The document discusses the binomial theorem, which provides a method for expanding binomial expressions to a power. Specifically: 1) The binomial theorem states that the terms in the expansion of (a + b)n follow a predictable pattern, with the power of the first term being n and decreasing by 1 for each subsequent term as the power of the second term increases from 0 to n. 2) Pascal's triangle can be used to find the coefficients of the terms in the expansion. For example, the coefficients of the terms in (a + b)3 are the numbers in the 4th row of Pascal's triangle. 3) A general formula is provided to calculate the coefficient of any term in the

1. The Binomial Theorem
Work by Namonda, Njamvwa and Anna

2. 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

3. (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?

4. 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.

5. A better way of expanding a binomial, is to use the
General formula:
Where = nC3 (combination)

6. To find the kth term:

7. 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