Binomial Theorem & Expansion Calculator

Binomial Theorem Guide

What's the key thing to understand about The Binomial Theorem?

(a + b)ⁿ = Σ C(n,r) × aⁿ⁻ʳ × bʳ for r from 0 to n. The (r+1)-th term (or term in xʳ): T_{r+1} = C(n,r) × aⁿ⁻ʳ × (bx)ʳ. Binomial coefficient C(n,r) = n! / (r!(n−r)!) = nCr. Example: (1+x)⁴. r=0: C(4,0)×1⁴×x⁰ = 1. r=1: C(4,1)×1³×x = 4x. r=2: C(4,2)×1²×x² = 6x². r=3: C(4,3)×1×x³ = 4x³. r=4: C(4,4)×x⁴ = x⁴. Expansion: 1 + 4x + 6x² + 4x³ + x⁴.

What do I need to know about Pascal's Triangle?

The binomial coefficients for each power n form rows of Pascal's triangle. Row 0: 1. Row 1: 1 1. Row 2: 1 2 1. Row 3: 1 3 3 1. Row 4: 1 4 6 4 1. Row 5: 1 5 10 10 5 1. Each entry = sum of two entries directly above. The r-th entry in row n = C(n,r) = nCr. Row sum: sum of row n = 2ⁿ (the total number of subsets of n items). Alternating sum: 0 for n > 0. The binomial coefficients appear in probability distributions, algebra, and combinatorics.

What do I need to know about Binomial Approximation?

For small |x| (|x| << 1): (1 + x)ⁿ ≈ 1 + nx. More precise: (1 + x)ⁿ ≈ 1 + nx + n(n−1)x²/2 + .... This works for any n (positive, negative, fractional). Example: (1.02)⁵ ≈ 1 + 5×0.02 = 1.10. Exact: 1.1041. Error: 0.37%. Very useful for: compound interest approximations. Relativistic mechanics (γ ≈ 1 + v²/2c²). Error propagation in physics. Fractional powers: (1 + x)^(1/2) ≈ 1 + x/2 − x²/8. (1 + x)^(−1) ≈ 1 − x + x² − x³... (geometric series).

What's the key thing to understand about Negative and Fractional Powers?

The binomial theorem extends to non-integer n for |x| < 1: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ... For negative n, this gives an infinite series (converges for |x| < 1). Example: (1 + x)^(−1) = 1 − x + x² − x³ + ... (geometric series with ratio −x). (1 − x)^(−1) = 1 + x + x² + x³ + ... This is how engineers and physicists quickly approximate complicated expressions — replace with the first few terms of the binomial expansion.