how would I find $x$ in the following: $$8.056=x+x^2+x^3+\ldots +x^{10}.$$

Well I know the right hand sides simplifies to $$x\cdot \frac{1-x^{10}}{1-x}$$ but I don't know how to proceed further.

-
Well you're trying to solve a degree $10$ polynomial equation. That is probably not possible in exact form. If you're looking just for numerical approximations of the roots then maybe you should say that in your question. In any case, if it is of any help, the roots are computed here. –  Adrián Barquero Nov 6 '12 at 20:00