The Binomial Theorem

Work by Namonda, Njamvwa and Anna

What is the binomial theorem?

If a binomial expression is the sum of two terms, for example ‘a + b’

Then the binomial theorem is a method for expanding a binomial expression to a power, for example (a + b)5

(a + b)n

It is found that (a + b)2 = a2 +2ab + b2

You will notice that the first term is an, the

second term is 2an-1b1 and the third term is an-2b2. (note: there are n+1 terms in the expansion)

That is the power of ‘a’ decreases from n to 0 and the power of ‘b’ increases from 0 to n as we go from left to right.But how do we find the coefficient?

Pascal’s triangle

Can be used to find the coefficients.For example, the coefficients of (a + b)3 are 1, 3, 3, 1 → the numbers in the 4th line of Pascal’s triangle.

A better way of expanding a binomial, is to use the

General formula:

Where = nC3 (combination)

To find the kth term:

Example

i) Find the first 3 terms in the expansion, in ascending powers of x, of (2 + x2)5

ii) Hence find the coefficient of x4 in the expansion of (1 + x2)2(2 + x2)5