Expanding powers of a binomial with the general term
The binomial theorem expands (a + b)ⁿ for a positive integer n without multiplying the bracket out repeatedly: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. The expansion has n + 1 terms, the powers of a fall from n to 0 while those of b rise from 0 to n, and the coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle. For example, (x + 2)³ = x³ + 6x² + 12x + 8.
The general term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ, which lets you find any single term directly. When n is even there is one middle term, the (n/2 + 1)th; when n is odd there are two. To find the term independent of x in an expansion such as (x + 1/x)⁶, write down the power of x in the general term, set it equal to zero and solve for r; here r = 3, giving ⁶C₃ = 20. Putting a = b = 1 shows that the binomial coefficients in row n add up to 2ⁿ.
Why it matters
The binomial theorem gives quick approximations, such as estimating (1.01)⁵ without a calculator, and it links algebra with counting, since ⁿCᵣ is also the number of ways to choose r objects from n. Probability uses the same coefficients in the binomial distribution, and the theorem is a standard Class 11 and JEE topic.
Key terms
- Binomial coefficient
- The number ⁿCᵣ that multiplies each term in the expansion of (a + b)ⁿ.
- General term
- The formula Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ that gives any term of the expansion.
- Pascal's triangle
- A triangular array in which each number is the sum of the two numbers above it.
- Middle term
- The term or pair of terms in the centre of a binomial expansion.