Binomial Theorem
The Binomial Theorem is used to expand a bracket to a given power.
𝑟=𝑛
𝑛
(𝑥 + 𝑦)𝑛 = ∑ ( ) 𝑥 𝑛−𝑟 𝑦 𝑟
𝑟
𝑟=0
The coefficient can be calculated using factorials or from Pascal’s Triangle.
Example
Expand (𝑥 + 𝑦)4
4
4
4
4
4
(𝑥 + 𝑦)4 = ( ) 𝑥 4 𝑦 0 + ( ) 𝑥 3 𝑦1 + ( ) 𝑥 2 𝑦 2 + ( ) 𝑥 1 𝑦 3 + ( ) 𝑥 0 𝑦 4
0
3
1
2
4
(𝑥 + 𝑦)4 = 𝑥 4 + 4𝑥 3 𝑦1 + 6𝑥 2 𝑦 2 + 4𝑥 1 𝑦 3 + 𝑦 4
The coefficients are from the fourth row of Pascal’s Triangle.