How do we solve for $x$ in $x^5-x-1=0$? What is the procedure to finding the simplest exact (or atleast a verifiable approximation to desired level of precision) correct answer to this quintic equation, and more generally to other polynomial equations of degree- 4 and higher yielding non-rational zeroes? And in this particular case, what are all the (exact or approximate) solutions? On desmos there appears to be one real x-intercept at ~1.167; I would imagine that there are up to four other non-real solutions (or fewer, should some have multiplicity greater than 1---is this possible for non-real complex roots?)

If you have a calculator that'll take fifth roots, the iteration
$$x_{n+1}=(1+x_n)^{1/5}$$
with $x_0=1$ gives
$$\begin{align}
x_1&\approx1.148698355\\
x_2&\approx1.1652928729\\
x_3&\approx1.1670872626\\
x_4&\approx1.1672806328\\
x_5&\approx1.1673014635\\
x_6&\approx1.1673037074\\
x_7&\approx1.1673039491\\
x_8&\approx1.1673039751\\
x_9&\approx1.1673039779\\
x_{10}&\approx1.1673039782\\
x_{11}&\approx1.1673039783
\end{align}$$
