Root condition for polynomials

Hi,

http://groups.google.com/group/sci.math.num-analysis/browse_thread/thread/d2b54d8a4d11f45f/98d955f9feef7a7f?tvc=2&q=Root+condition+for+polynomials#98d955f9feef7a7f

In the thread one of the authors give conditions
Polynomial is p = a0 x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4

CircleCriterion[p, x]

16*a0 + 16*a1 + 16*a2 + 16*a3 + 16*a4 > 0 &&
32*a0 + 16*a1 - 16*a3 - 32*a4 > 0 &&
640*a0^2 + 320*a0*a1 + 64*a1^2 - 384*a0*a2 -
64*a1*a2 - 576*a0*a3 + 64*a2*a3 - 64*a3^2 +
576*a1*a4 + 384*a2*a4 - 320*a3*a4 - 640*a4^2 > 0 &&
4096*a0^3 - 4096*a0^2*a2 + 4096*a0*a1*a3 -
4096*a0*a3^2 - 4096*a0^2*a4 - 4096*a1^2*a4 +
8192*a0*a2*a4 + 4096*a1*a3*a4 - 4096*a0*a4^2 -
4096*a2*a4^2 + 4096*a4^3 > 0 &&
(a0 - a1 + a2 - a3 + a4)*(4096*a0^3 - 4096*a0^2*a2 +
4096*a0*a1*a3 - 4096*a0*a3^2 - 4096*a0^2*a4 -
4096*a1^2*a4 + 8192*a0*a2*a4 + 4096*a1*a3*a4 -
4096*a0*a4^2 - 4096*a2*a4^2 + 4096*a4^3) > 0

How do you derive such a formula for quintic ?

I am not sure if this formula works for all polynomials of degree n
where n <=4.
( i tried applying the criterion for quadratic and it failed with a0 =
0 and a1=0).
Can someone give me similar formula for 3rd degree and 2nd degree.
How do i derive them ? If it can be done on mathematica what are the
commands ?

Thanks for the help.

Regards,
Terry

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