How can I go about finding the real roots of polynomial functions?

It depends on the degree of the polynomial. If it's linear, simply divide. If it's quadratic, use the formula x = (-b +/-√(b2 - 4ac))/2a. Cubic and quartic equations also have formulas to find the roots (although more complicated). However, it has been proven that there is no general, explicit formula for the roots of equations of degree 5 or higher (see Abel-Ruffini theorem). So you must try to factorize or use trial-and-error to find some roots. If they don't work, then by the above theorem you'll probably never find explicit solutions (unless you use elliptic functions). Of course, if you just want to analyze roots without getting actual values, just examine the graph with differentials.