If α and β are the roots of the equation x2−x+1=0,   then α2009+β2009=

x2−x+1=0         ⇒x=1+1−42 x=1+3i2 α=12+i32.      β=12−i32 α=cosπ3+isinπ3,β=cosπ3−isinπ3 α2009+β2009=2cos2009(π3) =2cos[668π+π+2π3]=2cos(π+2π3) =−2cos2π3=−2(−12)=1