分解因式(x-2)³-(y-2)³-(x-y)³①分解因式(x-2)³-(y-2)³-(x-y)³还有②分解因式a³+a²-a-1

Question

分解因式(x-2)³-(y-2)³-(x-y)³
①分解因式(x-2)³-(y-2)³-(x-y)³还有②分解因式a³+a²-a-1

Answer 1

①(x-2)³-(y-2)³-(x-y)³
=(x-y)(x²-4x+4+xy-2x-2y+4+y²-4y+4)-(x-y)³
=(x-y)(x²+y²-6x-6y+12+xy-x²+2xy-y²)
=(x-y)(3xy-6x-6y+12)
=3(x-y)(xy-2x-2y+4)
=3(x-y)(x-2)(y-2).
②a³+a²-a-1
＝（a³-a）+（a²-1）
＝a（a²-1）+（a²-1）
＝（a＋1）（a²-1）
＝（a＋1）（a＋1）（a－1）.