gen-briefly-explain-the-binomial-theorem-and-one-common-misconce-1f2b

Briefly explain the binomial theorem and one common misconception about it.

## The Binomial Theorem The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer: **(a + b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ** where C(n,k) are binomial coefficients (also written as "n choose k"). For example: (a + b)³ = a³ + 3a²b + 3ab² + b³ ## Common Misconception **Misconception: (a + b)ⁿ = aⁿ + bⁿ** Many students incorrectly assume you can simply distribute the exponent to each term. This only works when n = 1. For any other positive integer, there are middle terms with mixed powers of a and b that cannot be ignored. For instance, (a + b)² = a² + 2ab + b², **not** just a² + b².