +400 = x 4 –41 x 2 +400+ x 2 – x 2 = x 4 –40 x 2 +400– x 2 = ( x 2 –20 ) 2 – x 2 =[( x 2 –20 ) –x ] [( x 2 –20 ) +x ] =( x 2 –x–20 ) ( x 2 +x–20 ) =( x 2 –5x+4x–20 ) ( x 2 –4x+5x–20 ) =[x(x–5 ) +4(x–5 ) ] [x(x–4 ) +5(x–4 ) ] =(x–5 ) (x+4 ) (x–4 ) (x+5 )
  4. a 4 –2 a 2 b 2 + b 4 = ( a 2 – b 2 ) 2 = [(a–b ) (a+b ) ] 2 = (a–b ) 2 (a+b ) 2 =(a–b ) (a–b ) (a+b ) (a+b )
  5. x 5 + x 3 –2x =x( x 4 + x 2 –2 ) . 2 1 | 2 =x( x 4 – x 2 +2 x 2 –2 ) =x[ x 2 ( x 2 –1 ) +( x 2 –1 ) ] =x( x 2 –1 ) ( x 2 +1 ) =x(x–1 ) (x+1 ) ( x 2 +1 )
  6. 2 x 4 +6 x 3 –2x–6 =2 x 3 (x+3 ) –2(x+3 ) =(x+3 ) (2 x 3 –2 ) =2(x+3 ) ( x 3 –1 ) =2(x+3 ) (x–1 ) ( x 2 +x+1 )
  7. 3 x 4 –243 =3( x 4 –81 ) =3( x 2 +9 ) ( x 2 –9 ) =3( x 2 +9 ) (x–3 ) (x+3 )
  8. 16 x 4 –8 x 2 y 2 + y 4 = (4 x 2 – y 2 ) 2 = [(2x+y ) (2x–y ) ] 2 = (2x+y ) 2 (2x–y ) 2 =(2x+y ) (2x+y ) (2x–y ) (2x–y )
  9. 9 x 4 +9 x 3 y– x 2 –xy =9 x 3 (x+y ) –x(x+y ) =(x+y ) (9 x 3 –x ) =x(x+y ) (9 x 2 –1 ) =x(x+y ) (3x–1 ) (3x+1 )
  10. 12a x 4 +33a x 2 –9a =3a(4 x 4 +11 x 2 –3 ) . 12 1 | 12 =3a(4 x 4 – x 2 +12 x 2 –3 ) =3a[ x 2 (4 x 2 –1 ) +3(4 x 2 –1 ) ] =3a( x 2 +3 ) (4 x 2 –1 ) =3a( x 2 +3 ) (2x–1 ) (2x+1 )
  11. x 8 – y 8 =( x 4 + y 4 ) ( x 4 – y 4 ) =( x 4 + y 4 ) ( x 2 + y 2 ) ( x 2 – y 2 ) =( x 4 + y 4 ) ( x 2 + y 2 ) (x+y ) (x–y )
  12. x 6 –7 x 3 –8 = x 6 –8 x 3 + x 3 –8 . 8 1 | 8 = x 3 ( x 3 –8 ) +( x 3 –8 ) =( x 3 –8 ) ( x 3 +8 ) =(x–2 ) ( x 2 +2x+4 ) (x+2 ) ( x 2 –2x+4 )
  13. 64– x 6 =(8– x 3 ) (8+ x 3 ) =(2–x ) (4+2x+ x 2 ) (2+x ) (4–2x+ x 2 )
  14. a 5 – a 3 b 2 – a 2 b 3 + b 5 = a 3 ( a 2 – b 2 ) – b 3 ( a 2 – b 2 ) =( a 2 – b 2 ) ( a 3 – b 3 ) =(a–b ) (a+b ) (a–b ) ( a 2 +ab+ b 2 )
  15. 8 x 4 +6 x 2 –2 =2(4 x 4 +3 x 2 –1 ) . 4 1 | 4 =2(4 x 4 – x 2 +4 x 2 –1 ) =2[ x 2 (4 x 2 –1 ) +(4 x 2 –1 ) ] =2(4 x 2 –1 ) ( x 2 +1 ) =2(2x+1 ) (2x–1 ) ( x 2 +1 )
