ikerous wrote:Wow that is amazing. I'll have to play around with it some, see why it even works. Thanks a lot. That's very cool.

To see why it works, it helps to see where it does NOT work. Specifically, it doesn't work when there is a common factor that can be factored out first. And of course, as with any factoring method (besides quadratic formula and the like), it is difficult to apply if the factors have irrational or complex coefficients. I don't necessarily see that as a bad thing, though.

An example where the three-mistakes method fails:

20x

^{2}-2x-6 = x

^{2}-2x-120 = (x-12)(x+10) = (x-12/20)(x+10/20) = (x-3/5)(x+1/2) = (5x-3)(2x+1). As you can see, it's off by a factor of 2 - the common factor. Just as before, though, if all you are doing is looking for zeros, this can get you there since the constant factor doesn't come into play.

Something I always tell my students when they are learning factoring is to always look for common factors before doing any other factoring. If d=gcd(a,b,c), instead of listing all the factors of ac, you only need to list all the factors of ac/d

^{2}. Factoring whole numbers is alot easier for most humans if the numbers are smaller. Unfortunately, I don't have that data available for raptors.

Govalant wrote:I think you should anyway use the quadratic equation. There's no need to prove it, so it could be used easily.

There are a couple of problems I can see with using the quadratic formula to teach factoring. Primarily, the quadratic formula is usually proven using (a very special case of) factoring. Using the quadratic formula without proof seems to me to support the kind of rote memorisation that tends toward a lack of imagination in maths. The second problem is that

all quadratic expressions can be factored using the quadratic formula. That's a problem because that can interfere with the idea that a quadratic can be prime. Certainly no polynomial with degree>1 is prime in

C[x], but in

Q[x] any quadratic with irrational or complex zeros is prime. Using abstract algebra to explain that x

^{2}+4 and x

^{2}-5 are prime is a bit overkill. The fact that there is no pair of (rational) factors that have a product of 4 (or -5) and sum of 0 is a much more transparent way to make that point.

I apologise if that came across as confrontational. Using the quadratic formula as a tool for factoring is perfectly valid. I don't think it should be the first method learned for factoring, though.