Binomial product expansion

Question

I have seen the following mathematical identity in a book:

$$ \prod_{i=1}^{N}\left( 1 + ax_i \right)^c = \left( 1 + \sum_{i=1}^{N}{ax_i} + \cdots + a^N\prod_{i=1}^{N}{x_i} \right)^c $$

Is this a generalization of the Binomial theorem? What are the implicit terms within $\cdots$? Any explanations, proofs, references are appreciated!

Answer 1

This follows from the "Multi-Binomial Theorem". Quoted from the wiki page for the Binomial Theorem

$$(x_{1}+y_{1})^{n_{1}}\dotsm(x_{d}+y_{d})^{n_{d}} = \sum_{k_{1}=0}^{n_{1}}\dotsm\sum_{k_{d}=0}^{n_{d}} \binom{n_{1}}{k_{1}}\, x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\;\dotsc\;\binom{n_{d}}{k_{d}}\, x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}$$

Answer 2

These are simply Vieta's formulas, expressing the relations between roots and coefficients.