CAPITULO X
Descomposición Factorial
Descomposición Factorial
- Ejercicio 97
Factorar o descomponer en dos factores:
- x 4 +64 y 4 =( x 4 +16 x 2 y 2 +64 y 4 ) –16 x 2 y 2 = ( x 2 +8 y 2 ) 2 –16 x 2 y 2 =[ ( x 2 +8 y 2 ) +4xy ] [ ( x 2 +8 y 2 ) –4xy ] =( x 2 +4xy+8 y 2 ) ( x 2 –4xy+8 y 2 )
- 4 x 8 + y 8 =( 4 x 8 +4 x 4 y 4 + y 8 ) –4 x 4 y 4 = ( 2 x 4 + y 4 ) 2 –4 x 4 y 4 =[ ( 2 x 4 + y 4 ) –2 x 2 y 2 ] [ ( 2 x 4 + y 4 ) +2 x 2 y 2 ] =( 2 x 4 –2 x 2 y 2 + y 4 ) ( 2 x 4 +2 x 2 y 2 + y 4 )
- a 4 +324 b 4 =( a 4 +36 a 2 b 2 +324 b 4 ) –36 a 2 b 2 = ( a 2 +18 b 2 ) 2 –36 a 2 b 2 =[ ( a 2 +18 b 2 ) –6ab ] [ ( a 2 +18 b 2 ) +6ab ] =( a 2 –6ab+18 b 2 ) ( a 2 +6ab+18 b 2 )
- 4 m 4 +81 n 4 =( 4 m 4 +36 m 2 n 2 +81 n 4 ) –36 m 2 n 2 = ( 2 m 2 +9 n 2 ) 2 –36 m 2 n 2 =[ ( 2 m 2 +9 n 2 ) –6mn ] [ ( 2 m 2 +9 n 2 ) +6mn ] =( 2 m 2 –6mn+9 n 2 ) ( 2 m 2 +6mn+9 n 2 )
- 4+625 x 8 =( 4+100 x 4 +625 x 8 ) –100 x 4 = ( 2+25 x 4 ) 2 –100 x 4 =[ ( 2+25 x 4 ) –10 x 2 ] [ ( 2+25 x 4 ) +10 x 2 ] =( 2–10 x 2 +25 x 4 ) ( 2+10 x 2 +25 x 4 )
- 64+ a 12 =( 64+16 a 6 + a 12 ) –16 a 6 = ( 8+ a 6 ) 2 –16 a 6 =[ ( 8+ a 6 ) –4 a 3 ] [ ( 8+ a 6 ) +4 a 3 ] =( 8–4 a 3 + a 6 ) ( 8+4 a 3 + a 6 )
- 1+4 n 4 =( 1+4 n 2 +4 n 4 ) –4 n 2 = ( 1+2 n 2 ) 2 –4 n 2 =[ ( 1+2 n 2 ) –2n ] [ ( 1+2 n 2 ) +2n ] =( 1–2n+2 n 2 ) ( 1+2n+2 n 2 )
- 64 x 8 + y 8 =( 64 x 8 +16 x 4 y 4 + y 8 ) –16 x 4 y 4 = ( 8 x 4 + y 4 ) 2 –16 x 4 y 4 =[ ( 8 x 4 + y 4 ) +4 x 2 y 2 ] [ ( 8 x 4 + y 4 ) +4 x 2 y 2 ] =( 8 x 4 +4 x 2 y 2 + y 4 ) ( 8 x 4 +4 x 2 y 2 + y 4 )
- 81 a 4 +64 b 4 =( 81 a 4 +144 a 2 b 2 +64 b 4 ) –144 a 2 b 2 = ( 9 a 2 +8 b 2 ) 2 –144 a 2 b 2 =[ ( 9 a 2 +8 b 2 ) –12ab ] [ ( 9 a 2 +8 b 2 ) +12ab ] =( 9 a 2 –12ab+8 b 2 ) ( 9 a 2 +12ab+8 b 2 )