How can I solve this: x^4+x^3+x^2+x+1=0 ? thx

To solve x^4+x^3+x^2+x+1 =
0.

We know that 1+x+x^2+x^4 = 
(x^5-1)/(x-1).

So the given equation is therefore rewritten
as:

(x^5-1) /(x-1) =
0

Multiply by (x-1). So , mind x-1 is our
factor.

(x^5 -1) =
0

x^5 = 1.

We know 1 = cos2npi
+isin2npi.

Therefore ,

x^5 = 
cos2npi+isin2npi

Take the 5 th
root.

x = (co2npi+isin2npi)
^(1/5)

x = (cos(2npi)/5 +isin(2npi)/5 , for n =
0,1,2,3,4....by D'Moivres theorem.Actually after  n=4, for the next integral values the
roots repeat.

x0 = 1  is not the root as we have multiplied
by our factor (x-1) to the given expression (x^4+x^2+x^3+1
).

x1 = cos72+isin72

x2 =
cos144 + isin 144

x3 = cos216 +i sin
216

x4  = cos288 +isin288.

So
x1,x2,x3 and x4 are the solutions.