<s>[INST] Factor $z^5 + z +1 =0$ The problem stated in the title is:

Factor $z^5 + z + 1 = 0$
  (naturally without the use of computers or calculators)

How do I go about solving this? Is there a more systematic approach than simply guessing a root and then applying polynomial division etc. ? Thank you for any help!
 [/INST] If $z^5 + z + 1 = f(z)g(z)$, then:


*

*$\deg f =1, \deg g=4$, or

*$\deg f =2, \deg g=3$
In both cases, you can try to determine the coefficients. Perhaps simplify things by first trying monic polynomials with independent term $1$.
The second case gives 
$$
(z^2+a z+1)(z^3+bz^2+c z+ 1) =
z^5 + (a + b)z^4 + (a b + c + 1)z^3 +  (a c + b + 1)z^2 + (a + c)z + 1
$$
Forcing the coefficients of $z^4,z^3,z^2$ to be zero and $a+c=1$, we get
$$z^5 + z + 1 = (z^2 + z + 1) (z^3 - z^2 + 1)$$
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