Harristotelian Logic

Imagine you have a factorization of a polynomial

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

and you are told to multiply it by 7, and to pick one way, so you choose

7*P(x) = (7f(x) + 7)*(g(x) + 2)

can the value of the functions f(x) and g(x) invalidate your choice, turning it into something else?

That simplicity of multiplying by a constant lies at the heart of the non-polynomial factorization arguments.

To understand its importance let

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

and now multiply that polynomial times 7 and re-group the resultant terms:

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

to factor it as

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.

The functions now obscure how your multiplication proceeded, so let x=0, to clear them out, and you find

a^2 + a = 0

proving that your 7 went through as it did in my simple example above as one of the a's is 0 and the other -1 when you clear out the functions with x=0.

Simplicity is at the core where what complexity there is comes from the use of a creative construction:

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

to get the non-polynomial factorization.