TITLE: Show that this polynomial is positive
QUESTION [7 upvotes]: Consider the following polynomial in two variables :
$$
Q(k,x)=27x^6 - 144kx^4 + 80k^2x^3 + 240k^2x^2 - 192k^3x + (64k^4 - 128k^3)
$$
Then for any integer $k \geq 5$, the polynomial $Q(k,.)$ (in one variable $x$)
seems to be always positive (i.e, $Q(k,x) >0$ for any real number $x$).
  Prove or disprove.

REPLY [2 votes]: You can also use quantifier elimination algorithms. This gives you precisely for which $k$ the polynomial $Q(k,x)$ is positive. For example Mathematica implements such an algorithm :
In[1] := Resolve[ForAll[{x}, Q > 0]]
Out[1] := k < 0 || k > 4

I insist that you can trust this result, it is an exact algorithm not a numerical heuristic.