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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 = 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

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 rth term of the expansion of (x + y)2 contains xn-(r-1)yr-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)xn + C(n, 1)xn-1y + C(n, 2)xn-2y2 + ... + C(n, n - 1)xyn-1 + C(n, n)yn.
Each term r in the expansion of (x + y)n is given by C(n, r - 1)xn-(r-1)yr-1.


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

(x + y)7 = x7 +7x6y + 21x5y2 +35x4y3 +35x3y4 +21x2y5 +7xy6 + y7.

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  
 =16x4 +4(8x3)(3y) + 6(4x2)(9y2) + 4(2x)(27y3) + 81y4  
 =16x4 +96x3y + 216x2y2 +216xy3 +81y4.  


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


(5x - y)3=(5x)3 +3(5x)2(- y) + 3(5x)(- y)2 + (- y)3  
 =125x3 +3(25x2)(- y) + 3(5x)(y2) + (- y3)  
 =125x3 -75x2y + 15xy2 - y3.