m2+m-1=0,n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
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m2+m-1=0,n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
m2+m-1=0,n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
m2+m-1=0,n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
m2+m-1=0,n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
m2+m-1=0,n2-n-1=0
m^2+m=1,n^2-n=1
m3+n3+2m2-2n2+2008
=m^3+m^2+m^3-m^2+m^2-m^2+2008
=m(m^2+m)+n(n^2-n)+m^2-n^2+2008
=m+n+m^2-n^2+2008
=(m^2+m)-(n^2-n)+2008
=1-1+2008
=2008
250
m^2+m=1,n^2-n=1
m^3+n^3+2m^2-2n^2+2008
=m(m^2+m)+n(n^2-n)+m^2-n^2+2008
=m+n+m^2-n^2+2008
=(m^2+m)-(n^2-n)+2008
=1-1+2008
=2008
m^3+m^2-m=0;
n^3+n^2-n=0;
m^3+n^3+2m^2-2n^2+2008=
m^2-n^2+m+n=
(m^2+m-1)-(n^2-n-1)+1-1+2008=
0-0+1-1+2008=
2008
m^2+m-1=0
n^2-n-1=0
(-n)^2+(-n)-1=0
所以
若m=-n
则两式可以成立
则m^3+n^3+2m^2-2n^2+2008
=(-n)^3+n^3+2(-n)^2-2n^2+2008
=-n^3+n^3+2n^2-2n^2+2008
=2008
若m不等于-n
全部展开
m^2+m-1=0
n^2-n-1=0
(-n)^2+(-n)-1=0
所以
若m=-n
则两式可以成立
则m^3+n^3+2m^2-2n^2+2008
=(-n)^3+n^3+2(-n)^2-2n^2+2008
=-n^3+n^3+2n^2-2n^2+2008
=2008
若m不等于-n
则m和-n是方程x^2+x-1=0的两个根
则m+(-n)=-1,m*(-n)=-1
m-n=-1,mn=1
m^2+n^2=(m-n)^2+2mn=1+2=3
(m+n)^2=m^2+2mn+n^2=3+2=5
所以m+n=±√5
m^3+n^3+2m^2-2n^2+2008
=(m+n)(m^2-mn+n^2)+2(m+n)(m-n)+2008
=±√5*(3-1)+2*(±√5)*(-1)+2008
=±2√5-(±2√5)+2008
=2008
综上
m^3+n^3+2m^2-2n^2+2008=2008
收起
m^2=1-m,n^2=1+n
m3+n3+2m2-2n2+2008=(1-m)*m+n*(1+n)+2m^2-2n^2+2008=m+m^2+n-n^2+2008 =2008
由已知条件可得
(m2+m-1)*m=0 则m3+m2-m=0 所以m3+m2=m
故m3+2m2=m2+m=1
(n2-n-1)*n=0 则n3-n2-n=0 n3-n2=n n3-2n2=n-n2=-(n2-n)=-1
m3+n3+2m2-2n2+2008=1-1+2008=2008