If n is a odd integer greater than 3,but not a multiple of 3.Prove that x3+x2+x is a factor of (x+1)n-xn-1
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1 Answers
106x3+x2+x = x(x2+x+1)
This will be a factor of p(x) = (x+1)n-xn-1
iff x=0, x=ω and x=ω2 are factors of that.
putting x=0, we get p(0) = 0 so x=0 is a factor
putting x=ω, we get p(ω) = (-1)nω2n-wn-1
putting x=w2 p(w2) = (-1)nwn-w2n-1
the last two equations will be zero if n is an odd integer and not the multiple of 3