The Big Idea
The general binomial expansion lets you expand expressions like (1+x)n even when n isn't a whole number. Unlike the standard binomial expansion (which stops after n terms), this one gives you an infinite series — but you only need the first few terms for real answers. The catch: it only works when |x| < 1.
What We're Covering
The Four Core Topics
- General Binomial Expansion: The core formula and how to apply it to (1+x)n and (1+bx)n
- Subtleties: Handling messier expressions like (a+kx)n, roots, negative powers, and factoring
- Multiple Expansions: Breaking down compound rational expressions and combining their validities
- Approximation: Using the expansion to estimate values like √85 or ³√0.96
Topic 1: The General Binomial Expansion
What's Different from Standard Binomial Expansion?
In GCSE, you learned the binomial expansion for positive integer powers like (1+x)³ or (2+y)⁵. These expansions stop after finitely many terms — that's why (1+x)³ has exactly 4 terms.
The general binomial expansion removes that restriction. It works for ANY real number n — fractions, negatives, even irrationals. But the cost is that the series now runs forever (unless n is a positive integer, in which case it terminates like before).
Key insight: Even though the series is infinite, we only care about the first few terms. Higher powers of x become so small (when |x| < 1) that we can ignore them. That's why binomial expansions are so powerful for approximation.
The Formula: The General Binomial Theorem
(1+x)n = 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + ...
where n ∈ ℝ (any real number) and |x| < 1
Breaking Down the Formula
Let's decode what each term looks like:
- Constant term: Always 1
- x term: Coefficient is simply
n
- x² term: Coefficient is
n(n−1) / (2 × 1)
- x³ term: Coefficient is
n(n−1)(n−2) / (3 × 2 × 1)
- x⁴ term: Coefficient is
n(n−1)(n−2)(n−3) / (4 × 3 × 2 × 1)
Pattern to spot: Each coefficient is a fraction where:
- Numerator: Product of n descending by 1 each time: n × (n−1) × (n−2) × ...
- Denominator: Factorial: 1!, 2!, 3!, 4!, ...
The Validity Condition: |x| < 1
This is not optional — it's the region where the infinite series actually converges to a real value. Think of it this way:
- If |x| < 1, each successive power of x gets smaller and smaller. x⁴ is tiny, x¹⁰ is microscopic, and so on.
- If |x| ≥ 1, the powers don't shrink — they either stay big or grow. The series diverges to infinity (or oscillates forever).
The condition |x| < 1 means −1 < x < 1. Every expansion you write down MUST include this validity statement at the end.
How to Expand (1+x)n
Three simple steps:
- Write out the expansion formula (write it fresh each time — don't memorise blindly).
- Substitute your value of n and calculate the first few coefficients. Use brackets and be methodical.
- State the validity: |x| < 1.
Worked Example: Expand (1+x)1/2 up to x²
Step 1: Write the general formula
(1+x)n = 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + ...
Step 2: Substitute n = 1/2
(1+x)1/2 = 1 + (1/2)x + [(1/2)(1/2 − 1) / 2!] · x² + ...
= 1 + (1/2)x + [(1/2)(−1/2) / 2] · x² + ...
= 1 + (1/2)x + [−1/4 / 2] · x² + ...
Step 3:
|x| < 1
n
n
n
n
n
n