THE BINOMIAL THEOREM
1. Binomial Expansions
A binomial is an expression of the form . Consider the expansion of these binomials for
various values of n.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Notice the powers of a descend as the powers of b ascend, and the sum of the powers is always
n.
The triangle of numbers is known as Pascal’s triangle : the sum of each pair of adjacent numbers
gives the number underneath the pair. The numbers in Pascal’s triangle correspond to the
coefficients in the binomial expansions.
Example 1 : Use Pascal’s triangle to expand .
We need the next row in the table, and this has coefficients 1, 5, 10, 10, 5, 1.
1 = = 32
5 = =
10 = =
10 = =
5 = =
1 = =
And so .
Example 2 : Use Pascal’s triangle to expand .
We need the row with coefficients 1, 4, 6, 4, 1.
1 = =
4 = =
6 = = 150
4 = =
1 = =
And so .
, Example 3 : Find the coefficient of in the expansion of .
Here we do not need to find the full expansion, just the term in .
1
5
10
10 = =
5
1
The coefficient of is −3920.
Example 4 : Find the coefficient of in the expansion of .
We need to continue Pascal’s triangle down a few rows.
1
7
21 = =
The coefficient of is 5103.
Example 5 : Find the coefficient of in the expansion of .
We first expand …
1 = =
3 = =
3 = =
1 = =
And then multiply by , only bothering to find those terms involving .
1
The coefficient of is −585.
Note that if we were far-sighted enough, we would only bother finding the terms in x and in
the expansion of .
C2 p72 Ex 5A
1. Binomial Expansions
A binomial is an expression of the form . Consider the expansion of these binomials for
various values of n.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Notice the powers of a descend as the powers of b ascend, and the sum of the powers is always
n.
The triangle of numbers is known as Pascal’s triangle : the sum of each pair of adjacent numbers
gives the number underneath the pair. The numbers in Pascal’s triangle correspond to the
coefficients in the binomial expansions.
Example 1 : Use Pascal’s triangle to expand .
We need the next row in the table, and this has coefficients 1, 5, 10, 10, 5, 1.
1 = = 32
5 = =
10 = =
10 = =
5 = =
1 = =
And so .
Example 2 : Use Pascal’s triangle to expand .
We need the row with coefficients 1, 4, 6, 4, 1.
1 = =
4 = =
6 = = 150
4 = =
1 = =
And so .
, Example 3 : Find the coefficient of in the expansion of .
Here we do not need to find the full expansion, just the term in .
1
5
10
10 = =
5
1
The coefficient of is −3920.
Example 4 : Find the coefficient of in the expansion of .
We need to continue Pascal’s triangle down a few rows.
1
7
21 = =
The coefficient of is 5103.
Example 5 : Find the coefficient of in the expansion of .
We first expand …
1 = =
3 = =
3 = =
1 = =
And then multiply by , only bothering to find those terms involving .
1
The coefficient of is −585.
Note that if we were far-sighted enough, we would only bother finding the terms in x and in
the expansion of .
C2 p72 Ex 5A
