How to solve $x^4+8x-1=0 $ Any idea how to solve the following equation?

$$x^4+8x-1=0$$

I tried to find some obvious roots of this equation so it could help me to find the other roots (if it has more then $1$) but I had no success, and I would really appreciate some help.
There will be a real root a little above zero, as your polynomial has $f(0)=-1, f(1)=8$  You can use any root finding technique from here.  Bisection is easy-try $f(\frac 12)$ and replace the endpoint of the interval $(0,1)$ that has the same sign as $f(\frac 12)$.  This isn't the fastest, but it will work.  There are other techniques in any numerical analysis text.  The Newton-Raphson method will have an easy time here.  There is another root near $x=-2$.  Graphing the function will show that, then you can use the same technique.  The other two roots are complex.  

Here is the bisection $$\begin {array} {c c c c c c}a&amp;b&amp;(a+b)/2&amp;f(a)&amp;f(b)&amp;f((a+b)/2)\\0&amp;1&amp;0.5&amp;-1&amp;8&amp;3.0625\\0&amp;0.5&amp;0.25&amp;-1&amp;3.0625&amp;1.003906\\0&amp;0.25&amp;0.125&amp;-1&amp;1.003906&amp;0.000244\\0&amp;0.125&amp;0.0625&amp;-1&amp;0.000244&amp;-0.49998\\0.0625&amp;0.125&amp;0.09375&amp;-0.49998&amp;0.000244&amp;-0.24992\\0.09375&amp;0.125&amp;0.109375&amp;-0.24992&amp;0.000244&amp;-0.12486\\0.109375&amp;0.125&amp;0.117188&amp;-0.12486&amp;0.000244&amp;-0.06231\\0.117188&amp;0.125&amp;0.121094&amp;-0.06231&amp;0.000244&amp;-0.03103\\0.121094&amp;0.125&amp;0.123047&amp;-0.03103&amp;0.000244&amp;-0.0154\\0.123047&amp;0.125&amp;0.124023&amp;-0.0154&amp;0.000244&amp;-0.00758\\0.124023&amp;0.125&amp;0.124512&amp;-0.00758&amp;0.000244&amp;-0.00367\\0.124512&amp;0.125&amp;0.124756&amp;-0.00367&amp;0.000244&amp;-0.00171\\0.124756&amp;0.125&amp;0.124878&amp;-0.00171&amp;0.000244&amp;-0.00073\\0.124878&amp;0.125&amp;0.124939&amp;-0.00073&amp;0.000244&amp;-0.00024 \end {array}$$  

The root is always between $a$ and $b$.  The length of the interval gets cut in half each step.  You can see the root is just below $x=1/8$, which you might guess as $x^4$ is rather small, so you want $8x-1$ to be a tiny amount positive.
hint see this article from wolramalpha its irreducible over $Q$ http://www.wolframalpha.com/input/?i=is%20x%5E4%2B8%20x-1%20irreducible%3F&amp;lk=2 if you want to find roots they are hard to find . $x\approx -2,0.12,0.95-\pm\sqrt{3}i$