I am interested in factorizing RSA numbers as a coding challange. I understand the basic principle that $2$ large primes are multiplied together to generate the specific RSA number, however I also see that a polynominal can be used to represent the RSA number.
As a specific example, RSA-130 (https://en.wikipedia.org/wiki/RSA_numbers#RSA-130) can be represented as $2$ large prime numbers or as the polynominial:
$5748302248738405200 x_5 + 9882261917482286102 x_4 - 13392499389128176685 x_3 + 16875252458877684989 x_2 + 3759900174855208738 x_1 - 46769930553931905995$
where the root $(x) = 12574411168418005980468$
This could be simplified to $(ax^5) + (bx^4) - (cx^3) + (dx^2) + (ex^1) - (f)$
My question is what makes the polynomial valid? Additionally what is the link between the polynomial and the prime factors, eg how do I get from a polynomial result to a set of primes?
For example could the root $(x), a,b,c,d,e$ be all equal to $2$ with $f$ simply as a large remainder? That does not seem correct to me as it would be trivial to factorise any large number so I assume there are a set of rules that need to be followed to validate the polynomial.
Could you please provide an explanation (or link) of the rules that need to be followed preferably in laymans terms so that I can use the explanation to implement an algorithum.