Solve $x^5 + x - 1 = 0$

Question

Solve $x^5 +x - 1 = 0$

I am simply curious to see how the solution would go, since it is a quintic; it cannot be done by regular methods.

I'm just curious to see what people would come up with, and I can not solve the equation.

Thanks!

Answer 1

You can factor it: $$x^5+x-1=(x^2-x+1)\cdot (x^3+x^2-1)=0$$

Then solving these separately gives $$x=e^{\pm \frac{\pi i}{3}}$$ and some horrible solutions for the cubic.