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The Binomial Theorem

A binomial is a polynomial that has two terms. The Binomial Theorem explains how to raise a binomial to certain non-negative power.

The theorem states that in the expansion of ( x + y ) n ,

( x + y ) n = x n + n x n 1 y + ... + n C r x n r y r + ... + n x y n 1 + y n , the coefficient of x n r y r is

n C r = n ! ( n r ) ! r !

The symbol ( n r ) is often used in place of n C r to denote binomial coefficient.

The expansion is expressed in the sigma notation as ( x + y ) n = r = 0 n n C r x n r y r .

Note that, the sum of the degrees of the variables in each term is n .

Example:

What is the coefficient of a 4 in the expansion of ( 1 + a ) 8 .

Use the binomial theorem to determine the general term of the expansion.

The general term in the expansion of ( x + y ) n is n ! ( n r ) ! r ! x n r y r .

Here, x = 1 , y = a and n = 8 . The term that has the fourth power of the variable a will be the fourth term in the expansion. Therefore, substitute r = 4 in the binomial coefficient of the general term and evaluate.

8 ! ( 8 4 ) ! 4 ! = 8 × 7 × 6 × 5 4 × 3 × 2 × 1 = 70

Therefore, the coefficient of a 4 in the expansion is 70 .