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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:

  1. 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).
  2. Provide a proof that this problem is not, in general, tractable.
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  • $\begingroup$ I solved this problem, I will post my answer later today. $\endgroup$ – Void Star Mar 30 '16 at 16:21
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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.

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  • $\begingroup$ Keeping this open for now in case someone has a better answer. $\endgroup$ – Void Star Mar 30 '16 at 23:55

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