Question

先化简，再求值。2x(x+y)-(x+y)^2,其中x=根号2012,y=根号2011

（2）a^2(x-y)-2a(x-y)^2+(x-y)^3

Answer 1

解
2x(x+y)-(x+y)²
=(x+y)[2x-(x+y)]
=(x+y)(x-y)
=x²-y²
=(√2012)²-(√2011)²
=2012-2011
=1

a²(x-y)-2a(x-y)²+(x-y)³
=(x-y)[a²-2a(x-y)+(x-y)²]
=(x-y)[a-(x-y)]²
=(x-y)(a-x+y)²

追问

可以在回答一个吗。。就是对于任意数x，多项式2x-x^2-1的值是
一定是整数、不可能为正、一定是负数、不可能为负

追答

解
2x-x²-1
=-(x²-2x+1)
=-(x-1)²
∵(x-1)²≥0
∴-(x-1)²≤0
∴不可能为正的

Answer 2

(1)2x(x+y)-(x+y)^2=（x+y)(2x-x-y)=(x+y)(x-y)=x²-y²=(√2012)²-(√2011)²=2012-2011=1
(2)a^2(x-y)-2a(x-y)^2+(x-y)^3=(x-y)[a²-2a(x-y)+(x-y)²]=(x-y)[a-(x-y)]²=(x-y)(a-x+y)²