# Binomial Expansion

### The Binomial Theorem

#### The Binomial Theorem

Here is the expansion of (x + y)n for n = 0, 1,…, 5 :

(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x 2 +2xy + y 2
(x + y)3 = x 3 +3x 2 y + 3xy 2 + y 3
(x + y)4 = x 4 +4x 3 y + 6x 2 y 2 +4xy 3 + y 4
(x + y)5 = x 5 +5x 4 y + 10x 3 y 2 +10x 2 y 3 +5xy 4 + y 5

Look familiar? The coefficients of each expansion are the entries in Row n of Pascal's Triangle. Thus, the coefficient of each term r of the expansion of (x + y)n is given by C(n, r - 1) . The exponents of x descend, starting with n , and the exponents of y ascend, starting with 0, so the r th term of the expansion of (x + y)2 contains x n-(r-1) y r-1 .

This information can be summarized by the Binomial Theorem:

For any positive integer n , the expansion of (x + y)n is C(n, 0)x n + C(n, 1)x n-1 y + C(n, 2)x n-2 y 2 + ... + C(n, n - 1)xy n-1 + C(n, n)y n .
Each term r in the expansion of (x + y)n is given by C(n, r - 1)x n-(r-1) y r-1 .

Example: Write out the expansion of (x + y)7 .

(x + y)7 = x 7 +7x 6 y + 21x 5 y 2 +35x 4 y 3 +35x 3 y 4 +21x 2 y 5 +7xy 6 + y 7 .

When the terms of the binomial have coefficient(s), be sure to apply the exponents to these coefficients.

Example: Write out the expansion of (2x + 3y)4 .

 (2x + 3y)4 = (2x)4 +4(2x)3(3y) + 6(2x)2(3y)2 +4(2x)(3y)3 + (3y)4 = 16x 4 +4(8x 3)(3y) + 6(4x 2)(9y 2) + 4(2x)(27y 3) + 81y 4 = 16x 4 +96x 3 y + 216x 2 y 2 +216xy 3 +81y 4.

Example: Write out the expansion of (5x - y)3 .

 (5x - y)3 = (5x)3 +3(5x)2(- y) + 3(5x)(- y)2 + (- y)3 = 125x 3 +3(25x 2)(- y) + 3(5x)(y 2) + (- y 3) = 125x 3 -75x 2 y + 15xy 2 - y 3.

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