Factor doesn't factor out a common multiplier; a feature or a bug?

It's usual that Factor extracts a common multiplier:

1.512080625457108*^21 + 3.06237456725561*^20*x - 5.6448211439959745*^19*x^2 // Factor

(* -5.64482*10^19 (-8.55592 + 1. x) (3.13082 + 1. x) *)


This doesn't seem to always work, though. Consider following carefully picked polynomials:

{-4.627573651832992*^26 + 1.2080145470422454*^27*x - 5.963005423058937*^26*x^2,
-7.023191604602253*^25 + 2.3914405755063114*^25*x - 1.2176143437609178*^24*x^2,
3.917815028990508*^26 + 5.635303072532717*^25*x - 1.3273075954855617*^25*x^2,
-9.598625545617204*^25 + 2.8233044513990762*^25*x - 2.0725025122730105*^24*x^2,
-8.057893949471401*^25 + 2.053438258009868*^25*x - 1.3072226910287313*^24*x^2}


Results for these are quite different, and no shared multiplier can be observed:

% // Factor

(* {-(-3.69437*10^13 + 2.44193*10^13 x) (-1.2526*10^13 + 2.44193*10^13 x),
-(-1.77056*10^13 + 1.10346*10^12 x) (-3.96664*10^12 + 1.10346*10^12 x),
-(-2.89847*10^13 + 3.64322*10^12 x) (1.35168*10^13 + 3.64322*10^12 x),
-(-1.02134*10^13 + 1.43962*10^12 x) (-9.39806*10^12 + 1.43962*10^12 x),
-(-9.22832*10^12 + 1.14334*10^12 x) (-8.7317*10^12 + 1.14334*10^12 x)} *)


Should this be considered a feature, or a bug? Factor does work, but its output in these cases is not really useful for Simplify for taking out constants from e.g. square roots. On a quick look it would seem these polynomials are not particularly common, but some intermediate results produce lots of them.

• Can't you use CoefficientList? – Feyre Sep 16 '16 at 16:04
• @Feyre Yes, it's possible to replicate this feature using CoefficientList and massage the results on that basis, but I'm somewhat puzzled over some "failure modes" of Factor; are those intentional or not? – kirma Sep 16 '16 at 16:05
• "Factor applies only to the top algebraic level in an expression. You may have to use Map, or apply Factor again, to reach other levels.", see Factor—Wolfram Language Documentation – creidhne Sep 16 '16 at 16:14
• Bug fix, actually. The current behavior was put in around March 2009 to improve on a situation wherein factors might be removed in a way that gives rise to numerically bad behavior. A particular manifestation was that coefficients could become zero from application of Chop. This was a source of numerous problems. There are some size heuristics that determine at what point to avoid pulling out numbers, and this might play a role in the different behaviors observed in examples under discussion. – Daniel Lichtblau Sep 16 '16 at 17:00
• Possibly related: (123565) – Mr.Wizard Sep 18 '16 at 7:49

The handling of the examples in this thread resulted from changes early on in the series, dating to March 2009. The goal was to improve on a situation wherein factors might be removed in a way that gives rise to numerically bad behavior. A particular manifestation was that coefficients could become zero from application of Chop; this has been a source of numerous problems.