I'm asking if there is a way to generalize or develop the product of a finite number of linear polynomials like $(x+k)$.
I know that $$(x+a)(x+b)=x^2+x(a+b)+ab=x^2+x\sum\limits_{i\in a,b}i+\prod\limits_{i\in a,b}i$$
Computing $(x+a)(x+b)(x+c)$ we obtain $$(x^2+x(a+b)+ab)(x+c)=x^3+x^2(a+b+c)+x(ab+c(a+b))+abc$$
But if I want to compute it over $i\in M=\{a,b,c,d...n\}$? Is there a simplified formula with sums and products directly over the set of $M$? Or do I have to compute every single products? Moreover if I change the sign of the constant? $$(x-a)(x-b)\cdot\ldots\cdot(x-n)$$ Can we develop a generic formula also in this case?
P.S. Obviously every element in $M$ have multiplicity equal to 1.
The elementary symmetric polynomials are what you are looking for. For example, if we have a set $\, M =\{a,b,c\}, \,$ then $$ e_1(M) = a+b+c, \quad e_2(M) = ab+ac+bc, \quad e_3(M) = abc. $$
For your question about changing signs, each symmetric polynomial is homogeneous and thus changing sign results in a factor of $\,(-1)^d\,$ where $\, d \,$ is the degree of the polynomial.
By the way, the elementary symmetric polynomials also work with multisets. The elements of $\,M\,$ can have multiplicities greater than $1$. For example if $\, M =\{a,a,a\}, \,$ then $\, e_1(M) = 3a, \,$ $ e_2(M) = 3a^2, \,$ $ e_3(M) = a^3.\, $ This gives us $\, (x+a)(x+a)(x+a) = x^3 + 3ax^2 + 3a^2x + a^3. \,$