Lesson 5 of 10 · 7 min
The binomial theorem
NCERT §7.2.1
The stall now puts up a new card: expand (x + 2)⁶. With the coefficients known as nCr, the whole expansion can be written in one line.
The lesson in notes
In short
For any positive integer n: (a + b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + nC(n − 1) abⁿ⁻¹ + nCn bⁿ.
In sigma notation, (a + b)ⁿ = Σ nCk aⁿ⁻ᵏ bᵏ, the sum running over k = 0 to n, with b⁰ = 1 and a⁰ = 1.
The numbers nCr are called binomial coefficients. The expansion has n + 1 terms.
The power of a starts at n and falls by 1 each term down to 0; the power of b starts at 0 and rises to n. In every term the two powers add to n.
Example: (x + 2)⁶ = x⁶ + 6·x⁵·2 + 15·x⁴·4 + 20·x³·8 + 15·x²·16 + 6·x·32 + 64 = x⁶ + 12x⁵ + 60x⁴ + 160x³ + 240x² + 192x + 64.
Our own example: (2x + 1)⁴ = 16x⁴ + 32x³ + 24x² + 8x + 1. Putting x = 1 checks it: both sides equal 3⁴ = 81.
Negative and fractional powers of x can sit inside the binomial. With a = x² and b = 3/x, (x² + 3/x)⁴ = x⁸ + 12x⁵ + 54x² + 108/x + 81/x⁴ for x ≠ 0.