Question

分解因式 （a+b-2ab)(a+b-2)+(1-ab)² 用换元法

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Answer 1

（a+b-2ab)(a+b-2)+(1-ab)²
令a+b=x,ab=y
(x-2y)(x-2)+(1-y)²
=x²-2x-2xy+4y+1-2y+y²
=x²-2xy+y²-2x+2y+1
=（x-y）² -2(x-y)+1
=(x-y-1)²
=(a+b-ab-1)²
=[(a-1)(1-b)]²
=(a-1)²(1-b)²

Answer 2

解：
令a+b=x
ab+1=y
(a+b-2ab)(a+b-2)+(1-ab)^2

=(x-2y+2)(x-2)+(2-y)^2
=x^2-2xy+4(y-1)+(2-y)^2
=x^2-2xy+(4y-4+4-4y+y^2)
=x^2-2xy+y^2
=(x-y)^2
=[(a+b)-(ab+1)]^2

=[(a-1)(1-b)]^2
=(a-1)^2(b-1)^2

Answer 3

a+b=x
ab=y
则（x-2y)(x-2)+(1-y)²
=x²+4y-2x-2xy+y²-2y+1
=x²-2xy+y²+2y-2x+1
=（x-y)²-2(x-y)+1
=(x-y-1)²
=（a+b-ab-1)²
=[(a-1)(1-b)]²
=(a-1)²(b-1)²

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