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?)

Using Newton - Raphson
$$f(x) = x^5 - x - 1$$
$$\text{At}\quad x=   1.00,\space f(x)  =  -1.00 $$
$$\text{At}\quad x=   2.00 ,\space f(x)  =  29.00 $$
$$f′(x)=5x^4-1$$
$$x_0=2$$
$$x_1=x_0-\frac{f\left(x_0\right)}{f'\left(x_0\right)}$$
$$x_1≈1.6329$$
$$x_2≈1.3731$$
$$x_3≈1.2236$$
$$x_4≈1.1727$$
$$x_5≈1.1674$$
$$x_6≈1.167$$
