5) c2 the binomial expansion

Introduction• Binomial Expansion is a way of expanding

the same bracket very quickly, whatever power it is raised to

• It can be used in probability, in determining how many ways an event can happen

• We will see an example of this once we have understood the process which we go through in expanding the bracket

The Binomial ExpansionYou can work out brackets ‘the long way’, by multiplying out 2, then multiplying by the next bracket and so on…

5A

1a b 2a b 3a b

a b a b a b a b a b

2 22a ab b 3 2 2 33 3a a b ab b

a b

The Binomial ExpansionYou end up with this pattern…

5A

1a b

2a b

3a b

0a b

4a b

1

1a 1b

21a 21b2ab

23ab

31a

34ab

23a b 31b

2 26a b41a 34a b 41b

There is a pattern in the coefficients (the numbers at the front of each term)

The Binomial ExpansionThe coefficients make a pattern known as Pascal’s triangle

5A

1

1 1

2 11

1 3 3 1

4 161 4

10

510

1 5 1

You work out each number by adding

the 2 above it. Number 1s always go down the edges

The Binomial Expansion

1

a b

2a 2b2ab

23ab

3a

34ab

23a b 3b

2 26a b4a 34a b 4b

For (a + b)n

n = 0n = 1n = 2n = 3n = 4

Find the expansion of (x + 2y)3

a x 2b y 3n

3a 23a b 23ab 3b

3x 23( ) (2 )x y 23( )(2 )x y 3(2 )y

3x 26x y 23 (4 )x y 38y

3x 26x y 212xy 38y

n = 3, so use the relevant row

Sub in for a and b

Simplify Careful, (2y)2 =

4y2

Now fully simplify

5A


1

a b

2a 2b2ab

23ab

3a

34ab

23a b 3b

2 26a b4a 34a b 4b

For (a + b)n Find the expansion of (2x - 5)4

2a x 5b 4n

4a 34a b 2 26a b 4b

n = 4, so use the relevant row

Work out each part carefully

5A

34ab

4(2 )x 34(2 ) ( 5)x 2 26(2 ) ( 5)x 4( 5) 34(2 )( 5)x

416x 34(8 )( 5)x 26(4 )(25)x 4(2 )( 125)x 625

Careful with negatives!

416x 3160x 2600x 1000x 625

416x 3160x 2600x 1000x 625Simplify as

much as you can


1

a b

2a 2b2ab

23ab

3a

34ab

23a b 3b

2 26a b4a 34a b 4b

For (a + b)n The coefficient of x2 in the expansion of (2 - cx)3 is 294. Find

the value of c.Using n = 3

The ‘x’ will be substituted in for b, so we want the term which has

b223ab

23(2)( )cx

2 26c x

6c2 is the coefficient of x2, so must be equal to 294

26 294c 2 49c

7c 5A

Sub in for a and b

Careful with negatives!

Divide by 6

2 possible answers

The Binomial ExpansionYou can use combinations and factorials to work out the Binomial Expansion. This is quicker for higher indices.

Suppose that 3 people are running a race. There are 6 different possible outcomes for their final positions.

This can be calculated as;

3 x 2 x 1

3 x 2 x 1 can be written as 3! (3 factorial)

n! = n x (n-1) x (n-2)…………x 3 x 2 x 15B

A, B, CA, C, BB, A, CB, C, AC, A, BC, B, A

3 possibilities for 1st

After 1st is decided, 2

possibilities for 2nd

After 1st and 2nd, only 1

runner is left

0! = 1 (by definition)


Suppose we want to choose 2 letters from X, Y and Z, where order does not matter. There are 3 possible outcomes.

This can be written as:

To calculate it, you would work out the following.

5B

X, YX, ZY, Z

32C

32

or (2 items to choose from 3

options)

3!2!1!

62 1 3


In general, to work out how many ways of choosing ‘r’ items from a group of n items is written as:

It can be calculated using this general formula;

5B

nrC

nr

or (r items to choose from n

options)

!( )! !n

n r r

Calculate the number of ways of choosing 2

items from a selection of 5

!( )! !n

n r r

5!(5 2)!2!

1206 2

5 items, so n = 5. 2 Choices, so r = 2

5 – 2 = 3 3! = 6

The Binomial ExpansionYou can use to work out the coefficients in the Binomial Expansion

This method will seem more complicated at first, but with higher powers it is easier.

You will most likely need a calculator to work out some of the factorials

5C

nr

( ) ( )( )......n timesna b a b a b

The Binomial Expansion is;

0n nC a 1

1n nC a b 2 22nnC a b ......n n

nC b

0n

1na b 2 2na b ...... nbna 1n

2

n

nn

You will need the formula from the previous section

!( )! !n

n r r

!( 1)!1!n

n

r is effectively the ‘position’ in the

expansionn is the power which the

bracket is raised to


5C

!( )! !n

n r rnr

Calculate the Binomial Expansion of (2x + y)4

4(2 )x

n = 4 a = 2x

b = y

3(2 ) ( )x y41

2 2(2 ) ( )x y42

3(2 )( )x y43

4( )y4!

3!1!41

=

4!2!2!

42

=

4!1!3!

43

=

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

416x 332x y 2 224x y 38xy 4y


5C

!( )! !n

n r rnr

Calculate the Binomial Expansion of (3 – 2x)5

53

n = 5 a = 3 b = -2x

43 ( 2 )x51

3 23 ( 2 )x52

2 33 ( 2 )x53

43( 2 )x54

5( 2 )x5!

4!1!51

=

5!3!2!

52

=

5!2!3!

53

=

5!1!4!

54

=

53 43 ((5) 2 )x 3 23 ((10) 2 )x 2 33 ((10) 2 )x 43((5) 2 )x 5( 2 )x

243 810x 21080x 3720x 4240x 532x

243 810x 21080x 3720x 4240x 532x

You will always get either all positives, or a positive/negative alternating pattern…

The Binomial ExpansionYou will not always be asked to expand the whole thing!

Find the first 4 terms in ascending powers of x of

10

14x

5C

The Binomial ExpansionThere is a shortened version of the expansion when one of the terms is 1

Whatever power 1 is raised to, it will be 1, and can therefore be ignored

The coefficients give values from Pascal’s triangle.

For example, if n was 4…

0n

1 11n x 2 21n x ...... 1n r rx1n 1n

2

n

nr

5D

(1 )nx

1 nx 2( 1)2!

n n x 3( 1)( 2)3!

n n n x ......

1 4x 26x 34x 4x


5D

Find the first 4 terms of the Binomial expansion of (1 + 2x)5

1 nx 2( 1)2!

n n x 3( 1)( 2)3!

n n n x ......

1 5(2 )x 25(4) (2 )2

x 35(4)(3) (2 )6

x ......

1 5(2 )x 210(2 )x 310(2 )x ......

1 10x 240x 380x ......

Put the numbers in

Work out the fractions

Simplify


5D

Find the first 4 terms of the Binomial expansion of (2 - x)6

1 nx 2( 1)2!

n n x 3( 1)( 2)3!

n n n x ......

1 62x

26(5)2 2

x

36(5)(4)6 2

x

......

Put the numbers in

Work out the fractions

Simplify

6(2 )x 2 12x

[ ]6

6

64 12x

662 1

2x

162x

2308x 3120

48x ......

1 3x 23.75x 32.5x ......

1 192x 2240x 3160x ......

Remember to multiply

by 64!

Summary• We have learnt how to expand many

brackets when raised to a power

• We have seen several ways to do this, which all fit certain circumstances

• We also saw various questions on working out coefficients, as well as the brackets themselves

5) c2 the binomial expansion

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