Arrange the polynomial to be factored so the terms are in decreasing order of exponents. For example, the expression Z - 3Z^2 + 2 + Z^3 should be written Z^3 - 3Z^2 + Z + 2. This polynomial has a constant term -- the 2. An example of a polynomial without a constant term would be X^2 - 5X. If the polynomial has no constant term, the variable is one factor and the polynomial divided by the variable is another factor so X^2 - 5X = X(X - 5). So if X^2 - 5X = 0, either X = 0 or X - 5 = 0. X =0 and X = 5 are both possible solutions to the equation.
Find all of the factors of the constant term of the polynomial. Let n be one of these factors. If the variable of the polynomial is Z, then Z - n and Z + n are candidates for factors of the polynomial. For example, if the polynomial Z^2 - 2Z -15 is to be factored, the factors of 15 are 1, 3, 5 and 15. Candidates for factors of Z^2 - 2Z -15 are Z - 1, Z + 1, Z -3, Z + 3, Z - 5, Z + 5, Z - 15 and Z + 15. Trying them one at a time reveals that Z + 3 is a factor. Z^2 - 2Z -15 = (Z + 3)(Z - 5).
If the initial coefficient is greater than one, factor both this initial coefficient and the constant term and consider all possible monomials that are possible candidates for factors. For example, to factor 2Z^2 + 7Z + 3, consider Z - 1, Z + 1, Z -3, Z + 3, 2Z - 1, 2Z - 3 and 2Z + 3. Try these one at a time. If none of the candidates divides the polynomial, it is prime -- it cannot be factored. If you find that, for example, 2Z + 1 divides the polynomial, then divide to get the other factor: 2Z^2 + 7Z + 3 = (2Z + 1)(Z + 3).
If the highest exponent in the polynomial is 3 and there are no monomial factors, the polynomial is prime -- not factorisable. If the highest exponent is greater than 3, and there is no monomial factor, there is no easy way to factor it.