12.6
The binomial theorem
Expand each:
(a + b)1
(a + b)2
(a + b)3
(a + b)4
(a + b)5
TABLE for coefficients:
(a + b)0
(a + b)1
(a + b)2
(a + b)3
(a + b)4
(a + b)5
Expand (a + b)8
Expand (x − 2y)5
For
a natural number, define
as
Also define 0! =
So 6! =
Let
and
be nonnegative integers with
The
is
defined by
EXAMPLES Evaluate the expression.
1.
⎛ 8 ⎞
⎜⎝ 3 ⎟⎠
2.
⎛ 5 ⎞⎛ 4 ⎞
⎜⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠
⎛ n ⎞
⎜⎝ r ⎟⎠ =
FACT:
Binomial coefficients and Pascal’s triangle:
Pascal’s Identity:
⎛ k ⎞ ⎛ k ⎞
⎜⎝ r − 1 ⎟⎠ + ⎜⎝ r ⎟⎠ =
THEOREM (Binomial Theorem)
`
for nonnegative integers
and
with
EXAMPLE Use the Binomial Theorem to expand (2x − y)7 .
EXAMPLE Find the first four terms in the expansion of (x 2 + 1)20 .
EXAMPLE Find the term containing y 3 is the expansion of ( 2 + y)12 .