Mathematical logic & distributive property

There have been some long going and somewhat strange arguments on the newsgroups which boil down to misunderstandings from some about the distributive property when functions are involved, so here is a short discourse that explains the mathematical logic in simple but exhaustive detail.

The distributive property embodies the concept that when you multiply a group you multiply the elements within that group in turn as well, so

a*(b+c) = a*b+a*c

is just a simple way of mathematically saying that, and notice that if you have functions, so that you have

a*(f(x) + b) = a*f(x) + a*b

you have a situation where the value of the function is irrelevant to that operation, which is to say that even if a function is the member of a group being multiplied it still gets multiplied with the rest of the group regardless of its value.

You see, the function kind of gets dragged along with everything else. Being a function gives it no special powers in that situation!

So, if at some value of x you have that the function equals 0, that doesn't change the distributive property, asf or example at that point with what I have above you just have

a*(0 + b) = a*0 + a*b

which is just a*b = a*b which is, of course, still true, as the VALUE of the function is irrelevant, as the key fact is that it is a member of a group that is being multiplied.

Now to the type of example that has sparked SO MUCH ARGUING and debate from primarily the sci.math newsgroup:

On the complex plane given the polynomial P(x), and functions f(x) and g(x)

where f(0)=0 and g(0) = 0

let

7*P(x) = (f(x) + 7)*(g(x) + 1)

which I'll emphasize is true for all x, it must be true by the distributive property that 7 has multiplied through only the factor

f(x) + 7

where by "multiplied through" I mean 7 has multiplied times only one of the two factors of P(x), where the product is f(x)+7.

Now given that there MUST be two factors it can seem reasonable to suppose that 7 split between those factors, except that if it did, like say sqrt(7) multiplies times each, then by the distributive property, what must happen?

Obviously then sqrt(7) would show in the factors so you would have something like

7*P(x) = (f(x) + sqrt(7))*(g(x) + sqrt(7))

where the logical principle is easy—the value of the functions inside the factors of P(x) is irrelevant to the operation of the distributive property, so you can check at x=0, to see how that occurred, just like with

a*(f(x) + b) = a*f(x) + a*b

when I could consider a value where f(x) is 0.

Some posters have proclaimed that the 7 itself must be the product of two functions, so that how it multiplies through varies as x varies, but, um, how do you even know about x with those expressions?

Answer: From the functions.

So their claim is that the functions are somehow reaching outside of their groups to control how their group is being multiplied in direct contradiction to the logical principle that with

a*(f(x) + b) = a*f(x) + a*b

the value of the function has no impact on the how the distributive property operates.

And remember the distributive property is just telling you that if you multiply a group, you end up multiplying the elements within the group as the multiplication distributes through.

One way to look at the arguments of posters claiming the functions can force how 7 multiplies with

7*P(x) = (f(x) + 7)*(g(x) + 1)

is to use the expression "tail wagging the dog" to emphasize the point that the functions just don't have the capacity to affect the simple principle that when you multiply a group you multiply the elements within that group.

So if it's that easy, why would people argue against what I just said?

Well, consider what I've called the Decker example—in the ring of algebraic integers

7 Q(x) = 7((x^2 + x)(5^2) + (-1 + x)(5) + 7) = 7(25 x^2 + 30 x + 2)

and

7 Q(x) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are defined by

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

NOW we have some complexity!!!

But is it really all that complicated?

I'll leave it as an exercise to see if posters think that with this simple example the rules have changed.

Think of it as a test. Given what you have above, what must be true with the Decker example? 