I have a question about the zeros (roots) of polynomials. I have been sick the past couple of days and have a math test tomorrow, so asking my teacher may not be an option tomorrow.

The question(s) is, Is there a possibility that a polynomial function may not have any real solutions or real roots?

I assume you mean "real polynomial" , and the answer is yes: $\displaystyle x^2+1\,,\,\,x^4+25, x^2+2x+2$ are just a few examples of an infinity of real polynomial functions without any real root .
If so, is there a way of finding the complex roots without a real root to begin with?

Sometimes there is (the root formula for quadratics, for example), sometimes there isn't a direct one. It all depends on the particular polynomial we're dealing with. Tonio
Thanks for reading

PLEASE HELP!!!