Solve for zero by simple mathematical rules if the degree is less than two. If the degree is zero, the only way the polynomial can have a zero is if the polynomial is zero. If the degree is one, use the ordinary rules of algebra to solve the equation formed by setting the polynomial to zero --- if the polynomial is P then solve the equation P = 0. The solution will be a zero for this polynomial. For example, 2x + 4 = 0. Subtract 4 from both sides to get 2x = -4. Then, divide both sides by 2 to get x = -2. This is the zero.
Solve by using the quadratic equation if the degree is exactly two. There can be two solutions, though there might be only one. It is also possible that the solutions will be complex numbers. If the solutions are complex, there will be two solutions and the solutions will be complex conjugates of each other. Whatever the solutions --- even if they are complex --- the solutions will be zeros for the polynomial.
Solve by factoring if the degree is greater than two. Look first for a factor of the form of a binomial (a two-term polynomial). The first term of the binomial is just the variable, and the second term of the binomial should be a factor of the constant term (the term where the variable power is zero). The terms of the binomial are connected by either a plus or a minus (check both). If you find a binomial that divides the polynomial, then reverse the sign on the second term of the binomial to get a zero of the polynomial. Then divide the polynomial by the binomial to get a new polynomial. The degree of the new polynomial will be one less than the original polynomial, and zeros of the new polynomial will also be zeros of the original polynomial.