Question

分解x(a-x)(a-y)-y(x-a)(y-a)

为什么不可以是原式=x(a-x)(a-y)+y(a-x)(a-y)=(x+y)(a-x)(a-y)

Answer 1

x(a-x)(a-y)-y(x-a)(y-a)=x(a-x)(a-y)-y(a-x)(a-y)=(x-y)(a-x)(a-y)

【x(a-x)(a-y)+y(a-x)(a-y)=(x+y)(a-x)(a-y)】是错误的！
因为x(a-x)(a-y)-y(x-a)(y-a)=x(a-x)(a-y)-y[-(a-x)][-(a-y)]=x(a-x)(a-y)-y(a-x)(a-y)=(x-y)(a-x)(a-y)

追问

为什么不能那样变号呢？

追答

你注意这个地方(画线处)：-y[-(a-x)][-(a-y)]，这里有三个负号连乘，因此其
积只能取负号，即：-y[-(a-x)][-(a-y)]=-y[(a-x)][(a-y)]。于是：
x(a-x)(a-y)-y(x-a)(y-a)=x(a-x)(a-y)-y[-(a-x)][-(a-y)]=x(a-x)(a-y)-y(a-x)(a-y)
=(x-y)(a-x)(a-y) ———————