The binomial theorem is a useful formula for determining the algebraic expression that results from raising a binomial to an integral power. It provides one with a quick method for finding the coefficients and literal factors of the resulting expression.
The binomial theorem is stated as follows:
where n! is the factorial function of n, defined as
n! = n (n-1) (n-2) ..... 1
and 0! = 1 by convention.
EX.
The binomial theorem can also be used to find the rth term of the expansion of (a + b)n.
The first term of this expansion is an, and the (n+1)th term is bn. By looking at the statement of the binomial theorem, we can see that the literal factors of the rth term are an-r+1 · br-1, where the sum of the exponents of a and b must be equal to n. Since all terms in the expansion of (a + b)n can be written as
we can set p = r - 1 so that the literal factors of the above term will be an-r+1 · br-1 . Thus, the rth term of the expansion of (a + b)n is
EX. The 7th term of ( x + y )13 is computed as
