consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials.
now my first question is does this set together with the operation forms a monoid or at least a semi-group?
my second is, consider the polynomials, $x^2 + 2x + 1 $ and $x + 1$ which are both in the set, now if i take the product of the two then it would be $x^3 + 2x^2 + x + x^2 + 2x + 1$. my question is can i write it as $x^3 + 3x^2 + 3x + 1$, even if i don't have the operation "addition"? what could be the justification on why i can write it or cant write it that way?
Polynomial multiplication is associative (not to mention commutative). The constant polynomial 1 would be your identity element. So it looks like you have a monoid.
I would say that you should write the product in the latter form since you're just following the rules for polynomial multiplication. For a related example, consider the group of all $n \times n$ nonsingular real matrices with the operation of multiplication. In order to find the product of two matrices we need to add (aand multiply) real numbers together even though real numbers aren't our elements and addition isn't our operation.