I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ and $D$ is a second order polynomial of $x$ with real roots $\lvert r \rvert >1$. I want to factor this expression into the product of two second order polynomials of $x$ with real coefficients. I've solved this problem with complex coefficients - that isn't so hard - but I believe that factoring this quadratic form into the product of two second order polynomials of $x$ is a tractable problem!
A good answer will either:
- Give a method for computing the coefficients of the two polynomials of $x$ (without relying on root finding of polynomials of order greater than 2).
- Provide a proof that this problem is not, in general, tractable.
You can factor the quadratic form using the quadratic formula but, in general, the roots will be complex. The monomials the quadratic form factors into will be in the form $(D+cN)$, so the polynomial of $x$ one gets by distributing $c$ into $N$ and adding $D$ will have complex coefficients. Since real coefficients are desired, the problem is not yet solved. However, I found that by factoring these complex polynomials, the roots can be matched with their complex conjugates and expanded back into second order polynomials, but now with real coefficients.