HSSLIVE Plus One Maths Chapter 8: Binomial Theorem Notes

The Binomial Theorem provides an elegant formula for expanding expressions of the form (a+b)^n without direct multiplication. This chapter explores binomial expansions, properties of binomial coefficients, and practical applications in various mathematical contexts. Students will learn to find specific terms in expansions, understand Pascal’s triangle, and connect these concepts to combinatorial problems. This theorem serves as a bridge between algebra and more advanced topics like calculus, probability distributions, and mathematical analysis.

Chapter 8: Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ.

Binomial Expansion:

(a + b)ⁿ = nC0 × aⁿ × b⁰ + nC1 × aⁿ⁻¹ × b¹ + nC2 × aⁿ⁻² × b² + … + nCn × a⁰ × bⁿ

Or written with summation notation: (a + b)ⁿ = Σ(k=0 to n) nCk × aⁿ⁻ᵏ × bᵏ

Binomial Coefficients:

The coefficient of xᵏ in the expansion of (1 + x)ⁿ is nCk.

Pascal’s Triangle:

A triangular array where each number is the sum of the two numbers directly above it:

Each row represents the coefficients in the binomial expansion of (a + b)ⁿ.

Properties of Binomial Coefficients:

1. nC0 + nC1 + nC2 + … + nCn = 2ⁿ
2. nC0 – nC1 + nC2 – … + (-1)ⁿnCn = 0
3. nC0² + nC1² + nC2² + … + nCn² = ²ⁿCn

General Term:

The (r+1)th term in the expansion of (a + b)ⁿ is: Tr+1 = nCr × aⁿ⁻ʳ × bʳ

Applications:

• Probability distributions (Binomial probability)
• Approximation methods
• Series expansions