> Factoid (as our local TV station calls them): Of all possible quadratic > polynomials whose coefficients are less than or equal to 20, only 7.6%
> of them factor.
The question is an interesting one, if only because there are several ways to interpret it. Should we, for example, identify two polynomials that differ only by a constant factor--for example,
x^2 + 2 x + 1
2 x^2 + 4 x + 2?
What about the quadratic polynomial 3 x^2? Does it count, or is it too trivial to bother with?
Here are some facts: There are exactly 28000 quadratic polynomials of the form a x^2 + b x + c with integer coefficients a, b, c for which 0 < a <= 20, -20 <= b, c <= 20, a, b, and c are relatively prime, and b^2 + c^2 > 0. Of these,
Sqrt[b^2 - 4 a c]
is integral in 2237 cases. That's 7.9893%.
If we drop the requirement that a, b, c be relatively prime, and allow that 0 < Abs[a] <= 20, then there are 67200 such polynomials. Of these, the discriminant has integer square root in 6312 cases. That's 9.328%.