Question

How many zeros does 6z^7+iz have and how many are real?

Answer 1

Well
6z^7+iz = 0
if and only if
z (z^6 - (-i/6)) = 0

Hence there's one clear real zero, z = 0

But now, how many complex roots of z^6 = -i/6 are real?

Answer 2

what

Answer 3

Do you agree with this first:
6z^7+iz = 0
if and only if
z (z^6 - (-i/6)) = 0

Answer 4

yes

Answer 5

Now
z (z^6 - (-i/6)) = 0
iff
z = 0 or z^6 - (-i/6) = 0

Answer 6

Hence
z = 0 is a zero, which makes sense.

Now there are six other roots from the equation
z^6 - (-i/6) = 0

Answer 7

ok