Binomial Expansion

BINOMIAL THEOREM

OBJECTIVES
Recognize patterns in binomial expansions.
Evaluate a binomial coefficient.
Expand a binomial raised to a power.
Find a particular term in a binomial expansion
Understand the principle of mathematical induction.
Prove statements using mathematical induction.

Definition: BINOMIAL THEOREM

Patterns in Binomial Expansions
A number of patterns, as follows, begin to appear when we write the binomial expansion of a  b n, where n is a positive integer.

a  b   a  b
a  b 2  a 2  2ab  b 2
a  b 3  a3  3a 2b  3ab 2  b3
a  b 4  a 4  4a3b  6a 2b2  4ab3  b 4
5
a  b   a5  5a 4b  10a3b 2  10a 2b3  5ab4  b5
1

and so on.

In each expanded form above, the following can be observed: n 1. The first term is a , and the exponent on a decreases by 1 in each successive term.

2. The last term is b n and the exponent on b decreases by 1 in each successive term.
3. The sum of the exponents on the variables in any term is equal to n. n n  1 terms in the expanded form of a  b .
4. There are

Definition:
Binomial Coefficients
An interesting pattern for the coefficients in the binomial expansion can be written in the following triangular arrangement n=0 n=1 n=2 n=3 n=4 n=5 n=6 a  bn

1
1
1
1

1
1

1

1
2

3

4

1
3

6

1

4

1

5
10
10
5
1
6 15
20
15
6
1

This triangular array of coefficients is called the Pascal’s Triangle.
When n is small, the use of Pascal’s triangle is advantageous.
However, if n is large or a specific term is desired, the use of Binomial Theorem is more appropriate.

Definition :
THE BINOMIAL THEOREM
The Binomial Theorem provides a formula for expanding expressions of the form a  b n , where n is a natural number.
For any binomial a  b  and any natural number n ,

n n1 nn  1 n2 2 nn  1n  2...n  r  1 nr r
a  b  a  a b  a b  ...  a b  ...  b n
1!
2!