BINOMIAL EXPANSION

BINOMIAL EXPANSION
By Intermediate Algebra : Donald S Ruscell and George E Lanning


The operation ( opә’reΙ∫n ) of raising a binomial to some power ( paυә(r) ) is called expanding ( Ik’spændiŋ ) the binomial. By actual multiplication we can obtain the following :
( a + b )0 = 1 ( a plus b to the power zero is equal one )
( a + b )1 = a + b ( a plus b to the power one is equal a plus b )
( a + b )2 = a2 + ab + b2( a plus b to the power two is equal square of a plus a times b plus square of b )
( a + b )3 = a3 + 3a2 b + 3ab2 + b3 ( a plus b to the power three is equal cube of a plus three time square of a time b plus three time a time square of b plus cube of b )
We shall assume that this definite pattern ( ‘pætn ) holds true for all positive integral values of n. This assumption gives us Eq. 1, known a binomial theorem ( θIәrәm ).
( a + b )n = an + nan-1 b + an-2 b2 + an-3 b3 +………….. + nabn-1 + bn

In denominator ( dI’na mIneItә(r) ) of Eq. 1 we find 2 ! and 3 !. The symbol (!) is called factorial, and n ! is called n factorial. The expression n factorial (n!) is defined as the product of n and all the positive integers less than n. Thus, 2! = 2.1 = 2 and 3! = 3.2.1 = 6.
Example 1 :
By use the binomial theorem, obtain the expansion ( Ik’spæn∫n ) of ( x + y )3
( x + y )3 = x3 + 3x2 y + 3xy2 + y3
( cube of x plus y is equal to cube of x plus three time square of x time y plus three time x time square of y plus cube of y )