General Solution to Pseudo-Geometric Polynomial Product?

Question

Is there a general, fixed number of operations simplification to the expression $$\prod_{i=0}^{n} (a+bx^i)$$ in the same vein as $$\prod_{i=0}^n ax^i = (\sqrt {a^2 x^n})^{n+1}$$

Answer 1

I don't see what we can do except developing the product as following : $$ \prod_{k=1}^{n}{\left(a+bx^{k}\right)}=\sum_{k=0}^{n}{a^{n-k}b^{k}\sum_{1\leq i_{1}<\cdots <i_{k}\leq n}{x^{i_{1}+\cdots +i_{k}}}} $$ which is a generalisation of the binomial theoreme, thanks to it we can find the second result you gave by setting $ a=0 \cdot $