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Question

The sum of 162th power of the roots of the equation x3-2x2+2x-1=0 is


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Solution

Step 1: Find the roots of the equation :

Let the roots of x3-2x2+2x-1=0 be α,β,γ

x3-1-1(2x2-2x)=0(x-1)x2+x+1-2x(x-1)=0x-1x2+x-2x+1=0(x-1)(x2-x+1)=0x-1=0x2-x+1=0

Roots of x3-2x2+2x-1=0 are 1,ω,ω2

Step 2: Compute the required sum.

α162+β162+γ162=1162+ω162+ω2162α162+β162+γ162=1+ω354+ω3108α162+β162+γ162=1+154+1108ω3=1α162+β162+γ162=3

Hence, the sum of 162th power of the roots of the equation x3-2x2+2x-1=0 is 3.


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