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(r1)}y^{r1}.
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^{n1}y + C(n, 2)x^{n2}y^{2} + ^{ ... } + C(n, n  1)xy^{n1} + C(n, n)y^{n}.
Each term r in the expansion of (x + y)^{n} is given by C(n, r  1)x^{n(r1)}y^{r1}.
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}. 
