Harristotelian Logic

Polynomial factorization can be considered to be one of the more boring areas of modern mathematics because so much of what most people think of in factoring polynomials was figured out centuries ago.

But turns out there is so much more to do, where creativity and a lot of imagination are required.

This post is over advanced techniques in polynomial factorization, moving away from polynomial factors to non-polynomial ones, as consider

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

a rather basic non-monic quadratic, and the factorization

P(x) = (f(x) + 2)*(g(x) + 1)

where f(0) = g(0) = 0, but neither function is a polynomial one—that is, they are NOT simple linear functions.

So to re-cap, I have

175x^2 - 15x + 2 = (f(x) + 2)*(g(x) + 1)

where f(0) = g(0) = 0, so you can see the constant terms on both sides work out to 2, and also notice you do not see factors of 7 anywhere!!!

I emphasize that because here is where these ideas become EXTREMELY advanced and where a lot of confusion has arisen in the past as now I'm going to do something creative and multiply both side by 7:

7*(175x^2 - 15x + 2) = 7*(f(x) + 2)*(g(x) + 1)

where now I re-order in a special way on the left side and pick one way to multiply by 7 on the right:

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

with my re-ordering on the left side it is clear that I have 5 and 7 looking like variables in yet another polynomial!

Advanced techniques indeed!!!

But, things still don't quite look right so I need some substitutions for the functions:

Let

f(x) = 5a_1(x) + 5

and

7g(x) =5a_2(x)

so now I have

(49x^2 - 14x)5^2 + (7x-1)(7)(5) + 49 = (5a_1(x) + 7)*(5a_2(x) + 7)

which may not seem like a big deal to you, but now I can solve for the functions!!!

They are then just roots of

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

Now here's where these advanced techniques run against conventional thinking relying on some recent ideas in mathematics as they are a bit over a hundred years old, having come into the field after Gauss, as remember above?

Remember how I started with

175x^2 - 15x + 2 = (f(x) + 2)*(g(x) + 1)

where there are no factors of 7 anywhere, and I multiplied by 7 to bring it into the picture?

Well, if you pick a non-zero integer x, you find you get a polynomial irreducible over Q, but the analysis above shows what only one of the roots was multiplied by 7, but you can prove that in the ring of algebraic integers NEITHER of the roots can have 7 itself as a factor!!!

Some might try to shoe-horn factors of 7 back into the base factorization, by claiming that the result from the ring of algebraic integers shows that somehow 7 got split up, so let's consider that notion by introducing more functions:

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

(5a_1(x) + 7)*(a_2(x) + 7) = (5b_1(x)*w_1(x) + 7)*(5b_2(x)*w_2(x) + 7)

and go backwards—reversing how I started this post—by now dividing 7 out:

(5a_1(x) + 7)*(a_2(x) + 7)/7 = (5b_1(x) + w_2(x))*(5b_2(x) + w_1(x))

and notice it isn't gone on the right! The functions w_1(x) and w_2(x) are still there as a part of the factorization, but

175x^2 - 15x + 2 = (f(x) + 2)*(g(x) + 1)

doesn't have w_1(x) and w_2(x) in it, and 7 is not a part of that factorization, as why should it be?

I could have multiplied by 13 or some other number.

So there is no way LOGICALLY that 7 can be trapped into the original factorization.

The trouble here though is that people who resist the results above resist them because they see the mathematical result as a human failure which they can't accept because they get taught in school that mathematics is perfect and that past mathematicians were perfect on all the important details, and couldn't have made a mistake that would let it be true.

So they throw out the math to hang out to a flawed view of human perfection, and I have the arguments on Usenet for years to show how dedicated people can be in their error, as well as publication in a math journal covering some of these ideas—and retraction and later the journal dying.

Some people are not up to the challenge of handling mathematical truth,.

But the reality cannot be denied mathematically—some people screwed up with some complex ideas about numbers over a hundred years ago, and the fallout is huge, so some people today decided that mathematics may be good as a concept, but too painful as a reality, so they have switched to pretend mathematics.

And they are angry people who will call you names, and insult you for years if you try to push what is mathematically correct against their very human need to believe.