For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

A finite product is an object of the form $$ \prod _{k=1}^{n} a_k = a_1 \times a_2 \times \ldots \times a_n $$The $a_k$ are often complex numbers or functions. For finite products the only issue of convergence is that none of the $a_k$ are identically zero.

Examples of finite products include:

- The (perhaps misnamed) Fundamental Theorem of Algebra asserts that every polynomial in $\mathbb{C}[x]$ can be written as $$ p(x) = c \prod _{k=1}^n (x-\alpha_k), $$where $c$ is a constant and $\alpha_k$ are the roots of $p$ listed with multiplicities.
- The falling factorial $(x)_n$ is $$ (x)_n = \prod_{k=0}^{n-1}(x-k) $$
- The Lagrange interpolation formula: let $\{(x_i,y_i)\}_{i=0}^{n}$ be a set of $n+1$ points with the $x_i$ distinct. The unique polynomial of degree at most $n$ $P_n(x)$ passing through them is given by $$ L_i(x)=\prod\limits_{\substack{k=0 \\ k\neq i}}^n \frac{(x-x_k)}{(x_i-x_k)} $$ $$ P_n(x) = \sum_{k=0}^{n} y_i L_i(x) $$