1. The binomial theorem describes the algebraic expansion of powers of a binomial expression, i.e., expressions of the form $(a+b)^n$, where $n$ is a non-negative integer. 2. The theorem states that: $$ (a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k $$ where ${n \choose k}$ is the binomial coefficient representing the number of ways to choose $k$ items from $n$, calculated as: $$ {n \choose k} = \frac{n!}{k!(n-k)!} $$ 3. This expansion has $n+1$ terms, starting from $a^n$ (when $k=0$) to $b^n$ (when $k=n$), and the coefficients are symmetric. 4. For example, when $n=3$: $$(a+b)^3 = {3 \choose 0} a^3 b^0 + {3 \choose 1} a^2 b^1 + {3 \choose 2} a^1 b^2 + {3 \choose 3} a^0 b^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ 5. The binomial theorem is fundamental in combinatorics, algebra, probability, and calculus for expanding expressions and calculating coefficients efficiently.