how to solve polynomial equation using c# code

Not to put too fine a point on it, I'd drop the current idea. A quick check on Wolfram Alpha exceeded their computation limit trying to solver that one by rational means, so your "random number" idea it likely to have duration problems...like the Sun dying of old age. :laugh:

If you need to solve polynomial expressions than you need to be a lot, lot more sophisticated than that.
A quick Google[^] gives nearly 5,000,000 hits, and several of them look interesting: Polynomial Equation Solver[^] for example, but I don't think that allows for trig and log functions. You could look though, and you may get enough ideas to start using it as a basis - or look at some of the other Googler hits, they could also be very relevant.

But...I suspect you have bitten off rather more than you can chew with this one...it's not a project I would want to leap into without a heck of a lot of thought!

This is a really big problem which cannot be adequately solved in a Quick Answer. The problem is: this is not a numeric problem. This is a problem of creation or using of a Computer Algebra System:
http://en.wikipedia.org/wiki/Computer_algebra_system[^].

Such systems perform analytic (symbolic), not numeric solutions. "Symbolic" does not mean they have anything to do with strings representing expression. Instead, they operate on expression trees. I developed such solution but not ready to give you any detail, by many reasons. One of such reasons is: I don't believe you have enough knowledge/experience at this moment. If you think you do, it would be great: you could do some good research and use some available system (which is not easy at all, by, again, many reasons), or develop you own.

Besides, what you show in your string, is not an equation. At best, this is only its left of right part. That said, your solution is simply undefined. (That's why I though that you skills of solving such big problems are way insufficient; sorry if I'm wrong.) Besides, chances are, the solution may become unresolvable analytically. If you give us missing part of equation, we may help to solve it or tell you if it seems unsolvable.

Besides, you can solve it digitally. Instead of pointless string s ="cos(x)+ln(x)-e^x"; write the function:

using System;
//...

(x) => { return Math.Cos(x) + Math.Log(x) - Math.Exp(x); }

// or

double LeftPart(dounle x) { return Math.Cos(x) + Math.Log(x) - Math.Exp(x); }

and try to use it to find the solution (again, it will depend on what do you have on right/left part of your equation).

There are many numeric methods for solving equations which are already implemented, you can learn them. This is a whole big topic. In all cases, you can create some solution by yourself (can you? :-))…

—SA