Lesson 70: Binomial Theorem
The Binomial Theorem provides a formula for expanding powers of a binomial expression (a + b)^n. It uses binomial coefficients, which can be found using factorials or Pascal's Triangle.
Binomial Theorem Formula
(a + b)^n = Σ (n choose k) a^(n−k) b^k, where k = 0, 1, 2, …, n
The binomial coefficient is defined as: (n choose k) = n! / [k!(n−k)!]
Examples
Example 1: Expand (x + 2)^3
Solution: (x + 2)^3 = (3 choose 0)x^3·2^0 + (3 choose 1)x^2·2^1 + (3 choose 2)x^1·2^2 + (3 choose 3)x^0·2^3 = 1·x^3 + 3·x^2·2 + 3·x·4 + 1·8 = x^3 + 6x^2 + 12x + 8
Example 2: Find the 4th term of (2 + y)^5
Solution: The 4th term corresponds to k = 3 → (5 choose 3)·2^(5−3)·y^3 = 10·4·y^3 = 40y^3
Practice Problems
- Expand (x + 3)^4
- Find the 3rd term of (1 + 2y)^5
- Expand (2a − b)^3
- Find the coefficient of x^2 in (x + 5)^4
- Use the Binomial Theorem to expand (y − 1)^5