The Binomial Theorem provides a fast, algebraic method for expanding expressions of the form $(a + b)^n$ without having to perform tedious repeated polynomial multiplication.
1. The Core Formula
For any non negative integer $n$:
$(a + b)^n = \sum {k=0}^{n} \binom{n}{k} a^{n k} b^k$
Written out term by term:
$(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n 1} b^1 + \binom{n}{2} a^{n 2} b^2 + \dots + \binom{n}{n} a^0 b^n$
2. Key Components of the Formula
The Binomial Coefficient: $\binom{n}{k}$