  16. a 4 –25 a 2 +144 = a 4 –9 a 2 –16 a 2 +144 . 144 72 36 18 9 1 | 2 2 2 2 9 = a 2 ( a 2 –9 ) –16( a 2 –9 ) =( a 2 –9 ) ( a 2 –16 ) =(a–9 ) (a+3 ) (a+4 ) (a–4 )
  17. a 2 x 3 – a 2 y 3 +2a x 3 –2a y 3 = a 2 ( x 3 – y 3 ) +2a( x 3 – y 3 ) =( a 2 +2a ) ( x 3 – y 3 ) =a(a+2 ) (x–y ) ( x 2 +xy+ y 2 )
  18. a 4 +2 a 3 – a 2 –2a = a 4 – a 2 +2 a 3 –2a = a 2 ( a 2 –1 ) +2a( a 2 –1 ) =( a 2 +2a ) ( a 2 –1 ) =a(a+2 ) (a–1 ) (a+1 )
  19. 1–2 a 3 + a 6 = (1– a 3 ) 2 = [(1–a ) (1+a+ a 2 ) ] 2 = (1–a ) 2 (1+a+ a 2 ) 2
  20. m 6 –729 =( m 3 +27 ) ( m 3 –27 ) =(m+3 ) ( m 2 –3m+9 ) (m–3 ) ( m 2 +3m+9 )
  21. x 5 –x =x( x 4 –1 ) =x( x 2 +1 ) ( x 2 –1 ) =x( x 2 +1 ) (x–1 ) (x+1 )
  22. x 5 – x 3 y 2 + x 2 y 3 – y 5 = x 3 ( x 2 – y 2 ) + y 3 ( x 2 – y 2 ) =( x 2 – y 2 ) ( x 3 + y 3 ) =(x+y ) (x–y ) (x+y ) ( x 2 –xy+ y 2 )
  23. a 4 b– a 3 b 2 – a 2 b 3 +a b 4 = a 3 b(a–b ) –a b 3 (a–b ) =(a–b ) ( a 3 b–a b 3 ) =ab(a–b ) ( a 2 – b 2 ) =ab(a–b ) (a+b ) (a–b )
  24. 5 a 4 –3125 =5( a 4 –625 ) =5( a 2 +25 ) ( a 2 –25 ) =5( a 2 +25 ) (a–5 ) (a+5 )
  25. ( a 2 +2a ) 2 –2( a 2 +2a ) –3 = ( a 2 +2a ) 2 –3( a 2 +2a ) +( a 2 +2a ) –3 . 3 1 | 3 =( a 2 +2a ) [( a 2 +2a ) –3 ] +[( a 2 +2a ) –3 ] =[( a 2 +2a ) –3 ] [( a 2 +2a ) +1 ] =( a 2 +2a–3 ) ( a 2 +2a+1 ) =( a 2 –a+3a–3 ) (a+1 ) 2 =[a(a–1 ) +3(a–1 ) ] (a+1 ) 2 =(a+3 ) (a–1 ) (a+1 ) 2 =(a+3 ) (a–1 ) (a+1 ) (a+1 )
  26. a 2 x 3 +2a x 3 –8 a 2 –16a =a x 3 (a+2 ) –8a(a+2 ) =(a+2 ) (a x 3 –8a ) =a(a+2 ) ( x 3 –8 ) =a(a+2 ) (x–2 ) ( x 2 +2x+4 )
  27. 1– a 6 b 6 =(1– a 3 b 3 ) (1+ a 3 b 3 ) =(1–ab ) (1+ab+ a 2 b 2 ) (1+ab ) (1–ab+ a 2 b 2 )
  28. 5a x 3 +10a x 2 –5ax–10a =5a x 3 –5ax+10a x 2 –10a =5ax( x 2 –1 ) +10a( x 2 –1 ) =( x 2 –1 ) (5ax+10a ) =5a(x–1 ) (x+1 ) (x+2 )
