求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy) 分解因式

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求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy)分解因式求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy)分解因式求解一道数学题;(xy-1)^2+(x+y+2)(x

求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy) 分解因式
求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy) 分解因式

求解一道数学题;(xy-1)^2+(x+y+2)(x+y+2xy) 分解因式
令x+y=a,xy=b
原式化为:
(b-1)^2+(a+2)(a+2b)
=b^2-2b+1+a^2+2a+2ab+4b
=b^2+2b+1+a^2+2a+2ab
=a^2+2ab+b^2+2a+2b+1
=(a^2+2ab+b^2)+2(a+b)+1
=(a+b)^2+2(a+b)+1
=(a+b+1)^2
=(xy+x+y+1)^2
=[(x+1)(y+1)]^2
=(x+1)^2*(y+1)^2

(xy-1)^2+(x+y+2)(x+y+2xy)
=(xy-1)^2+4xy+(x+y)^2+2(xy+1)(x+y)
=(xy+1)^2+(x+y)^2+2(xy+1)(x+y)
=(xy+1+x+y)^2
=(x+1)^2(y+1)^2

分解因式:(xy-1)^2+(x+y+2)(x+y+2xy)

设x+y+2=r,x+y+2xy=s
原式
=((s-r)/2)^2+rs
=((s+r)/2)^2
=(x+y+xy+1)^2
=(x+1)^2*(y+1)^2