<s>[INST] Finding the product of real and imaginary roots separately If I am given a polynomial of nth degree and asked to fond the product of real and imaginary roots what steps should I take? 
I know how to calculate the sum or product of all roots of a polynomial of nth degree but how to separately find the product of real and imaginary roots?
For example  if I'm asked to find the product of real roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$.
What steps should I take?
 [/INST] This is too long for a comment.  Just found a partial  answer for $x^4+a_3x^3+a_2x^2+a_1x+a_0=0$.  Let $y$ and $x$ denote the product of the real and complex roots and $b$ and $a$ the sums of the real and complex roots resp.  Playing Vieta with the coefficients and those sums and product, you will find the following system of equations.
$$\begin{align*}
yx&=a_0\\
a+b&=a_3\\
y+x+ab&=a_2\\
xb+ya&=a_1
\end{align*}
$$
As this system is symmetric in $a\leftrightarrow b$ and in $x\leftrightarrow y$ we expect six solutions.  For the given polynomial Mathematica confronts us with:
$$
\left\{\left\{a\to 0,b\to \frac{3}{2},x\to -3,y\to
   -\frac{5}{2}\right\},\left\{a\to \frac{3}{2},b\to 0,x\to -\frac{5}{2},y\to
   -3\right\},\left\{a\to \frac{1}{2} \left(5-2 \sqrt{3}\right),b\to
   \sqrt{3}-1,x\to -\frac{5 \sqrt{3}}{2},y\to -\sqrt{3}\right\},\left\{a\to
   -1-\sqrt{3},b\to \frac{1}{2} \left(5+2 \sqrt{3}\right),x\to \sqrt{3},y\to
   \frac{5 \sqrt{3}}{2}\right\},\left\{a\to \sqrt{3}-1,b\to \frac{1}{2}
   \left(5-2 \sqrt{3}\right),x\to -\sqrt{3},y\to -\frac{5
   \sqrt{3}}{2}\right\},\left\{a\to \frac{1}{2} \left(5+2 \sqrt{3}\right),b\to
   -1-\sqrt{3},x\to \frac{5 \sqrt{3}}{2},y\to \sqrt{3}\right\}\right\}
$$
Only the first solution fits to the given equation.  At the moment I haven't a clue to de-symmetrize the conditions.  Alas, it's a beginning.
Michael
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