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If-x-1-x-1-prove-that-x-n-x-n-2-x-n-4-0-




Question Number 66197 by Rasheed.Sindhi last updated on 11/Aug/19
If  x+(1/x)=1,prove that:            x^n +x^(n−2) +x^(n−4) =0
Ifx+1x=1,provethat:xn+xn2+xn4=0
Commented by mr W last updated on 10/Aug/19
VERY NICE!
VERYNICE!
Commented by Rasheed.Sindhi last updated on 11/Aug/19
          x^n +x^(n−2) +x^(n−4)            =x^(n−2) (x^2 +1+(1/x^2 ))           =x^(n−2) (x^2 +2+(1/x^2 )−1)           =x^(n−2) ( (x+(1/x))^2 −1)           =x^(n−2) ({1}^2 −1)           =x^(n−2) .0          =0
xn+xn2+xn4=xn2(x2+1+1x2)=xn2(x2+2+1x21)=xn2((x+1x)21)=xn2({1}21)=xn2.0=0
Commented by Rasheed.Sindhi last updated on 11/Aug/19
ThankSs for Encouraging Sir!
ThankSsforEncouragingSir!
Answered by som(math1967) last updated on 10/Aug/19
x+(1/x)=1  ((x^2 +1)/x)=1  x^2 −x+1=0⇒(x+1)(x^2 −x+1)=0  ⇒x^3 +1=0∴x^3 =−1 ⇒x^3 .x=−x  ∴x^4 =−x  Now x^n +x^(n−2) +x^(n−4)   =x^n (1+x^(−2) +x^(−4) )  =x^n (1+(1/x^2 )+(1/x^4 ))=x^n (((x^4 +x^2 +1)/x^4 ))  =x^n (((x^2 −x+1)/x^4 ))  ★  =x^n ((0/x^4 ))=0  ★[★  ★x^4 =−x        ★★x^2 −x+1=0
x+1x=1x2+1x=1x2x+1=0(x+1)(x2x+1)=0x3+1=0x3=1x3.x=xx4=xNowxn+xn2+xn4=xn(1+x2+x4)=xn(1+1x2+1x4)=xn(x4+x2+1x4)=xn(x2x+1x4)=xn(0x4)=0[x4=xx2x+1=0
Commented by Rasheed.Sindhi last updated on 11/Aug/19
THanks Sir!
THanksSir!
Answered by mr W last updated on 10/Aug/19
x≠0  x+(1/x)=1  ⇒x^2 −x+1=0  ⇒x^2 =x−1  ⇒x^4 =(x−1)^2 =x^2 −2x+1  ⇒x^4 +x^2 +1=2x^2 −2x+2  ⇒x^4 +x^2 +1=2(x^2 −x+1)  ⇒x^4 +x^2 +1=0  ⇒x^(n−4) (x^4 +x^2 +1)=0  ⇒x^n +x^(n−2) +x^(n−4) =0
x0x+1x=1x2x+1=0x2=x1x4=(x1)2=x22x+1x4+x2+1=2x22x+2x4+x2+1=2(x2x+1)x4+x2+1=0xn4(x4+x2+1)=0xn+xn2+xn4=0
Commented by Rasheed.Sindhi last updated on 11/Aug/19
Thαnks Sir!
ThαnksSir!
Answered by gunawan last updated on 11/Aug/19
x^2 +x^(−2) +2=1  x^2 +x^(−2) +1=0    x^n +x^(n−2) +x^(n−4) =0  x^(n−2) (1+x^2 +x^(−2) )  =x^(n−2) ×0=0
x2+x2+2=1x2+x2+1=0xn+xn2+xn4=0xn2(1+x2+x2)=xn2×0=0

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