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Given a polynomial equation:

$$x^4+Ax^3+(B+C+D)x^2+(AB+AC)x^2+BD=0$$

where $A$, $B$, $C$, $D$ are known. Numerically I know it has complex solutions. However, I tried but failed to analytically convert the polynomial to be like

$$(x^2+Ex+F)(x^2+Gx+H)=0 $$

I wonder whether it can be done analytically or it is purely a hard problem?

Many thanks for any replies.

user3703018
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1 Answers1

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You could use the Tchirnhaus rotation, which makes the coefficient of the $x^3$ term zero, this is done by considering $f(x-\frac{a_{n-1}}{na_n})$.

In this case $a_{n-1}=A$ and $a_n =1$ and $n=4$. Doing this you will arrive at a polynomial of the form:

$$x^4+Ex^2+F=0$$

Which should be nicer to factorise, so find the roots, then subtract $\frac{a_{n-1}}{na_n}$ from the result.

Ellya
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