I do not know exactly what you are asking, but if you have a polynomial of degree then by synthetic division (assuming the remainder polynomial is zero) will have degree . What you can do with this is the following useful trick.
Given a polynomial:
divide it synthetically by where is a zero of the polynomial. Your polynomial must reduce to:
Now to find the zeros of all need now is to find zeros of . This is especially useful when you are solving a cubic. Because the steps for solving a cubic a rather long and complicated if you can spot a trivial solution you can syntheticaly divide which will leave you with a quadradic equation. Now apply the quadratic formula and you have your 3 roots of the cubic.