Binomial Theorem and Coefficients Explained

Quick answer

(a + b)^n = Σ C(n, k) a^(n−k) b^k for k = 0 to n.

Introduction

Polynomial chapters reuse combination logic under new notation. The variable names change; the count does not.

If coefficients are unfamiliar, read what a binomial coefficient is before you chase exponents.

Triangle fans can cross-check small n on Pascal's triangle while algebra students practice theorem shortcuts here.

Choosing powers in the expansion

Each term picks b exactly k times and a exactly n − k times across n factors. The number of distinct exponent patterns is C(n, k).

That is why (x + y)^4 shows coefficients 1, 4, 6, 4, 1: they are C(4, k) for k = 0,…,4.

Theorem statement

• (a + b)^n = Σ_{k=0}^n C(n,k) a^(n−k) b^k
• Coefficient of a^s b^(n−s) is C(n, s)

Sign variants like (a − b)^n alternate signs on b^k terms while keeping the same absolute coefficients.

Step-by-step guide

1. Identify n from the outer exponent. That n drives every coefficient in the row.
2. Map the term you need to k. Power on b (or x in (1+x)^n) equals k.
3. Return C(n, k). Attach any sign pattern required by minus signs.

Coefficient of x^2 in (1 + x)^8

Here a = 1, b = x, n = 8, needed k = 2.

Coefficient is C(8, 2) = 28.

Full expansion is optional when the question asks for a single coefficient.