Reduce the third order polynomial to a second order polynomial. You will be producing a "factor list" where the first factor in the list can be divided into the order 3 polynomial to produce the order 2 polynomial. Note that either the third degree or second degree polynomial may be prime. Therefore, factoring a third order polynomial may result in a single third order polynomial, a first order and a second order polynomial or three first order polynomials. If every term of a polynomial contains a variable, put the variable on the factor list and divide the polynomial by the variable.
Reduce third order polynomials with a constant term by dividing with logical first order polynomials. For example, to factor Z^3 + 4Z^2 + 5Z + 2, you note that the factors of the constant term are 1 and 2, so the candidates for factors are Z - 2, Z + 2, Z - 1 and Z + 1. Trying these one at a time we see that Z + 2 is a factor of Z^3 + 4Z^2 + 5Z + 2 -- in fact (Z + 2)(Z^2 + 2Z + 1) = Z^3 + 4Z^2 + 5Z + 2. Note that some order 3 polynomials are prime -- they cannot be factored.
Factor the order 2 polynomial in the same way. The polynomial Z^2 + 2Z + 1 has only two candidates: Z - 1 and Z + 1. Dividing Z^2 + 2Z + 1 by Z - 1 leaves a remainder of 5, but dividing Z^2 + 2Z + 1 by Z + 1 leaves no remainder. (Z + 1)(Z + 1) = Z^2 + 2Z + 1. Note that the order 2 polynomial may not have factors.