A binomial is a polynomial with two terms. We're going to look at the Binomial Expansion Theorem, a shortcut method of raising a binomial to a power.
(x+y)0 = 1
(x+y)1 = x + y
(x+y)2 = x2 + 2xy + y2
(x+y)3 = x3 + 3x2y + 3xy2 + y3
(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x+y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5
There are several things that you hopefully have noticed after looking at the expansion
- There are n+1 terms in the expansion of (x+y)n
- The degree of each term is n
- The powers on x begin with n and decrease to 0
- The powers on y begin with 0 and increase to n
- The coefficients are symmetric
Pascal's Triangle
Pascal's Triangle, named after the French mathematician Blaise Pascal is an easy way to find the coefficients of the expansion.
Each row in the triangle begins and ends with 1. Each element in the triangle is the sum of the two elements immediately above it.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1