The sum of 162th power of the roots of the equation x3-2x2+2x-1=0 is
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)=0⇒x-1x2+x-2x+1=0⇒(x-1)(x2-x+1)=0⇒x-1=0⋮x2-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+ω2162⇒α162+β162+γ162=1+ω354+ω3108⇒α162+β162+γ162=1+154+1108∵ω3=1⇒α162+β162+γ162=3
Hence, the sum of 162th power of the roots of the equation x3-2x2+2x-1=0 is 3.