The most basic way to solve any polynomial is to factor it after setting it equal to zero. The zero product property can then be applied and each factor can be set equal to zero and solved for possible answers. For instance, the polynomial X^2 - X - 6 = 0 would be factored as (X-3)(X+2)=0 and then provide the solutions 3 and -2 for the variable of X.
Quadratic equations, or polynomials in which the highest power involved is the second power, can sometimes be solved with factoring. Many times, however, polynomials cannot be factored. In this case, two other techniques work on this particular type of polynomial: the quadratic equation and "completing the square."
The solutions to a polynomial represent the x-intercepts on its graph. Therefore, graphing a polynomial on a graphing calculator and using the calculator to identify the x-intercepts will give you the solutions to the equation.
Solutions to polynomial equations are called "roots." If the same answer appears twice when solving a polynomial, it is called a "double root" and represents a place where the graph becomes tangential to (bounces off) the x-axis. On the other hand, single solutions represent places where the graph actually crosses through the x-axis.
If a number (not a variable) can be factored out of a polynomial, it does not affect the solutions and x-intercepts. It only changes the shape of the graph. For instance, 3X^3 - 6X^2 + 3X - 9 would have the same solutions as X^3 - 2X^2 + X - 3.