## Polynomials, general factorization, distributive property

Here I will give an explanation with some basic examples, where I make the effort in the hope that it will clear the air about how I use the distributive property with certain factorizations of a polynomial.

Consider a polynomial C(x) with a constant term that is 1 on the complex plane.

Multiply it by 7 and factor it as

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

where f(0) = g(0) = 0.

The reason the polynomial is multiplied by a constant is that in some of my research you get expressions similar to the above, where I'm abstracting for simplicity.

Now if f(x) and g(x) are simple linear functions, it's easy enough to see how 7 distributes through, for instance, if C(x) = x^2 + 2x + 1 and g(x) = x, then it must be true that

f(x) = 7x

where it doesn't mean anything in the complex plane to say that f(x) has 7 as a factor as so does g(x), though it equals x.

The mathematical terminology hasn't been developed for these types of arguments, so I usually just say that 7 multiplied through one factor of C(x), in this case, 7x+7.

I doubt many would argue that for any polynomial C(x) on the complex plane, where f(x) and g(x) are linear functions of x that 7 would have to have multiplied through the factor where 7 is visible, with the requirement that f(0) = g(0) = 0.

Well, let C(x) = x^2 + x - 1, and

g(x) = sqrt(x^2 + x) + 1

and notice that though it's not a polynomial it still must be true that 7 multiplied through the first factor, and again, as these results are valid in the complex plane it's meaningless to say that 7 is a factor of f(x).

But, of course, for this example

f(x) = 7*sqrt(x^2 + x)

and the question is how to describe the general case with

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

where f(0) = g(0) = 0.

Now I can give specific cases repeatedly, but it's better to try and abstract out what's happening—why is it true that 7 multiplies through the one factor?

Can anyone give a counterexample where it does not?

I like these examples because I not only show the simple polynomial case—which should be very familiar—but I also give an explicit non-polynomial factorization where again you can actually see the result.

The question then for those who disagree with me is, can you have an explicit solution where the seemingly simple result that 7 multiplied through f(x) + 7 is violated?

Can you break my toy examples?

If so, give the counterexample.

If not, why not?