Make a list of factor polynomials -- each one of which has one of the roots. When you have all the factor polynomials corresponding to each root in the list, the product of all these small polynomials is the polynomial you want. Suppose the list of roots is just the pair 1 and 2. The factor polynomials that have these roots are Z - 1 and Z - 2, because he solution to Z - 1 = 0 is 1 and the solution to Z - 2 = 0 is 2. The desired polynomial is the product of Z - 1 and X - 2, or Z^2 - 3Z + 2.
Modify the process for fractional roots. If a/b is one of the roots, the simple polynomial that has a/b for a solution is bX - a. So if 3/4 is a root, 4X - 3 is a simple solution with a 3/4 root: 4X -3 = 4(3/4) - 3 = 3 - 3 = 0.
Include both roots if there duplications. For example, if 5 is in the roots of the solution, X - 5 is one of the factor polynomials you are looking for. If the root 5 is in the list of roots two times, the X - 5 factor polynomial will be used twice.
Multiply all the factors together and collect terms to get the desired polynomial. For example, if the factors are Z + 2 and Z + 3, the multiplication would go like this: (Z + 2)( Z + 3) = Z^2 + 2Z + 3Z + 6 = Z^2 + 5Z + 6. The entire process goes from the roots (-2 and -3) to factors that have these roots -- (Z + 2) and (Z+3) -- to the polynomial that has these roots: the product of (Z + 2) and (Z + 3), which is Z^2 + 5Z + 6.