Term 1 : Unit 3

Binomial Theorem

3.1 The Binomial Expansion of (1 + b) n

3.2 The Binomial Expansion of (a + b) n

r

n

Objectives

3.1 The Binomial Expansion of (1 + b) n

In this lesson, you will use Pascal’s triangle or to find the

binomial coefficient of any term. You will use the Binomial Theorem to expand (1 + b)n for positive integer values of n and identify and find a particular term in the expansion (1+ b)n using the

result, .1r

r br

nT

Binomial Theorem

2b

2a b

A square of side

( a + b ).

2 2 22a b a ab b

Binomial Theorem

a

a b

b

2a

ab

ab

a b

a b

Split the square in four.

Separate the component

parts.

Binomial expansion for n = 2.

3a b

3b

A cube of side a + b

Binomial Theorem

2ab

2ab

2a b

2ab

2a b

2a b

3a

Split the cube up as

shown

A small cube with volume a3

A cuboid with volume

a2b

Another cuboid with volume a2b

And another

A cuboid with volume

ab2

And another

And another

Finally, a cube of

volume b3.

3

3 2 2 33 3

a b

a a b ab b

Add two adjacent terms to make the

term below.

Binomial Theorem

111

1 1211 3 3

1 14 4611 55 10 10

11 66 15 152011 35217 72135

1 170 56 28 88 28 5611 126 84 36 99 36 84 126

Pascal’s Triangle

Now, we will apply the

triangle to the binomial

expansion.

Binomial Theorem

Using Pascal’s Triangle to expand (1 + b) 6

1 14 4611 55 10 10

11 66 15 152011 35217 72135

1 170 56 28 88 28 5611 126 84 36 99 36 84 126

b 2b 3b 4b 6b5b 0b

Write ascending powers of b from b0 to

b6.

Take the 6th row of Pascal’s Triangle.

Use these numbers as coefficients.

Form into a series.

Binomial Theorem

Take the 5th row of Pascal’s Triangle.

Use these numbers as coefficients.

Write ascending powers of b from b0 to b5.

5Expand 1 b

Example 1

2 3 4 51 b b b b b 5 2 3 4 51 1 5 10 10 5b b b b b b

11 55 10 10

.

Binomial Theorem

5 2 3 4 5

5

Use the result 1 1 5 10 10 5

to find 1

b b b b b b

b

2 3 4 51 5 10 10 5b b b b b

551 1b b

2 3 4 51 5 10 10 5b b b b b

Take care of the minus signs here.

Notice how the signs alternate

between odd and even terms.

Binomial Theorem

5 2 3 4 5

5

Use the result 1 1 5 10 10 5

to find 1 2

b b b b b b

x

5 2 3 4 51 2 1 5 2 10 2 10 2 5 2 2x x x x x x

2 3 4 51 10 40 80 80 32x x x x

2 3 4 51 5 2 10 4 10 8 5 16 32x x x x x

Remember to include the coefficients inside the

parentheses.

Binomial Theorem

11 55 10 10

5

5

5

0

5

4

5

1

5

2

5

3

The fifth row of Pascal’s Triangle was

Using Binomial Coefficient notation, these numbers are

In the expansion of 1 the coefficient of is n r n

b br

n is the row and r is the position

(counting from 0).

Binomial Theorem

1 2 1

1 2 3 2 1

n n n n n r

r r r r

1 2 1 1, 1,

0 !

n n n n n n n r

n r r

5 5 4 3 10

3 3 2 1

8 8 7 6 5 4 56

5 5 4 3 2 1

The binomial coefficient can be found from this formula.

The number of terms in the numerator and denominator

is always the same.r! – r factorial

Binomial Theorem

0 1 2 31 0 1 2 3

n nn n n n nb b b b b b

n

2 31 1 21 1

2! 3!

n nn n n n nb nb b b b

The Binomial Theorem

Binomial Theorem

Using this result

82Find the first four terms in the expansion of 1 x

8 2 32 2 2 28 8 81 1

1 2 3x x x x

2 4 68 7 8 7 61 8

2 1 3 2 1x x x

2 4 61 8 28 56x x x 8

Estimate the value of 1.01

88 21.01 1 0.1

2 4 61 8 0.1 28 0.1 56 0.1 1 0.08 0.002 8 0.000 056 1.082 856

Example 3

.

Binomial Theorem

Using this result

122 5Find the terms in and in the expansion of 1

2

xx x

212

2 2

x

12

3 3 2 533 993 2 1 3 2

2 2 4

xx x x x

5 3 299 333 2

4 2x x x

212 11

2 1 4

x

233

2x

512

5 2

x

512 11 10 9 8

5 4 3 2 1 2

x 599

4x

12

5 3Find the coefficient of in the expansion of 3 2 12

xx x

5165

4x

Example 5

.

Binomial Theorem

79 2

Find the first four terms, in ascending powers of in the

expansions of 1 2 and 1 2 .

x

x x

9 2 39 8 9 8 71 2 1 9 2 2 2

2 1 3 2 1x x x x

2 31 18 144 672x x x

2 4 61 14 84 280x x x

7 2 32 2 2 27 6 7 6 51 2 1 7 2 2 2

2 1 3 2 1x x x x

Exercise 6.1, qn 3(d), (g)

Binomial Theorem

In this lesson, you will use the Binomial Theorem to expand (a + b) n for positive integer values of n. You will identify and find a particular term in

the expansion (a + b) n, using the result .1T .n r rr

na b

r

3.2 The Binomial Expansion of (a + b) n

Objectives

Binomial Theorem

2 3

1 1 2 3

nn n n n nb b b b

ana a a a

The Binomial Theorem

1 1n n

n nb ba b a a

a a

1 2 2 3 3 1 2 3

n n n n n nn n na b a a b a b a b b

Binomial Theorem

Take the 6th row of

Pascal’s Triangle.

Use these numbers as coefficients.

Write ascending

powers of b from b0 to

b5.

5Expand .a b

b 5 5 4 3 2 2 3 4 55 10 10 5a b a a b a b a b ab b

11 55 10 10

3b 4b 5b2b3a 2a4a5a a

Write descending powers of a

from a5 to a0.

The combined total of powers is always

5.

Example

Binomial Theorem

6

2

Find, in descending powers of , the first four terms of

1.

x

xx

6 33

206 15x x

x

6 2 36 5 4 3

2 2 2 2

1 1 1 16 15 20x x x x x

x x x x

Don’t try to simplify yet – not until the

next stage.

Notice that the third term is

independent of x.

Example 6(b)

Binomial Theorem

Looking at the

combined powers of

x

12

15 2

1 1Find the term in in the expansion of 2 .x

x x

12

2

12 1The general term is 2 .

rr

xr x

15

1For the term in , 12 2 15r r

x 9r

318

1220 8x

x

15

1760

x

9

3

15 2

121 1The term in is 2

9x

x x

There is no need to find all the

terms.

Be careful with negative values.

Example 8(b)

Binomial Theorem

9Find the fourth term in the expansion of 3 2 .x

684 729 8x

9 6 393 2 3 2

3x x

There is no need to

find all the terms.

6The fourth term is 489888 .x

Exercise 6.2, qn 6(b)

Binomial Theorem

103

2

2Find the constant term in the expansion of .x

x

1212

64210x

x

The combined

powers of x are 0.

The constant term is 13440.

1032

10 2The general term is

rr

xr x

3 10 2 0r r 6r

6

40 32

10 2The term in is

6x x

x

Exercise 6.2, qn 7(c)