SparkNotes Shopping Cart  |     |  Checkout
Brought to you by Barnes and Noble
  Home : Math & Science : Math Study Guides : Algebra II : Binomial Expansion : The Binomial Theorem
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 = 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.  

Help | Feedback | Make a request | Report an error | Send to a friend
 
Beat the SAT II Biology test with the latest book from the experts at SparkNotes.
More...
 
Feeling hopelessly behind? No Fear Algebra will help you catch up in no time.
More...
 
 
Go to top