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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
题目详情
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
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
▼优质解答
答案和解析
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).
=(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).
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