This question is from TYS complex numbers, p. 65, q. 24, in my
Dyna edition 2009, yellow cover.
Write down, in any form, all the roots of the equation
z5 − 1 = 0.
Hence find all the roots of the equation
(w − 1)4 + (w − 1)3 + (w − 1)2 + w = 0
and deduce that none of them is real.
Find the arguments of the two roots which have the smaller
modulus.
This question is quite interesting. I have a question about the
last part - find the arguments. The TYS answer is 2π/5 and
3π/5, but my answer is 2π/5 and −2π/5.
Mr. Wee (or anyone else), if you can check this for me, I'd be
grateful.
Just for comparison, my answer to the first part is z =
e^(i 2kπ /
5)
where k = 0, 1, 2, 3, 4.
And for second part, I get w = 1 + e^(i 2kπ /
5)
where k = 1, 2, 3, 4.
From the Argand diagram, the two roots with the smaller modulus
should fall in the first and fourth quadrant. Or did I slip up
somewhere?
