Harristotelian Logic

The equations should be familiar enough by now:

P(x) = 175x^2 - 15x + 2

7*P(x) = (49x^2 - 14x)5^2 + (7x-1)(7)(5) + 49

with the non-polynomial factorization

7*P(x) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

Several posters have tried to invoke weird functions to try and explain what happens with factors of 7 above, but I can use a simple equivalency argument that mirrors their claims.

With the factorization divide both sides by 7, but assume 7 divides one factor, and now use b's as the resultant function to get:

P(x) = (5b_1(x) + 1)(5b_2(x)+ 7)

But posters claim that 7 is not a factor of only one so introduce w's where

w_1(x)*w_2(x) = 7, so

P(x) = (5b_1(x)/w_2(x) + 1)(5b_2(x)*w_2(x)+ 7)

and their claims are just equivalent to a function b_1(x)/w_2(x) that is in a sense a fraction and now multiply by 7, and you get

7*P(x) = (5b_1(x)*w_1(x) + 7)(5b_2(x)*w_2(x)+ 7)

which is how you can APPEAR to get around the distributive property using functions—one of the functions just needs to be like a fraction.

BUT some might argue, so what? How do I know then that there exists functions so that the factorization

P(x) = (5b_1(x) + 1)(5b_2(x)+ 7)

does not contain functions that are like fractions?

Easy. Notice that if w_1(x) and w_2(x) behave like any kind of normal function you could get values where they both have the SAME factors in common with 7. That is, if they had any continuity whatsoever.

But

7*P(x) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

and looking at the middle term and specifically at 7x-1 which is the sum of the roots, you find that it is coprime to 7, unless—x is is something like a fraction which can divide out factors in common with 7.

So there is no way that if w_1(x) and w_2(x) are functions of x that they are anything like what you and I would consider a function to be, unlike the a's which can be found using the quadratic formula:

a(x) = (7x - 1 +/- sqrt((7x-1)^2 -4*(49x^2 - 14x)))/2

where you get a continuous function—and yeah, sure take the top half or whatever, like just use

a(x) = (7x - 1 + sqrt((7x-1)^2 -4*(49x^2 - 14x)))/2

assuming the square root only gives the positive root if that makes you happy fitting into the standard definition of a function.

So these supposed factor of 7 functions have to be rather bizarre, and don't change the distributive property argument I've made anyway, as if you multiply 7 times

P(x) = (5b_1(x)/w_2(x) + 1)(5b_2(x)*w_2(x)+ 7)

you get the SAME answer anyway, so there is the equivalency.

So yes, the distributive property works the same way—even with functions—but if the factors don't seem to work right then hey, you have some functions that are dividing out factors of what is being multiplied times them, which isn't rocket science or some bizarrely complicated idea.

So how can you tell definitively that there are not these really weird functions?

Well, made up "functions" are one thing but when you stick a variable like x in with REAL functions you get functional behavior, like continuity. Yeah, you can get creative with wacky functions, but having actual variables in there with some kind of mathematical expression, like with

a(x) = (7x - 1 + sqrt((7x-1)^2 -4*(49x^2 - 14x)))/2

means they have to BEHAVE like mathematical expressions. Notice in arguments with posters they'll do weird things like say, well the function equals 6 at this value and x+1 or something else at some other value, as they sit there and deliberately make up something that can't behave like

a(x) = (7x - 1 + sqrt((7x-1)^2 -4*(49x^2 - 14x)))/2.

Oh, so why argue? These posters here are the bottom of the barrel, which is why they bother to argue on Usenet. I've communicated with the people supposedly at the top—Barry Mazur, Andrew Granville, and even a leading professor at my own alma mater Vanderbilt University.

There was none of the nonsense crap that the bottom feeders of math newsgroups on Usenet come up with, they just willfully did NOTHING. So, there isn't really any point. I'm just kind of spinning my wheels and at some times looking for a bit of entertainment in my misery.

Watching bottomfeeders jump through hoops may be weak and silly fun, but it's the limited fun I have.

And they do jump, now don't they?