z^5 = -1
z<sub>k</sub>^5 = cis(pi + 2kpi)
z<sub>k</sub> = cis( (pi + 2kpi)/5 )
k = 0, z = cis (pi/5)
k = 1, z = cis 3pi/5
k = -1, z = cis (-pi/5)
k = 2, z = cis pi = -1
k = -2, z = cis(-3pi/5)
making a quadratic equation with roots cis pi/5, cis -pi/5
sum of roots = cis(pi/5) + cis(-pi/5) = 2cospi/5 [remember z + z conjugate =2Rez]
product of roots : cispi/5 * cis(-pi/5) = cis0 = 1
.'. quadratic equation roots cispi/5 , cis(-pi/5)
z^2 + (-sum of roots)z + (product of roots)
= z^2 -2zcospi/5 + 1
similarly for cis 3pi/5 and cis -3pi/5
sum of roots = 2cos3pi/5
product of roots = cis0 = 1
.'. quadratic equation roots cis 3pi/5 , cis(-3pi/5)
z^2 + (-sum of roots)z + (product of roots)
= z^2 -2zcos(3pi/5) + 1
.'. equation with roots cis(+- pi/5), cis(+- 3pi/5) and -1
= (z+1)(z^2-2zcos(3pi/5)+1)(z^2-2zcos(pi/5)+1), as required