21. 当n为自然数时，有x^6n+1/x^6n=2 ..

来源：百度知道 编辑：UC知道 时间：2024/05/13 10:03:38
21. 当n为自然数时，有x^6n+1/x^6n=2
证明：(1)x+1/x=-1 ==>x^6n+1/x^6n=2
(2)x+1/x=1 ==>x^6n+1/x^6n=2

22.证明：(1)a^2+b^2+c^2+d^2-ab-bc-cd-da=0 ==>a=b=c=d
(2)a^4+b^4+c^4+d^4-4abcd=0 ==>a=b=c=d

21.x^2+x+1=0
x^3-1=(x-1)(x^2+x+1)=(x-1)*0=0
==>x^3=1
==>x^6n=(x^3)^2n=1^2n=1
==>x^6n+1/x^6n=1+1=2
同理第2个,x^2-x+1=0
==>
x^3+1=(x+1)(x^2-x+1)=(x+1)*0=0
==>x^3=-1
==>x^6n=(x^3)^2n=(-1)^2n=[(-1)^2]^n=1^n=1
==>x^6n+1/x^6n=1+1=2

22(1)a^2+b^2+c^2+d^2-ab-bc-cd-da=0
==>2(a^2+b^2+c^2+d^2-ab-bc-cd-da=0
==>(a-b)^2+(a-d)^2+(b-c)^2+(c-d)^2=0
==>a=b=c=d

(2)a^4+b^4+c^4+d^4-4abcd=0
==>(a^2-b^2)^2+2a^2*b^2+(c^2-d^2)^2+2c^2*d^2-4abcd=0
==>(a^2-b^2)^2+(c^2-d^2)^2+2(ab-cd)^2=0
==>a^2=b^2;c^2=d^2;ab=cd
==>a=b=c=d