The Binomial Theorem

The binomial theorem expands powers of a sum (a + b)ⁿ without multiplying everything out. It appears in IB Mathematics: Analysis & Approaches (AA) at both SL and HL.

The expansion

(a + b)ⁿ = Σ ₙCᵣ · aⁿ⁻ʳ · bʳ for r = 0 to n, where ₙCᵣ = n! / (r!(n−r)!) is the binomial coefficient (on your GDC, "nCr").

For small n you can read the coefficients from Pascal's triangle.

(x + 2)³ = x³ + 3x²(2) + 3x(2²) + 2³ = x³ + 6x² + 12x + 8

Finding a specific term

You rarely need the whole expansion. The general term is ₙCᵣ · aⁿ⁻ʳ · bʳ.

Worked example

Find the term in x² in (2x + 3)⁵.

We need x², so the power on 2x is 2 → r = 3.

₅C₃ · (2x)² · 3³ = 10 · 4x² · 27 = 1080x².

IB strategy

For "find the coefficient of xᵏ" questions, do not expand everything — set up the general term and solve for r first. On Paper 1 (non-calculator) compute ₙCᵣ from the factorial formula or Pascal's triangle; on Paper 2 use the GDC's nCr. HL extends this to fractional and negative powers (infinite series).