Binomial Expansion

The process of expanding a binomial uses the following pattern

(a + b)2 = a2 + 2ab + b2 (a+b)3 = a3 + 3a2b + 3ab2 + b3 (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Binomial Expansion

If we continue the expansion all the way to (a+b)n

1. There are n+1 terms in the expansion 2. The exponents of a decrease by one starting with an and

the exponents of b increase by one starting with b0

3. The sum of the exponents in each term will equal the value of n

(a + b)2 = a2 + 2ab + b2 (a+b)3 = a3 + 3a2b + 3ab2 + b3 (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Binomial Expansion

Pascal’s Triangle

To fill in the questions marks you can use Pascal’s triangle

Expand (a+b)7 a7 + ?a6b + ?a5b2 + ?a4b3 + ?a3b4 + ?a2b5 + ?ab6 + b7

a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7

Using Pascal’s Triangle

Expand (2x + 1)4 4 0 3 1 2 2 1 3 0 4

4 3 2 4

4 3 2

(1)(2 ) (1) (4)(2 ) (1) (6)(2 ) (1) (4)(2 ) (1) (1)(2 ) (1)

(1)(16 )(1) (4)(8 )(1) (6)(4 )(1) (4)(2 )(1) (1)(1)(1)

16 32 24 8 1

x x x x x

x x x x

x x x x

(3+1)2 + (4+1)2 + (5+1)2 + (6+1)2 = 126

Arithmetic with d = 0.7 10 is the 11th term

S10 = (10/2)(3+10) = 65

(2x)4 + 4(2x)3(y) + 6(2x)2(y)2 + 4(2x)(y)3 + y4

16x4 + 4(8x3)(y) + 6(4x2)(y)2 + 4(2x)(y)3 + y4

16x4 + 32x3y + 24x2y2 + 8xy3 + y4

Expand

Binomial Coefficient

Since Pascal’s Triangle might not always be a feasible option you need another way to figure out the value of each coefficient in your expansion.

To do this you will use the following

Binomial Coefficient

Using the method to find the binomial coefficient allows us to write the expansion of (a+b)n as

Binomial Coefficient

Expand (x + 2y)5

5 0 4 1 3 2 2 3 1 4 0 5

5 4 3 2 2 3 4 5

5 4 3 2

5 5 5 5 5 5( ) (2 ) ( ) (2 ) ( ) (2 ) ( ) (2 ) ( ) (2 ) ( ) (2 )

0 1 2 3 4 5

(1)( )(1) (5)( )(2 ) (10)( )(4 ) (10)( )(8 ) (5)( )(16 ) (1)(1)(32 )

10 40 8

x y x y x y x y x y x y

x x y x y x y x y y

x x y x y

2 3 4 50 80 32x y xy y

General Term

To find the r th term an expansion use

General Term

Find the 20th term in the expansion of (a+b)25

General Term

Find the 6th term of (a+2b)8

8 (6 1) 6 1

3 5

3 5

3 5

8( ) (2 )

6 1

8( ) (2 )

5

56( )(32 )

1792

a b

a b

a b

a b

General Term

The term that contains ar in the expansion of (a+b)n is

General Term

Find the term that contains x4 in the expansion (x+2y)10

4 6

4 6

4 6

5

10( ) (2 )

4

210( )(64 )

13440

r

x y

x y

x y