# Paradox running times of Reduce for polynomial inequalities in Mathematica 8 [closed]

I found something odd about the time needed by Reduce to solve a polynomial inequality (under Wolfram Mathematica 8).

Consider the following:

Timing[Do[Reduce[-(519/197) - (257 x)/121 + (13 x^2)/20 - (16 x^3)/307 - x^4/
194 - x^5/4426 < 0, x, Reals], {1000}]]


On my computer it clocks at about 2.1 seconds. However the following:

Timing[Do[Reduce[-(947/350) - (557 x)/249 + (247 x^2)/350 < 0, x,
Reals], {1000}]]


takes around 9.7 seconds.

In other words, the solving of an inequality of order 2 is five times slower then the solving of one of order 5!

To make the things even more interesting, I can "reproduce" this paradoxical effect (i.e. that a more complex equation is solved faster). If I modify my latter example by adding a third-order term (no matter how big or small), the solving immediately accelerates (!), for example this

Timing[Do[Reduce[-(947/350) - (557 x)/249 + (247 x^2)/350 + 10^(-30)*x^3 < 0,
x, Reals], {1000}]]


as well as this

Timing[Do[Reduce[-(947/350) - (557 x)/249 + (247 x^2)/350 + 10^(30)*x^3 < 0,
x, Reals], {1000}]]


Either I am overlooking something trivial about how Mathematica solves these inequalities or something very strange is going on...