Binomial Expansion
Using Pascal's Triangle
Explore how Pascal's Triangle can be used to expand binomial expressions. Select different powers to see the corresponding row in Pascal's Triangle and the expanded binomial form.
(a + b)⁰ = 1
Observation:
The numbers in Pascal's Triangle correspond to the coefficients in binomial expansion. Each number is the sum of the two numbers directly above it.
The Mathematics Behind Binomial Expansion
Key Concepts:
Binomial Theorem describes the algebraic expansion of powers of a binomial:
- Pascal's Triangle: Each row gives the coefficients for (a + b)ⁿ
- Binomial Coefficients: The numbers in Pascal's Triangle are combinations (n choose k)
- Pattern: Each term follows the pattern C(n,k) · aⁿ⁻ᵏ · bᵏ
Real-world Applications:
Binomial expansion is used in probability theory, algebra, calculus, and many areas of science and engineering where polynomial approximations are needed.
