Beyond polynomial mental barrier

Thinking about ways of looking at problems intrigues me, where I recently posted about something rather simple to try, which has remarkable consequences:

k*P(x) = k*(g₁(x) + 1)(g₂(x) + 2)

where P(x) is a quadratic with integer coefficients, g₁(0) = g₂(0) = 0, but g₁(x) does not equal 0 for all x. And k is any nonzero non-unit integer.

And:

k*P(x) =  (f₁(x) + k)(f₂(x) + k)

These two expressions force you in certain directions while including the polynomial case which I think is the one the human brain by design prefers. So then:

 g₁(x) = x and g₂(x) = x

That forces P(x) = x² + 3x + 2

Which means, switchable by indices of course:

 f₁(x) =  kx and f₂(x) = x - (k-2)

Which is the simplest case of it. Polynomial cases are easy. But I suggest to you there is a mental barrier to accepting that the g's are not limited to what is familiar to a human mind! But if you push beyond polynomials you break what most people think they know about mathematics and numbers.

Which is why I like this example.

Test yourself--find another set of f's for a different set of g's.

Polynomials are boring. There are more of those of course, an infinity more but that pattern is easy.

If you're stuck, welcome to the limitations of your brain. Some may be incapable of seeing anything other than polynomials here. You may have an inherent limitation in the wiring of your brain. And if so that's not my fault! Don't get mad at me because of it.

Notice I didn't specify a ring. See how that matters. Try different rings, or even the complex field! The f's and g's will still exist there of course.

Push beyond your mental barriers--if you can.

Are you smart enough to find any other answer beyond the basic your mind already knows?

But some of you don't need to be so enticed, as you've been looking for a door into something beyond, maybe your entire life. For those people, you're welcome to the fun!


James Harris