Factoring a Quartic Polynomial into Two Quadratics 2

  The previous blog neglected to mention what happens if b=0 which has to be treated as a special case since the formula for d results in division by zero. When b=0 the initial sets of constraints are simplified slightly and the formula for d changes.

One can then proceed as before to find the best values for a, b, c and d.

There is only one remaining constraint which gives the same zeros.

We need only one solution to factor the quartics since the various zeros correspond to alternative permutations of the monomials in the quartic. Another special case occurs when in addition a=0 and the altered constraints tell us that d=A₀ and c=A₃. Zero coefficients warn about the occurrence of the special cases. One can use the same general method to factor a cubic equation into a quadratic and a monomial where one also finds a special case for a=0 since the formula for b involves division by a.