  29. a 2 x 2 + b 2 y 2 – b 2 x 2 – a 2 y 2 = a 2 x 2 – a 2 y 2 – b 2 x 2 + b 2 y 2 = a 2 ( x 2 – y 2 ) – b 2 ( x 2 – y 2 ) =( x 2 – y 2 ) ( a 2 – b 2 ) =(x+y ) (x–y ) (a+b ) (a–b )
  30. x 8 + x 4 –2 = x 8 – x 4 +2 x 4 –2 . 2 1 | 2 = x 4 ( x 4 –1 ) +2( x 4 –1 ) =( x 4 –1 ) ( x 4 +2 ) =( x 2 –1 ) ( x 2 +1 ) ( x 4 +2 ) =(x–1 ) (x+1 ) ( x 2 +1 ) ( x 4 +2 )
  31. a 4 + a 3 –9 a 2 –9a = a 3 (a+1 ) –9a(a+1 ) =(a+1 ) ( a 3 –9a ) =a(a+1 ) ( a 2 –9 ) =a(a+1 ) (a–3 ) (a+3 )
  32. a 2 x 2 + a 2 x–6 a 2 – x 2 –x+6 = a 2 ( x 2 +x–6 ) –( x 2 +x–6 ) =( x 2 +x–6 ) ( a 2 –1 ) =( x 2 –2x+3x–6 ) ( a 2 –1 ) =[x(x–2 ) +3(x–2 ) ] ( a 2 –1 ) =(x+3 ) (x–2 ) (a+1 ) (a–1 )
  33. 16 m 4 –25 m 2 +9 =16 m 4 –25 m 2 +9+ m 2 – m 2 =16 m 4 –24 m 2 +9– m 2 = (4 m 2 –3 ) 2 – m 2 =[(4 m 2 –3 ) –m ] [(4 m 2 –3 ) +m ] =(4 m 2 –m–3 ) (4 m 2 +m–3 ) =(4 m 2 –4m+3m–3 ) (4 m 2 +4m–3m–3 ) =[4m(m–1 ) +3(m–1 ) ] [4m(m+1 ) –3(m+1 ) ] =(m–1 ) (4m+3 ) (m+1 ) (4m–3 )
  34. 3ab x 2 –12ab+3b x 2 –12b =3ab x 2 +3b x 2 –12ab–12b =3b x 2 (a+1 ) –12b(a+1 ) =(a+1 ) (3b x 2 –12b ) =3b(a+1 ) ( x 2 –4 ) =3b(a+1 ) (x–2 ) (x+2 )
  35. 3 a 2 m+9am–30m+3 a 2 +9a–30 =3m( a 2 +3a–10 ) +3( a 2 +3a–10 ) . 10 5 1 | 2 5 =( a 2 +3a–10 ) (3m+3 ) =3( a 2 –2a+5a–10 ) (m+1 ) =3[a(a–2 ) +5(a–2 ) ] (m+1 ) =3(a–2 ) (a+5 ) (m+1 )
  36. a 3 x 2 –5 a 3 x+6 a 3 + x 2 –5x+6 = a 3 ( x 2 –5x+6 ) +( x 2 –5x+6 ) . 6 3 1 | 2 3 =( x 2 –2x–3x+6 ) ( a 3 +1 ) =[x(x–2 ) –3(x–2 ) ] ( a 3 +1 ) =(x–3 ) (x–2 ) ( a 3 +1 ) =(x–3 ) (x–2 ) (a+1 ) ( a 2 –a+1 )
  37. x 2 ( x 2 – y 2 ) –(2x–1 ) ( x 2 – y 2 ) =( x 2 – y 2 ) [ x 2 –(2x–1 ) ] =(x–y ) (x+y ) ( x 2 –2x+1 ) =(x–y ) (x+y ) (x–1 ) 2 =(x–y ) (x+y ) (x–1 ) (x–1 )
  38. a( x 3 +1 ) +3ax(x+1 ) =a[( x 3 +1 ) +3x(x+1 ) ] =a[(x+1 ) ( x 2 –x+1 ) +3x(x+1 ) ] =a[(x+1 ) {( x 2 –x+1 ) +3x } ] =a[(x+1 ) { x 2 –x+1+3x } ] =a(x+1 ) ( x 2 +2x+1 ) =a(x+1 ) (x+1 ) 2 =a(x+1 ) (x+1 ) (x+1 )