For example, consider the polynomial above::formula_28Extracting its rational roots and factoring it, we end with::formula_33But that is not completely factored over C. If we need to factor our polynomial to a product of linear factors, we must deal with that quadratic factor:formula_36The easiest way is to use quadratic formula, which gives us:formula_37and the solutions:formula_38:formula_39So the completely-factored polynomial over C will be::formula_40However, it should be noted that we cannot in every case expect things to be so easy; the quadratic formula's analogue for fourth-order polynomials is very messy and no such analogue exists for 5th-or-higher order polynomials.
For example, consider the polynomial above::formula_28Extracting its rational roots and factoring it, we end with::formula_33But that is not completely factored over C. If we need to factor our polynomial to a product of linear factors, we must deal with that quadratic factor:formula_36The easiest way is to use quadratic formula, which gives us:formula_37and the solutions:formula_38:formula_39So the completely factored polynomial over C will be::formula_40However, it should be noted that we cannot in every case expect things to be so easy; the quadratic formula's analogue for fourth-order polynomials is very messy and no such analogue exists for 5th-or-higher order polynomials.