Question

已知m平方=n+2,n平方=m+2.求m的三次方-2mn+n的三次方

Answer 1

m^2=n+2
n^2=m+2
相减：
(m+n)(m-n)=-(m-n)
(m+n+1)(m-n)=0

(1)若m-n=0,m=n,
则m^2-m-2=0
m1=2,m2=-1
∴m=n=2或m=n=-1
m^3-2mn+n^3=8或-4

(2)若m+n=-1,
则n=-1-m
m^2=-m+1
m^2+m-1=0
同理n^2+n-1=0
m,n为方程t^2+t-1=0两根，
m+n=-1,mn=-1
m的三次方-2mn+n的三次方
=(m+n)(m^2-mn+n^2)-2mn
=(-1)(1+3)-2*(-1)
=-2

综上所述，m的三次方-2mn+n的三次方=-2或-4或8.