Can we determine the existence of real solutions of a sixth order polynomial?

Question

I am developing an algorithm. In the algorithm, I look for the real root of a sixth order polynomial. In the code, I used Newton's method to find the roots. I try to limit the roots, (because Newton's depends on the initial point) this makes the method very expensive. Do you suggest any other method which is cheaper than Newton? Do you know a good way to find suitable initial point for Newton?