How to approximately guess the roots of a function My question is : How to approximately guess the root of a function...
By root i mean is the starting point guess when used in case of Newton's method or any other root formulating methods. (Without calculators!)

I searched for the answers and came across the following solutions:


*

*Graphs: Of course! The point where the curve intersect with the $x-$axis...We find an integer or a simple decimal number close to it and define it as the starting point!

*Bisection method: We find out the interval $[a,b]$ where the root lies and if it follows certain conditions (link given) we can use $a$ or $b$ as starting points.



But suppose 1) I cannot plot the graph and 2) It is a very complex function (consists of trig., log, etc etc) and guessing the roots is not possible...not even the intervals...

So is there a better method to find the starting point?


For example the function is something like : $$\dfrac{\sin{x}}{x}=\log{x}$$ or $$x^8+24x^6+32x^3+12x+1=0$$ (I guess i exaggerated!)
Thanks!

P.S. - This may seem like a duplicate. But in all the questions i have seen in SE, none answer these questions (or i may have looked over some) concretely! 

I don't think there is a magic answer-you have to think about the specific problem.  For your first example, I would note that $\frac {\sin 1}1 \gt 0 = \log 1, \frac {\sin e}e \lt \frac 1e \lt 1=\log e$, so I have a bracket.  For the second I would note that $f(0)=1$ and the only terms that can go negative are $32x^3+12x$ and I would note that $f(-1)=-18$ and again I have a bracket.  Since this goes to $+\infty$ as $x \to -\infty$ there will be another root below $-1$. $x^8$ is so big that I would try $f(-2)$, find it is positive, and have a bracket on that root.
