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BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

ClassNotes Team 8 MIN READUPDATED 13 JUN 2026

FURTHER MATHEMATICS SSS2 THIRD TERM

WEEK 3

BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

Content

PASCAL’S TRIANGLE

Consider the expressions of each of the following:

(x + y)0;  (x + y )1; (x + y)2; (x + y)3; (x + y)4

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3= 1x3 + 3x2y + 3xy2 + 1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1x4

The coefficient of x and y can be displayed in an array as:

                                      1

BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

Example 3

Using Pascal’s triangle, simplify, correct to 5 decimal places (1.01)4

Solution

We can write (1.01)4 = (1 + 0.01)4

(1 + 0.01)4 = 1 + 4(0.01) + 6(0.01)2 + 4(0.01)3+(0.01)4

= 1 + 0.04 + 0.0006 + 0.000004 + 0.00000001

= 1.04060401

= 1.04060 (5 d.p)

The Binomial Expansion Formula

Consider the expansion of (x + y)5 again

(x + y)5 = (x + y)(x + y)(x + y)(x + y)(x + y)

The first term is obtained by multiplying the xs in the five brackets. there is only one way to do this

(x + y)n = xn + nxn – 1y + n(n-1)2!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER   xn – 2y2 + nn-1( n-2)3!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER    xn-3y3 + …. nn-1n-2….(n-r+1)r!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

xn-ryr + …. yn

It can be shown that the binomial expansion formula holds for positive, negative, integral or any rational value of n, provided there is a restriction on the values of x and y in the expansion of (x + y)n

We shall however consider only the binomial expansion formula for a positive integral n

Example 4:

  1. Write down the binomial expansion of  1+ 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER6 simplifying all the terms
  2. Use the expansion in (a) to evaluate (1.0025)6 correct to five significant figures.

Solution

1+ 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER6  = 1 + 6C1 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER1    +   6C214xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER2

    +   6C3  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER3    +    6C4  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

    +   6C5  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER5    +    6C6  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER6

1+ 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER6=  1 + 6114BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  x   +  6 x 51 x 214xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER3   +  6 x 5 x 4 x 31 x 2 x 3BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x

14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER4    +    6 x 5 x 4 x 3 x 21 x 2 x 3 x 4BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER5   +  14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER6

= 1 + 32BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx   +   1516BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx2+   516BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx3  +15256BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx4  +   3512BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx5 +   14096BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx6

  1. (1.0025)6   =  (1 + 0.0025)6

= 1+ 2510000BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )6

= 1+ 1400BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )6

Put 14BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx  =1400BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER

x = 1400BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  x 4   = 1100BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER=  0.01

therefore   (1.0025)6   =  1  +  32BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  (0.01)  + 1516BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  (0.01)2  + 516BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER(0.01)  +15256 BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER(0.01)4 + …

= 1 + 0.015 + 0.00009375 + 0.0000003125

= 1.0150940625

= 1.0151 (5 s.f.)

EVALUATION

Expand ( 2 + 4x )4 simplifying the terms

 

Example 5
(a) Using the binomial theorem, obtain the expansion of (1 + 3x)6 + (1 – 3x)6  simplifying all the terms

(b)Use the above result to calculate the value of (1.03)6 + (0.97)6, correct to five decimal places

Solution:

(1 + 3x)6 = 1 + 6C1 (3x) + 6C2 (3x)2 + 6C3 (3x)3 _ 6C4 (3x)4 _ 6C5 (3x)5 + 6C6 (3x)6          ….. (1)

(1 - 3x)6 = 1 - 6C1 (3x) + 6C2 (3x)2 - 6C3 (3x)3 + 6C4 (3x)4 _ 6C5 (3x)5 + 6C6 (3x)6          ….. (2)

Adding (1) and (2)

(1 + 3x)6 +(1 - 3x)6  = 2 + 2 x  6C2 (3x)2  + 2 x 6C4 (3x)4  + 2 x 6C6 (3x)6

= 2 + 2 x 6 x 51 x 2BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  9x2 + 2 x 6 x 5 x 4 x 31 x 2 x 3 x 4BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  x 81x4  + 2  x 729x6

= 2 + 270x2 + 2430x4 + 1458x6

(1.03)6    =   (1 + 0.03)6

(0.97)6    =   (1 – 0.03)6

Put 1 + 0.03 = 1 + 3x

Therefore    3x   =   0.03

Therefore     x    =   0.01

Hence

(1.03)6 + (0.97)6   = 2 + 270(0.01)2 + 2430(0.01)4 + 1458(0.01)

= 2 + 0.027 + 0.0000243 + 2.0270243

= 2.02702    (5 d.p)

Example 6

  1. Using the binomial theorem, expand (1 + 2x)5, simplifying all the terms
  2. Use your expansion to calculate the value of 1.025, correct to six significant figures

If the first three terms of the expansion of (1 + px)n in ascending powers of x are 1 + 20x + 160x,

Find the values of n and p

Solution:

  1. (1 + 2x)5 = 1 . 5C1(2x) + 5C2(2x)2 + 5C3(2x)2 + 5C4(2x)4 + 5C5(2x)5

= 1 + 5.(2x) + 5.41.2 BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  . 4x2 + 5.4.31.2 .3BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  . 8x3  +5.4.3.21.2 .3.4BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  . 16x   + 32x5

= 1 + 10x + 40x2 + 80x3 + 80x4 + 32x5

  1. (1.02) = (1 + 0.02)

Put 1 + 0.02 = 1 + 2x

Therefore 2x = 0.02

x = 0.01

Hence:

(1.02)5 = 1 + 10(0.01) + 40(0.01)2 + 80(0.01)3 + 80(0.01)4 + 32(0.01)5

= 1 + 0.1 + 0.004 + 0.0008 + 0.00000008

= 1.10408 (6.s.f.)

