Chapter 7
The binomial series
7.1 Pascal’s triangle
A binomial expression is one which contains two terms
connected by a plus or minus sign. Thus ( p +q), (a +
x) 2 , (2x + y) 3 are examples of binomial expressions.
Expanding (a + x) n for integer values of n from 0 to 6
gives the results as shown at the bottom of the page.
From these results the following patterns emerge:
(i) ‘a’ decreases in power moving from left to right.
(ii) ‘x’ increases in power moving from left to right.
(iii) The coefficients of each term of the expansions are
symmetrical about the middle coefficient when n
is even and symmetrical about the two middle
coefficients when n is odd.
(iv) The coefficients are shown separately in Table 7.1
and this arrangement is known as Pascal’s triangle. A coefficient of a term may be obtained
by adding the two adjacent coefficients immediately above in the previous row. This is shown
by the triangles in Table 7.1, where, for example,
1 + 3 = 4, 10 + 5 = 15, and so on.
(v) Pascal’s triangle method is used for expansions of
the form (a + x) n for integer values of n less than
about 8.
(a + x) 0 =
1
(a + x) 1 = a + x
a + x
(a + x) 2 = (a + x)(a + x) =
a 2 + 2ax + x 2
(a + x) 3 = (a + x) 2 (a + x) =
a 3 + 3a 2 x + 3ax 2 + x 3
(a + x) 4 = (a + x) 3 (a + x) =
a 4 + 4a 3 x + 6a 2 x 2 + 4ax 3 + x 4
(a + x) 5 = (a + x) 4 (a + x) =
a 5 + 5a 4 x + 10a 3 x 2 + 10a 2 x 3 + 5ax 4 + x 5
(a + x) 6 = (a + x) 5 (a + x) = a 6 + 6a 5 x + 15a 4 x 2 + 20a 3 x 3 + 15a 2 x 4 + 6ax 5 + x 6
Table 7.1
1
1
1
1
(a 1 x) 0
(a 1 x) 1
(a 1 x)
2
(a 1 x)
3
(a 1 x)
4
(a 1 x)
5
(a 1 x) 6
3
3
4
4
5
6
6
15
15
20
5
10
10
6
1
1
1
1
1
1
1
1
1
2
Problem 1. Use the Pascal’s triangle method to
determine the expansion of (a + x) 7 .
From Table 7.1, the row of Pascal’s triangle corresponding to (a + x) 6 is as shown in (1) below. Adding
adjacent coefficients gives the coefficients of (a + x)
7
as shown in (2) below.
1
1
1
(1)
(2)
1
6
6
15
15
20
7
7
21
21
35
35
The first and last terms of the expansion of (a + x) 7 are
a 7 and x 7 respectively. The powers of ‘a’ decrease and
the powers of ‘x’ increase moving from left to right.
The binomial series
7.1 Pascal’s triangle
A binomial expression is one which contains two terms
connected by a plus or minus sign. Thus ( p +q), (a +
x) 2 , (2x + y) 3 are examples of binomial expressions.
Expanding (a + x) n for integer values of n from 0 to 6
gives the results as shown at the bottom of the page.
From these results the following patterns emerge:
(i) ‘a’ decreases in power moving from left to right.
(ii) ‘x’ increases in power moving from left to right.
(iii) The coefficients of each term of the expansions are
symmetrical about the middle coefficient when n
is even and symmetrical about the two middle
coefficients when n is odd.
(iv) The coefficients are shown separately in Table 7.1
and this arrangement is known as Pascal’s triangle. A coefficient of a term may be obtained
by adding the two adjacent coefficients immediately above in the previous row. This is shown
by the triangles in Table 7.1, where, for example,
1 + 3 = 4, 10 + 5 = 15, and so on.
(v) Pascal’s triangle method is used for expansions of
the form (a + x) n for integer values of n less than
about 8.
(a + x) 0 =
1
(a + x) 1 = a + x
a + x
(a + x) 2 = (a + x)(a + x) =
a 2 + 2ax + x 2
(a + x) 3 = (a + x) 2 (a + x) =
a 3 + 3a 2 x + 3ax 2 + x 3
(a + x) 4 = (a + x) 3 (a + x) =
a 4 + 4a 3 x + 6a 2 x 2 + 4ax 3 + x 4
(a + x) 5 = (a + x) 4 (a + x) =
a 5 + 5a 4 x + 10a 3 x 2 + 10a 2 x 3 + 5ax 4 + x 5
(a + x) 6 = (a + x) 5 (a + x) = a 6 + 6a 5 x + 15a 4 x 2 + 20a 3 x 3 + 15a 2 x 4 + 6ax 5 + x 6
Table 7.1
1
1
1
1
(a 1 x) 0
(a 1 x) 1
(a 1 x)
2
(a 1 x)
3
(a 1 x)
4
(a 1 x)
5
(a 1 x) 6
3
3
4
4
5
6
6
15
15
20
5
10
10
6
1
1
1
1
1
1
1
1
1
2
Problem 1. Use the Pascal’s triangle method to
determine the expansion of (a + x) 7 .
From Table 7.1, the row of Pascal’s triangle corresponding to (a + x) 6 is as shown in (1) below. Adding
adjacent coefficients gives the coefficients of (a + x)
7
as shown in (2) below.
1
1
1
(1)
(2)
1
6
6
15
15
20
7
7
21
21
35
35
The first and last terms of the expansion of (a + x) 7 are
a 7 and x 7 respectively. The powers of ‘a’ decrease and
the powers of ‘x’ increase moving from left to right.