Question

因式分解 x方+y方-xy+2x-y+1

Answer 1

设x^2+y^2-xy+2x-y+1=(ax+by+m)(cx+dy+n)
=ac*x^2+bd*y^2+(ad+bc)xy+(an+mc)x+(bn+md)y+mn
得
ac=1 (1)
bd=1 (2)
ad+bc=-1 (3)
an+mc=2 (4)
bn+md=-1 (5)
mn=1 (6)
得
c=1/a
d=1/b
n=1/m
代入到(3)和(4)得
a/b+b/a=-1
a/m+m/a=2
解得
a/b=(-1±√3i)/2
a=m
不妨令a=m=1,a/b=(-1+√3i)/2,则
a=c=m=n=1
b=-(1+√3i)/2
d=1/b=(-1+√3i)/2
所以
x^2+y^2-xy+2x-y+1=(x-(1+√3i)y/2+1)(x+(-1+√3i)y/2+1)