因式分解問題

(a2 + b2 + c2 )3 – a6 – b6 – c6
= ( a2 + b2 )3 + 3 ( a2 + b2 )2 ( c2 ) + 3 ( a2 + b2 )( c4 ) + c6 – a6 – b6 – c6
= ( a2 + b2 )3 + 3 ( a2 + b2 )2 ( c2 ) + 3 ( a2 + b2 )( c4 ) – ( a6 + b6 )
= ( a2 + b2 )3 + 3 ( a2 + b2 )2 ( c2 ) + 3 ( a2 + b2 )( c4 ) – ( a2 + b2 )( a4 – a2b2 + b4 )
= ( a2 + b2 ) { ( a2 + b2 )2 + 3c2 ( a2 + b2 ) + 3c4 – a4 + a2b2 – b4 }
= ( a2 + b2 )( 3a2b2 + 3a2c2 + 3b2c2 + 3c4 )
= 3 ( a2 + b2 ){ a2 ( b2 + c2 ) + c2 ( b2 + c2 ) }
= 3 ( a2 + b2 )( a2 + c2 )( b2 + c2 )

2007-08-10 13:55:22 補充：
(a^2+b^2+c^2)^3不等於a^6 + b^6 + c^6

(a^2+b^2+c^2)^3-a^6-b^6-c^6
=a^6+b^6+c^6-a^6-b^6-c^6
=a^6-a^6+b^6-b^6+c^6-c^6
=0

(a^2+b^2+c^2)^3-a^6-b^6-c^6
=a^6+b^6+c^6-a^6-b^6-c^6
=0