6.3 The Binomial Theorem for any index

The Binomial expansion formula is also applicable to any index n, where n can be a positive or negative integer or even a fraction

If /x/ <BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER 1, then:

(1 + x)n = 1 + nx + nn-1x22!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  + 3!          nn-1(n-2)x3BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  + nn-1n-2n-34!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x4  + … where n may be a negative integer or a fraction.

 

Example 7

Use the Binomial expansion formula to obtain the first five terms of the expansion of  (1 +  12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)-2

Solution:

(1 +  12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)-2  = 1 + (-2) (12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x) + (-2(-3)2!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER (12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)2 + -2-3(-4)3!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER (12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)3 + -2-3-4(-5)4!BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER (12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)4 +  ….

= 1 – x + 3. x4 BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER2  - 4.x8 BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER3  + 5.x16 BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER4

 

  1. (1 + px)n = 1 + 20x + 160x2 + …

(1 + px)n = 1 + nc1 (px) + nc1 (px)2 + …

            = 1 + npx   + nn-12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  p2x2 …

            = 1 + 20x + 160x2 + …

By equating coefficients

np = 20                                     …       (1)

nn-12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERp2  = 160                           …       (2)

From (1) p = 20nBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER                          …       (3)

Therefore p2 =  400n2BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER           …       (4)

Substituting (4) into (2)

nn-12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  x 400n2BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER   = 160

nn-12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERx 200   =  160

There 200(n – 1) = 160n

200n – 200           = 160n

200n – 160n         = 200

                40n         = 200

                     n         = 5

From (3)p =  205BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER  = 4

Hence, n = 5, p = 4

 

Example 8

Obtain the first four terms of the explanation of (2 +  12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)8in ascending powers of x. hence, find the value of (2.005)8, correct to five significant figures.

Solution:

(2 +  12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)8= 28(1+  14BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)8  

= 28[1 +8C1( 14BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)  + 8C2 ( 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )2  +  8C3 ( 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )3   + … ]

=   28[1 +8( 14BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x)  + 8.71.2BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER ( 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )2  + 8.7.61.2.3BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER ( 14xBINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER )3   + …]

=   28[1 + 2X + 74BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER X2+ 78BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWERX3  + …]

Write 2.0.005

Put 2 + 12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x    =  2 + 0.005

Therefore  12BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER x    =   0.005

Therfore  x   =  0.005 x 2

                       = 0.01

Hence,

(2.005)8   =   28[1 +2(0.01)  +74BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER (0.01)2+ 74BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITIVE AND FRACTIONAL POWER (0.01)3 ]

(2.005)8   = 28 + 29(0.01) + 26.7(0.01)2 + 25 x 7(0.01)3 + …

= 256 + 5.12 + 0.0448 + 0.000224

               = 261.165025

               = 261.17 (5 s.f.)

 

GENERAL EVALUATION

1) Write down and simplify all the terms of the binomial expansion of ( 1 – x )6. Use the expansion to evaluate  0.9976  correct to 4 dp

2) Write down the expansion of  ( 1 + ¼ x ) 5 simplifying all its coefficients

3) Use the binomial theorem to expand  ( 2 – ¼ x)5 and simplify all the terms

4) Deduce  the expansion of   ( 1 – x +x2 )6  in ascending powers of x

Reading Assignment

New Further Maths Project 2 pages 73 – 78

 

WEEKEND ASSIGNMENT

If the first three terms of the expansion of ( 1 + px )n in ascending powers of  x are   1 + 20v + 160x find  the value of

1)  n  a) 2  b) 3  c) 4  d) 5

2) p   a) 2  b) 3  c) 4  d) 5

3) In the expansion of  ( 2x + 3y )4  what is the coefficient of  y4   a) 16  b) 81  c) 216  d) 96

4) How many terms are in the expansion  of  ( 1 – 4x ) 5  a) 3  b) 5  c) 6  d) 8

5) What is the third term in the expansion of  ( 1 – 3x )6 in ascending powers of x  a) 18  b) -540  c) 135  d) 729

THEORY

1) Using the binomial theorem, write down and simplify the first seven terms of the expansion of  ( 1 + 2x )10 in ascending powers of x

2) Expand  ( 2 + x )5 ( 1 – 2x ) 6 as far as the term in x3   . Evaluate  ( 1.999 )5 ( 1.002 )6